year 6 fraction problem solving

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$year 6 fraction problem solving$

24 Fraction Questions For Year 6 (SATs): Complete With Answers

Emma Johnson

Here are 24 free fractions questions for Year 6, a perfect resource to use with pupils in the run up to the KS2 Maths SATs in May. In addition to the questions, we’ve also included useful background information on the fractions knowledge Year 6 pupils need to acquire by the end of KS2. This includes the key vocabulary, concepts and fraction knowledge from earlier Key Stage 2 year groups.

We’re focusing on fractions here as they are a key component of the National Curriculum and a topic pupils are guaranteed to see multiple times in the end of Key Stage 2 SATS.

Alongside decimals and percentages, fractions make up the third highest content domain across the arithmetic and reasoning papers.

$year 6 fraction problem solving$

This article is one of a series providing teachers with a bank of hundreds of SATs questions for Year 6 that they can continue to refer back to through the year.

What do Year 6 pupils need to know about fractions?

Throughout Key Stage 2, pupils cover a substantial amount of work on fractions. In Year 6, pupils continue to build on this knowledge from previous years, helping them to become proficient in the end of KS2 fractions standards.

National Curriculum expectations for fractions in KS2 maths:

In Year 3 pupils learn to:

Recognise unit and non-unit fractions
Identify equivalent fractions
Order and compare fractions
Add and subtract fractions with the same denominator

In Year 4 pupils learn to:

Build on their understanding of equivalent fractions
Continue adding and subtracting fractions with the same denominator
Count in tenths and hundredths

In Year 5 pupils learn to:

Compare and order fractions with different denominators
Add and subtract fractions with different denominators
Convert between improper fractions and mixed numbers
Begin to multiply fractions and mixed numbers
Explore relationships between equivalent fractions, decimals and percentages

By the time students reach Year 6, the expectation is that students will have a secure understanding of fractions.

Considerable exposure to concrete resources and visual images should ensure they have a good conceptual understanding of fractions.

In Year 6 pupils learn to:

Build on knowledge developed throughout KS2
Be confident adding subtracting, multiplying and dividing fractions and simplifying their final answer
Calculate equivalent fractions, decimals and percentages

$year 6 fraction problem solving$

200 arithmetic and reasoning questions for Year 6

Download 100 free arithmetic questions and 100 free reasoning questions for Year 6. Includes answers and mark scheme.

Thousands of pupils develop their fractions knowledge and understanding every week through our one to one online maths lessons. Our maths experts have developed hundreds of fraction questions for pupils to work through step by step with their tutor to build confidence. Try the one to one maths fraction lessons for yourself in this free downloadable SATs intervention pack .

What fractions questions appear in the KS2 SATs?

Predominantly, the KS2 SATs arithmetic paper consists of fluency style fraction questions. Reasoning Paper 2 and Paper 3 are made up of more word problem style questions.

Children can expect to answer questions on the four operations and fractions. Often, these question types increase in levels of difficulty throughout the paper.

For example, in the 2023 arithmetic paper, children were given three adding fractions questions. The first question required pupils to add fractions within 1 while the second question asked Year 6 pupils to add 3 fractions within 1. The third fractions question required pupils to add 3 fractions with a total greater than 1.

$year 6 fraction problem solving$

Fractions questions included in this article cover simple fractions questions and become increasingly harder.

Questions begin with fluency skills and progress to questions involving reasoning and problem solving.

Key Year 6 fractions terminology needed for these questions

Pupils should have a strong understanding of key fractions terminology by the time they reach Year 6.

Here are some of the key terms children should have a secure understanding of:

Denominato r: The bottom number of a fraction. This tells us how many equal parts make up a whole.
Numerator : The top number of a fraction. This tells us how many equal parts of the whole are being considered.
Unit fraction : A fraction with a numerator of 1.
Non-unit fraction : A fraction with a numerator greater than 1.
Whole number : An integer, that isn’t a fraction or a decimal. Whole numbers can be expressed as a fraction by placing the whole number in place of the numerator and a 1 in place of the denominator.
Decimal: Another representation of a fraction. The digits to the left of the decimal point represent the whole number and the digits after the decimal point indicate the fraction of the whole number. For example, 2.14
Improper fractions: Fractions with a numerator greater than the denominator. These fractions have a total value of more than one whole. For example, \frac{13}{4}
Mixed numbers : A combination of a whole number and a fraction. When a number is greater than one whole, it can be written as either an improper fraction or a mixed number. For example, 1 \frac{2}{3} = \frac{5}{3}
Equivalent fractions : Fractions that represent the same value but have a different numerator and denominator. For example, \frac{3}{4} = \frac{9}{12}
Multiples and factors: These refer to the common divisors and multiples of the numerator and denominator.
Highest common factor (HCF) is the largest number that the numerator and denominator can be divided by. For example, \frac{6}{12} both 6 and 12 can be divided by 6
Lowest common multiple (LCM) is the smallest number that both numbers are factors of. For example, \frac{2}{4} both 2 and 4 are factors of 4
Common denominator: The shared multiple of the denominators of two or more fractions. For example, \frac{3}{4} and \frac{2}{6} share a common denominator of 12
Simplify fractions: When fractions are reduced to their lowest and simplest form. For example, \frac{4}{20} can be simplified to \frac{1}{5}

24 fractions questions for Year 6

Fraction questions for year 6: comparing and ordering.

Fraction question 1

Compare the following pairs of fractions and put the > , < or = symbol between each pair:

\frac{5}{8} \frac{3}{4}

\frac{2}{3} \frac{5}{12}

\frac{7}{10} \frac{3}{5}

\frac{5}{8} < \frac{3}{4}

\frac{2}{3} > \frac{5}{12}

\frac{7}{10} > \frac{3}{5}

\frac{3}{4} = \frac{6}{8}

\frac{2}{3} = \frac{8}{12}

\frac{3}{5} = \frac{6}{10}

Fraction question 2

Order the following fractions from smallest to biggest:

\frac{2}{5} \frac{2}{3} \frac{2}{6} \frac{2}{4} \frac{2}{10}

Answer: \frac{2}{10} \frac{2}{6} \frac{2}{5} \frac{2}{4} \frac{2}{3}

Fraction question 3

Re-arrange the following fractions from greatest to smallest

\frac{4}{6} \frac{1}{3} \frac{7}{12}

Answer: \frac{4}{6} \frac{7}{12} \frac{1}{3}

\frac{4}{6} = \frac{8}{12}

\frac{1}{3} = \frac{4}{12}

\frac{7}{12} = \frac{7}{12}

Fraction question 4

Hamza ate \frac{3}{5} of his bag of 50 sweets, Jack ate \frac{4}{10} of his bag of 50 sweets. Who ate the most sweets:

Answer: Hamza

\frac{6}{10} is greater than \frac{4}{10}

Fraction question 5

Sophie, Ben and Alice were taking part in a running competition. With 10 minutes to go, Sophie had completed \frac{4}{5} of the route, Ben had completed katex]\frac{5}{6}[/katex] and Alice had completed \frac{2}{3} . Order them from 1-3, with the person who had run the most at the top.

Answer: 1st: Ben, 2nd: Sophie, 3rd: Alice

Sophie \frac{4}{5} = 2 \frac{24}{30}

Ben \frac{5}{6} = \frac{25}{30}

Alice \frac{2}{3} = \frac{20}{30}

Fraction question 6

Chloe, Ahmed and Mason have each ordered a pizza. Chloe ate \frac{3}{8} of her pizza, Ahmed ate \frac{5}{12} of his and Mason ate \frac{2}{6} . Who ate the most pizza? Order them from 1 to 3.

Answer: Ahmed at the most, followed by Chloe, then Mason ate the least.

Chloe: \frac{3}{8} = \frac{9}{24}

Ahmed: \frac{5}{12} = \frac{10}{24}

Mason: \frac{2}{6} = \frac{8}{24}

Fraction questions for Year 6: four operations with fractions

Fraction question 7

\frac{3}{8} + \frac{2}{4} =

Answer: \frac{7}{8}

\frac{3}{8} + \frac{4}{8} = \frac{7}{8}

Fraction question 8

\frac{9}{12} – \frac{1}{3} =

Answer : \frac{5}{12}

\frac{9}{12} – \frac{4}{12} = \frac{5}{12}

Fraction question 9

\frac{3}{5} x 7 =

Answer : 4 \frac{1}{5}

\frac{3}{5} x 7 = \frac{21}{5} = 4 \frac{1}{5}

Fraction question 10

\frac{12}{17} ÷ 3 =

Answer : \frac{4}{17}

Fraction question 11

\frac{7}{12} ÷ 4 =

Answer : \frac{7}{48}

\frac{7}{12} ÷ 4 = \frac{1}{4} of \frac{7}{12}

\frac{1}{4} x \frac{7}{12} = \frac{7}{48}

Fraction question 12

Masie, Chloe and Jess are sharing some pizzas. Maisie eats \frac{5}{8} of a pizza, Chloe eats \frac{3}{8} and Jess eats \frac{6}{8} . How much pizza have they eaten altogether?

Answer : 1 \frac{6}{8} or 1 \frac{3}{4}

\frac{5}{8} + \frac{3}{8} + \frac{6}{8} = \frac{14}{8} = 1 \frac{6}{8}

Fraction question 13

4 friends have 3 \frac{1}{5} pizzas to share between them. If they share it equally, how much pizza will each of the friends receive?

Answer : \frac{4}{5} of a pizza

3 \frac{1}{5} = \frac{16}{5}

\frac{16}{5} ÷ 4 = \frac{4}{5}

Fraction question 14

Adam is baking some cooking. He needs \frac{3}{4} of a cup of sugar to make 10 cookies. If he wants to bake 60 cookies, how many cups of sugar will he need?

Answer : 4 \frac{1}{2} cups

\frac{3}{4} x 6 = \frac{18}{4} = 4 \frac{2}{4} = 4 \frac{1}{2}

Fraction questions for Year 6: fractions, decimals and percentages

Fraction question 15

Order the following from smallest to largest:

35%, 0.3, \frac{34}{100}

Answer : 0.3, \frac{34}{100} , 35%

35% = \frac{35}{100}

0.3 = \frac{30}{100}

Fraction question 16

Circle three numbers which add up to 1:

0.5 15% \frac{1}{4} 10% 0.2 \frac{7}{10}

Answer : \frac{7}{10} 0.2 10%

0.2 = \frac{2}{10}

10% = \frac{1}{10}

Fraction question 17

Which of the following are equivalent to 20%?

0.02 \frac{40}{200} \frac{10}{50} 0.2 \frac{2}{10} 0.22 \frac{40}{100} \frac{2}{5}

Answer : \frac{40}{200} \frac{10}{50} 0.2 \frac{2}{10}

Fraction question 18

What is 35% of 240?

Answer : 84

Fraction question 19

Sam, Amy and Lucy have raised £560 in a sponsored swim.

Sam raised \frac{3}{8} of the total

Amy raised 30% of the total.

How much did Lucy raise?

Answer: £182

Sam: \frac{3}{8} of 560 = £210

Amy: 30% of 560 = £168

£210 + £168 = £378

Lucy raised £560 – £378 = £182

Fraction question 20

There are 60 sweets in a bag. The sweets are red, orange or yellow.

\frac{5}{12} of the sweets are red

40% are orange.

How many are yellow?

Answer : 11 yellow sweets

\frac{5}{12} of 60 = 25

40% of 60 = 24

15 + 24 = 49

60 – 49 = 11

Fraction questions for Year 6: fraction word problems

Fraction question 21

Max has 150 pages in his book. He has read \frac{3}{5} of the book. How many pages does he have left to read?

Answer : 60 pages left to read.

\frac{1}{5} of 150 = 30

\frac{2}{5} of 150 = 60

Fraction question 22

Sienna has half of a pizza left. She cuts the half a pizza into 4 equal sized pieces. What fraction is each piece?

Answer : \frac{1}{8}

\frac{1}{2} ÷ 4 = \frac{1}{8}

Fraction question 23

8 friends share some pizzas equally. Each person is given \frac{5}{8} of a pizza. How much pizza did the eat altogether?

Answer : 5 pizzas

\frac{5}{8} x 8 = \frac{40}{8} = 5

Fraction question 24

Oliver had some birthday money to buy some clothes. He spent \frac{1}{5} of the money on a T-shirt, \frac{1}{4} on a hoodie and \frac{1}{2} on jeans. If he spent £12.50 on the hoodie, how much did he have left after his shopping trip?

Answer : £2.50

\frac{1}{4} = £12.50

Total = £50

\frac{1}{5} = £10

\frac{1}{2} = £25

Total spent = £12.50 + £10 + £25 = £47.50

£50 – £47.50 = £2.50

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25 Year 6 Algebra Questions And Answers: KS2 Assessment Preparation

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250+ SATs Questions Free To Download [With Answers]

FREE KS2 Maths SATs Practice Papers (Set of 6)

Get ready for KS2 SATs tests with this set of 6 maths SATs practice papers.

Includes 2 Arithmetic and 4 Reasoning Papers, with mark schemes and answer booklets, that follow the National Curriculum Assessments.

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$year 6 fraction problem solving$

Compare and order fractions

Your child will compare fractions and put them in size order.

They will be able to do this with proper fractions (i.e. fractions less than one) and fractions that are greater than one (i.e. improper fractions or mixed numbers ). For example:

is greater than is greater than

Calculate with mixed numbers and fractions with different denominators

Your child will learn to add and subtract fractions with different denominators (bottom numbers). They will do this by finding equivalent fractions that have the same denominator. For example:

+ = + =

$1/2$

They will use objects, diagrams, and symbols to solve problems with mixed numbers and fractions, and to explain how they have solved a problem.

Multiply and simplify proper fractions

Your child will be able to multiply pairs of proper fractions (i.e. fractions smaller than 1) together, and then write the answer in its simplest form.

× = =

Your child will use objects, diagrams, and symbols in their calculations.

Divide proper fractions by whole numbers

Proper fractions are fractions with a value of less than 1. Your child will be able to divide these kinds of fractions by whole numbers. For example:

Your child will do this using objects, diagrams, and symbols.

Understand the link between fractions, division, and decimals

To find of 36cm, you would divide 36 by 4 (36cm ÷ 4 = 9 cm). If you know of an unknown length is 36cm and you want to find the whole length, you would multiply 36 by 4 (36cm × 4 = 144cm).

$3/8$

Use numbers with 3 decimal places

Your child will understand the value of each digit in numbers with 3 decimal places. They will also be able to multiply and divide numbers by 10, 100, and 1000, giving answers up to 3 decimal places. For example:

9 ÷ 1000 = 0.009 0·734 × 100 = 73.4

Your child will divide decimal numbers by 1-digit whole numbers. At first, they will learn this in practical contexts involving measures and money (for example, 0.65m ÷ 5 = 0.13m).

Multiply numbers with decimal places

Your child will learn to multiply 1-digit numbers with up to 2 decimal places by whole numbers. For example:

2.75 × 2 = 5.5

They will start with simple examples, such as 0.4 × 2 = 0.8, in practical contexts such as using money.

Use written division methods where appropriate

Your child will use written division methods (like long division and short division) in cases where the answer has up to 2 decimal places. You can find out more about these methods on our YouTube channel: see How to do long division and How to do short division .

Round numbers to estimate answers

Your child will round numbers and estimate answers to check their answers to decimal calculations. For example:

2.56 × 5.3 ≈ 2.5 × 5 = 12.5 So, when your child works out the question exactly, they will expect their result to be roughly 12.5. If they get an answer like 1350, they will know something has gone wrong!

Use equivalences between simple fractions, decimals, and percentages

$1/2$

How to help at home

There are lots of ways you can help your child to understand fractions. Here are just a few ideas.

1. Help your child calculate with fractions

$1/5$

They could separate the whole numbers and the fractional parts:

First, add the whole numbers: 3 + 2 = 5. Then, add the fractional parts: + = . Finally, add the whole number and fractional parts together to create a mixed number: 5 .

They could turn each mixed number into an improper fraction, then add the improper fractions together, and finally convert the answer back into a mixed number:

First, convert 3 into an improper fraction: . Then, convert 2 into an improper fraction: . Then, add the improper fractions together: + = . Finally, turn the answer back into a mixed number: = 5 .

The same strategies can be applied to subtracting fractions using mixed numbers.

2. Multiply and divide with fractions

$1/4$

In this example, the half is shown on the left by splitting the shape vertically into two equal pieces. The same shape will then be drawn to represent quarters , this time split horizontally into four equal pieces. The diagrams will then be added together to divide the shape into eighths:

$year 6 fraction problem solving$

The answer can be found where the half and the quarter overlap to make one eighth :

$year 6 fraction problem solving$

Fraction Calculations in School

$Pizza fractions$

3. Calculate with decimals

Your child will continue to practise adding and subtracting with decimal numbers up to 3 decimal places. They will have to understand the effect of multiplying and dividing decimal numbers by 10, 100, 1000, and so on.

Place value charts are a great way to visualise these changes:

$year 6 fraction problem solving$

Help your child to understand that dividing by 10 means getting ten times smaller, dividing by 100 means getting a hundred times smaller, and so on. The same concept applies to multiplication, but the other way around.

Calculate with numbers that have 3 decimal places in the real world. For example, when you fill the car with petrol, ask your child to tell you what the number is by explaining how many tenths, hundredths, or thousandths it has. Can they round the total price or the total amount of petrol to the nearest whole number, tenth, or hundredth? For example:

If you have 56.784 litres of petrol, you could round to the nearest litre (57L), tenth (56.8L), or hundredth (56.78L).

Money and measures are great for practising using numbers with 2 decimal places. You could show your child your shopping receipt with the total hidden. Ask your child to work out the total cost of the shopping receipt. How much change you would be given if you paid using, for instance, a £10 note?

4. Link fractions, decimals, and percentages

It is important your child understands that decimals and percentages are just other ways to show fractions.

Fraction, Decimal, and Percentage Treasure Hunt

$year 6 fraction problem solving$

Practise matching percentages, decimals, and fractions on a fun treasure hunt.

5. Multiply fractions and mixed numbers by whole numbers

Your child will multiply proper fractions and mixed numbers by whole numbers.

$1/4$

6. Percentages

Your child will continue to work with percentages. Talk about how ‘percent’ means ‘number of parts per hundred’.

$55/100$

Sales at the shops can provide great real-life opportunities to work with percentages. For example, you could ask your child to help you work out the sale price of an item. If there is 30% off a T-shirt and the full price is £9.99, what is 30% of the full price and what is the sale price?

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Year 6 Maths Worksheets UK Hub Page

Welcome to our Year 6 Maths Worksheets area.

Here you will find a wide range of free printable Year 6 Maths Worksheets for your child to enjoy.

Come and take a look at our rounding decimal pages, or maybe some of our adding and subtracting fractions worksheets. Perhaps you are looking for some worksheets about finding angles in a triangle, or need some ratio problem worksheets to help your child learn about ratio?

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Here are the instructions how to enable JavaScript in your web browser .

This page contains links to other Math webpages where you will find a range of activities and resources.
If you can't find what you are looking for, try searching the site using the Google search box at the top of each page.

Year 6 Maths Learning

Here are some of the key learning objectives for the end of Year 6:

know and use Place value up to 10 million
Counting on and back in steps of powers of 10 from any number up to 10 million
Round numbers to any given degree of accuracy.
Count forwards and backwards through zero with positive and negative numbers.
Read Roman numerals to 1000 and recognise years written in Roman numerals
solve multi-step problems using addition and subtraction in a range of contexts
identify multiples and factors including common factors
multiply and divide up to 4-digit numbers by up to 2 digits
Use their knowledge of the order of operations to carry out calculations involving the four operations.
Identify common factors, common multiples and prime numbers.
solve problems involving addition, subtraction, multiplication and division
simplify fractions
compare and order fractions including mixed numbers
add and subtract fractions with different denominators including mixed numbers
multiply simple fractions together and simplify the answer
divide proper fractions by whole numbers
recall and use equivalence between simple fractions, decimals and percentages.
Multiply and divide whole numbers and decimals up to 3dp by 10, 100 or 1000
read, write, order and compare numbers up to 3dp
round decimals with up to 3dp to the nearest whole
solve problems with numbers up to 3dp
work out percentages of different amounts
solve problems using percentages
use simple formulae
express missing number problems using algebra
find pairs of numbers that satisfy equations with two variables
solve problems involving simple ratios
solve problems involving similar shapes where the scale factor is known
use, read, write and convert between standard units of measure
measure, compare and calculate using different measures
know that shapes with the same area can have different perimeters
find the area of parallelograms and right triangles
find the volume of cubes and cuboids
convert between miles and km
name and understand the parts of circles - radius, diameter and circumference
draw 2D shapes accurately using dimensions and angles
compate and classify 2D shapes by a range of properties
find missing angles in triangles, quadrilaterals and regular shapes
use coordinates in all 4 quadrants
draw and translate simple shapes in all 4 quadrants
interpret and construct pie charts and line graphs
calculate the mean as an average

Please note:

Our site is mainly based around the US Elementary school math standards.

Though the links on this page are all designed primarily for students in the US, but they are also at the correct level and standard for UK students.

The main issue is that some of the spelling is different and this site uses US spelling.

Year 6 is generally equivalent to 5th Grade in the US.

On this page you will find link to our range of math worksheets for Year 6.

Quicklinks to Year 6 ...

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Coronavirus Stay At Home Support

For those parents who have found themselves unexpectedly at home with the kids and need some emergency activities for them to do, we have started to develop some Maths Grab Packs for kids in the UK.

Each pack consists of at least 10 mixed math worksheets on a variety of topics to help you keep you child occupied and learning.

The idea behind them is that they can be used out-of-the-box for some quick maths activities for your child.

They are completely FREE - take a look!

Free Maths Grabs Packs

Place Value & Number Sense Zone

Year 6 number worksheets.

Here you will find a range of Free Printable Year 6 Number Worksheets.

Using these Year 6 maths worksheets will help your child to:

use place value with numbers up to 10 million;
use place value with up to 3 decimal places;
understand how to use exponents (powers) of a number;
understand and use parentheses (brackets);
understand and use multiples and factors;
extend their knowledge of prime and composite (non-prime) numbers up to 100;
know and be able to use the PEMDAS (or PEDMAS) rule.
Place Value Worksheets to 10 million
Place Value to 3dp
Ordering Decimals Worksheets
PEMDAS Rule Support Page
PEMDAS Problems Worksheets
Balancing Math Equations
Roman Numerals worksheets

Ordering Large Numbers and Decimals to 3dp

The sheets in this section involve ordering lists of decimals to 3 decimal places and also large numbers up to 100 million.

There are sheets with decimals up to 10, and also sheets with numbers from -10 to 10.

Ordering Large Numbers up to 100 million
Ordering Decimals to 3dp

Rounding Decimals

Rounding to the nearest tenth
Rounding Decimal Places Sheets to 2dp
Rounding Decimals Worksheet Challenges

Year 6 Decimal Counting Worksheets

Using these sheets will support you child to:

count on and back by multiples of 0.1;
fill in the missing numbers in sequences;
count on and back into negative numbers.
Counting By Decimals

Year 6 Mental Maths Zone

Each worksheet tests the children on a range of math topics from number facts and mental arithmetic to geometry, fraction and measures questions.

A great way to revise topics, or use as a weekly math quiz!

Year 6 Mental Maths Tests

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Year 6 Addition Worksheets

add decimals including tenths and hundredths mentally;
add a columns of multi-digit numbers, including decimals.
Decimal Addition Fact Worksheets
5th Grade Addition Worksheets BIG Numbers
Decimal Column Addition Worksheets
Money Worksheets (randomly generated)

Year 6 Subtraction Worksheets

Using these sheets will help your child to:

subtract decimals including tenths and hundredths mentally;
subtract multi-digit numbers, including decimals using column subtraction.
Subtracting Decimals Worksheets (mental)
Subtraction Worksheets up to Billions (columns)
Column Subtraction with Decimals

Year 6 Multiplication Worksheets

extend their knowlege of multiplication to decimals;
use their multiplication tables to answer related facts, including decimals;
multiply a range of decimals with up to 2 decimal places (2dp) by a whole number;
multiply different money amounts by a whole number.
Multiplying Decimals by 10 and 100
Multiplication Fact Sheet Decimals
Decimal Multiplication Worksheets to 1dp
Decimal Multiplication Worksheets to 2dp
Free Multiplication Worksheets (randomly generated)
Multiply and Divide by 10 100 (decimals)
Multiplication & Division Worksheets (randomly generated)
Multiplication Word Problems

Division Worksheets 5th Grade

Using these Year 6 maths worksheets will help your child learn to:

divide any whole number up to 10000 by a two digit number;
express any division with a remainder in the form of a mixed number (a number with a fraction part).
Long Division Worksheets (whole numbers)
Long Division of Decimal Numbers
Decimal Division Facts
Division Facts Worksheets (randomly generated)

Year 6 Maths Problems

apply their addition, subtraction, multiplication and division skills;
apply their knowledge of rounding and place value;
solve a range of problems including "real life" problems and ratio problems.

These sheets involve solving one or two more challenging longer problems.

Year 6 Math Problems (5th Grade)

These sheets involve solving many 'real-life' problems involving data.

Year 6 Math Word Problems (5th Grade)

These sheets involve solving a range of ratio problems.

Year 6 Fraction Worksheets

Year 6 percentage worksheets, year 6 ratio worksheets.

compare and order fractions;
add and subtract fractions and mixed numbers;
understand how to multiply fractions by a whole number;
understand how to multiply two fractions together, including mixed fractions;
understand the relationship between fractions and division;
know how to divide fractions and mixed fractions;
convert decimals to fractions.
Comparing Fractions Worksheet page
Adding Fractions Worksheets
Adding Improper Fractions
Subtracting Fractions Worksheets
Adding Subtracting Fractions Worksheets
Improper Fraction Worksheets
Converting Decimals to Fractions Worksheets
Fractions Decimals Percents Worksheets
Multiplying Fractions Worksheets
Dividing Fractions by Whole numbers
Divide Whole numbers by Fractions
Simplifying Fractions Worksheets
Free Printable Fraction Riddles (harder)

Take a look at our percentage worksheets for finding the percentage of a number or money amount.

We have a range of percentage sheets from quite a basic level to much harder.

Percentage of Numbers Worksheets
Money Percentage Worksheets
Percentage Word Problems

These Year 6 Ratio worksheets are a great way to introduce this concept.

We have a range of part to part ratio worksheets and slightly harder problem solving worksheets.

Ratio Part to Part Worksheets
Ratio and Proportion Worksheets

Year 6 Geometry Worksheets

know how to find missing angles in a range of situations;
learn the number of degrees in a right angle, straight line, around a point and in a triangle;
know how to calculate the area of a triangle;
know how to calculate the area of a range of quadrilaterals.
learn the formulas to calculate the area of triangles and some quadrilaterals;
write and plot coordinates in all 4 quadrants.
(5th Grade) Geometry - Angles
Area of Quadrilaterals
5th Grade Volume Worksheets
Coordinate Worksheets (1st Quadrant)
Coordinate Plane Worksheets (All 4 Quadrants)
Parts of a Circle Worksheets

Measurement Zone, including Time & Money

Year 6 measurement worksheets.

Using these sheets will help your child understand how to:

learn how to read a standard scale going up in different fractions: halves, quarters, eighths and sixteenths;
learn how to read a metric scale going up in 0.1s, 5s, 10s, 25s, 50s & 100s;
learn how to estimate a measurement of length, weight or liquid;
convert temperatures in Celsius and Fahrenheit.
(5th Grade) Measurement Worksheets

Time Puzzles - harder

Here you will find our selection of harder time puzzles.

Time Word Problems Worksheets - Riddles (harder)

Using these sheets will help you to:

find the mean of up to 5 numbers;
find a missing data point when the mean is given.
Mean Worksheets

Fun Zone: Puzzles, Games and Riddles

Year 6 maths games.

Year 6 Math Games (5th Grade)

Year 6 Maths Puzzles

The puzzles will help your child practice and apply their addition, subtraction, multiplication and division facts as well as developing their thinking and reasoning skills in a fun and engaging way.

Printable Math Puzzles

Math Salamanders Year 6 Maths Games Ebook

Our Year 6 Maths Games Ebook contains all of our fun maths games, complete with instructions and resources.

This ebooklet is available in our store - use the link below to find out more!

Year 6 Maths Games Ebook

Other UK Maths Worksheet pages

See below for our other maths worksheets hub pages designed for children in the UK.

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$year 6 fraction problem solving$

Fraction word problems. Years 5-6

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

$year 6 fraction problem solving$

Last updated

18 August 2019

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$year 6 fraction problem solving$

These worksheets consist of two sets of word problems, one limited to halves and quarters. The other includes fifths, sixths and eighths. They are aimed at years 5-6 and include the answers.

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Perfect for working through fractions in context. Thanks!

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Fraction word problems for year 6

$Year 6 fraction word problems$

A fraction word problems worksheet for year 6. Children are asked to simplify fractions, to add and subtract fractions, to multiply fractions, to divide fractions and to solve problems to find fractions of amounts. Calculations feature proper fractions and include those with different denominators

This one-page problem solving maths worksheet is linked to national curriculum objectives and is ideal for classroom and home learning. Answers are included.

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Fractions KS2

This collection is one of our Primary Curriculum collections - tasks that are grouped by topic.

$year 6 fraction problem solving$

Fraction Match

A task which depends on members of the group noticing the needs of others and responding.

$year 6 fraction problem solving$

Here is a picnic that Petros and Michael are going to share equally. Can you tell us what each of them will have?

$year 6 fraction problem solving$

Round the Dice Decimals 2

What happens when you round these numbers to the nearest whole number?

$year 6 fraction problem solving$

Fractional Wall

Using the picture of the fraction wall, can you find equivalent fractions?

$year 6 fraction problem solving$

Would You Rather?

Would you rather: Have 10% of Â£5 or 75% of 80p? Be given 60% of 2 pizzas or 26% of 5 pizzas?

$year 6 fraction problem solving$

Round the Dice Decimals 1

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

$year 6 fraction problem solving$

Tumbling Down

Watch this animation. What do you see? Can you explain why this happens?

$year 6 fraction problem solving$

Linked Chains

Can you find ways to make twenty-link chains from these smaller chains? This gives opportunities for different approaches.

$year 6 fraction problem solving$

Fractional Triangles

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

$year 6 fraction problem solving$

Bryony's Triangle

Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle?

$year 6 fraction problem solving$

A4 Fraction Subtraction

This task offers opportunities to subtract fractions using A4 paper.

$year 6 fraction problem solving$

Matching Fractions

Can you find different ways of showing the same fraction? Try this matching game and see.

$year 6 fraction problem solving$

More Fraction Bars

What fraction of the black bar are the other bars? Have a go at this challenging task!

$year 6 fraction problem solving$

Fractions in a Box

The discs for this game are kept in a flat square box with a square hole for each. Use the information to find out how many discs of each colour there are in the box.

$year 6 fraction problem solving$

Extending Fraction Bars

Can you compare these bars with each other and express their lengths as fractions of the black bar?

$year 6 fraction problem solving$

Route Product

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

$year 6 fraction problem solving$

Fraction Lengths

Can you find combinations of strips of paper which equal the length of the black strip? If the length of the black is 1, how could you write the sum of the strips?

$year 6 fraction problem solving$

Light Blue - Dark Blue

Investigate the successive areas of light blue in these diagrams.

$year 6 fraction problem solving$

A4 Fraction Addition

Try adding fractions using A4 paper.

Forgot the Numbers

On my calculator I divided one whole number by another whole number and got the answer 3.125. If the numbers are both under 50, what are they?

$year 6 fraction problem solving$

Andy's Marbles

Andy had a big bag of marbles but unfortunately the bottom of it split and all the marbles spilled out. Use the information to find out how many there were in the bag originally.

$year 6 fraction problem solving$

Matching Fractions, Decimals and Percentages

Can you match pairs of fractions, decimals and percentages, and beat your previous scores?

$year 6 fraction problem solving$

Doughnut Percents

A task involving the equivalence between fractions, percentages and decimals which depends on members of the group noticing the needs of others and responding.

There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?

$year 6 fraction problem solving$

Free Printable Fraction Word Problems Worksheets for 6th Year

Fraction Word Problems: Discover a collection of free printable Math worksheets for Year 6 students, focusing on solving real-life scenarios involving fractions. Enhance learning and problem-solving skills with Quizizz's resources.

$year 6 fraction problem solving$

Explore Fraction Word Problems Worksheets by Grades

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Explore printable Fraction Word Problems worksheets for 6th Year

Fraction Word Problems worksheets for Year 6 are an essential resource for teachers looking to help their students master the challenging world of fractions in math. These worksheets provide a variety of math word problems that incorporate fractions, allowing students to practice their skills in a fun and engaging way. Teachers can use these worksheets to supplement their lesson plans, provide extra practice for struggling students, or even as a form of assessment to gauge their students' understanding of the material. With a wide range of topics covered, from addition and subtraction of fractions to more complex problems involving ratios and proportions, these Year 6 worksheets are a valuable tool for any math teacher looking to enhance their students' learning experience.

In addition to Fraction Word Problems worksheets for Year 6, teachers can also utilize Quizizz, an interactive platform that offers a variety of educational resources, including quizzes, games, and other engaging activities. Quizizz allows teachers to create custom quizzes and games based on the content they are teaching, making it a perfect complement to the worksheets they are already using in their classroom. With Quizizz, students can practice their math skills in a more interactive and enjoyable way, helping to solidify their understanding of the material. Furthermore, Quizizz provides teachers with valuable data and insights into their students' progress, allowing them to identify areas where additional support may be needed. By incorporating both Fraction Word Problems worksheets for Year 6 and Quizizz into their lesson plans, teachers can provide a well-rounded and effective learning experience for their students.

Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years.

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything.

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students.

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

$year 6 fraction problem solving$

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told them was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah.

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it."

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson: So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash.

Bill Whitaker: Did you look at the problem?

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

$year 6 fraction problem solving$

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter)

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

$year 6 fraction problem solving$

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics.

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch.

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh)

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah.

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans.

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates.

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

$year 6 fraction problem solving$

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in.

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven.

Bill Whitaker: What do you mean by that?

$year 6 fraction problem solving$

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter)

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

$year 6 fraction problem solving$

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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A Plan to Remake the Middle East

While talks for a cease-fire between israel and hamas continue, another set of negotiations is happening behind the scenes..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From New York Times, I’m Michael Barbaro. This is The Daily.

[MUSIC CONTINUES]

Today, if and when Israel and Hamas reach a deal for a ceasefire fire, the United States will immediately turn to a different set of negotiations over a grand diplomatic bargain that it believes could rebuild Gaza and remake the Middle East. My colleague Michael Crowley has been reporting on that plan and explains why those involved in it believe they have so little time left to get it done.

It’s Wednesday, May 8.

Michael, I want to start with what feels like a pretty dizzying set of developments in this conflict over the past few days. Just walk us through them?

Well, over the weekend, there was an intense round of negotiations in an effort, backed by the United States, to reach a ceasefire in the Gaza war.

The latest ceasefire proposal would reportedly see as many as 33 Israeli hostages released in exchange for potentially hundreds of Palestinian prisoners.

US officials were very eager to get this deal.

Pressure for a ceasefire has been building ahead of a threatened Israeli assault on Rafah.

Because Israel has been threatening a military offensive in the Southern Palestinian city of Rafah, where a huge number of people are crowded.

Fleeing the violence to the North. And now they’re packed into Rafah. Exposed and vulnerable, they need to be protected.

And the US says it would be a humanitarian catastrophe on top of the emergency that’s already underway.

Breaking news this hour — very important breaking news. An official Hamas source has told The BBC that it does accept a proposal for a ceasefire deal in Gaza.

And for a few hours on Monday, it looked like there might have been a major breakthrough when Hamas put out a statement saying that it had accepted a negotiating proposal.

Israeli Prime Minister Benjamin Netanyahu says the ceasefire proposal does not meet his country’s requirements. But Netanyahu says he will send a delegation of mediators to continue those talks. Now, the terms —

But those hopes were dashed pretty quickly when the Israelis took a look at what Hamas was saying and said that it was not a proposal that they had agreed to. It had been modified.

And overnight —

Israeli troops stormed into Rafah. Video showing tanks crashing over a sign at the entrance of the city.

— the Israelis launched a partial invasion of Rafah.

It says Hamas used the area to launch a deadly attack on Israeli troops over the weekend.

And they have now secured a border crossing at the Southern end of Gaza and are conducting targeted strikes. This is not yet the full scale invasion that President Biden has adamantly warned Israel against undertaking, but it is an escalation by Israel.

So while all that drama might suggest that these talks are in big trouble, these talks are very much still alive and ongoing and there is still a possibility of a ceasefire deal.

And the reason that’s so important is not just to stop the fighting in Gaza and relieve the suffering there, but a ceasefire also opens the door to a grand diplomatic bargain, one that involves Israel and its Arab neighbors and the Palestinians, and would have very far-reaching implications.

And what is that grand bargain. Describe what you’re talking about?

Well, it’s incredibly ambitious. It would reshape Israel’s relationship with its Arab neighbors, principally Saudi Arabia. But it’s important to understand that this is a vision that has actually been around since well before October 7. This was a diplomatic project that President Biden had been investing in and negotiating actually in a very real and tangible way long before the Hamas attacks and the Gaza war.

And President Biden was looking to build on something that President Trump had done, which was a series of agreements that the Trump administration struck in which Israel and some of its Arab neighbors agreed to have normal diplomatic relations for the first time.

Right, they’re called the Abraham Accords.

That’s right. And, you know, Biden doesn’t like a lot of things, most things that Trump did. But he actually likes this, because the idea is that they contribute to stability and economic integration in the Middle East, the US likes Israel having friends and likes having a tight-knit alliance against Iran.

President Biden agrees with the Saudis and with the Israelis, that Iran is really the top threat to everybody here. So, how can you build on this? How can you expand it? Well, the next and biggest step would be normalizing relations between Israel and Saudi Arabia.

And the Saudis have made clear that they want to do this and that they’re ready to do this. They weren’t ready to do it in the Trump years. But Mohammed bin Salman, the Crown Prince of Saudi Arabia, has made clear he wants to do it now.

So this kind of triangular deal began to take shape before October 7, in which the US, Israel, and Saudi Arabia would enter this three way agreement in which everyone would get something that they wanted.

And just walk through what each side gets in this pre-October 7th version of these negotiations?

So for Israel, you get normalized ties with its most important Arab neighbor and really the country that sets the tone for the whole Muslim world, which is Saudi Arabia of course. It makes Israel feel safer and more secure. Again, it helps to build this alliance against Iran, which Israel considers its greatest threat, and it comes with benefits like economic ties and travel and tourism. And Prime Minister Benjamin Netanyahu has been very open, at least before October 7th, that this was his highest diplomatic and foreign policy priority.

For the Saudis, the rationale is similar when it comes to Israel. They think that it will bring stability. They like having a more explicitly close ally against Iran. There are economic and cultural benefits. Saudi Arabia is opening itself up in general, encouraging more tourism.

But I think that what’s most important to the Crown Prince, Mohammed bin Salman, is what he can get from the United States. And what he has been asking for are a couple of essential things. One is a security agreement whose details have always been a little bit vague, but I think essentially come down to reliable arms supplies from the United States that are not going to be cut off or paused on a whim, as he felt happened when President Biden stopped arms deliveries in 2021 because of how Saudi was conducting its war in Yemen. The Saudis were furious about that.

Saudi Arabia also wants to start a domestic nuclear power program. They are planning for a very long-term future, possibly a post-oil future. And they need help getting a nuclear program off the ground.

And they want that from the US?

And they want that from the US.

Now, those are big asks from the us. But from the perspective of President Biden, there are some really enticing things about this possible agreement. One is that it will hopefully produce more stability in the region. Again, the US likes having a tight-knit alliance against Iran.

The US also wants to have a strong relationship with Saudi Arabia. You know, despite the anger at Mohammed bin Salman over the murder of the Saudi dissident Jamal Khashoggi, the Biden administration recognizes that given the Saudis control over global oil production and their strategic importance in the Middle East, they need to have a good relationship with them. And the administration has been worried about the influence of China in the region and with the Saudis in particular.

So this is an opportunity for the US to draw the Saudis closer. Whatever our moral qualms might be about bin Salman and the Saudi government, this is an opportunity to bring the Saudis closer, which is something the Biden administration sees as a strategic benefit.

All three of these countries — big, disparate countries that normally don’t see eye-to-eye, this was a win-win-win on a military, economic, and strategic front.

That’s right. But there was one important actor in the region that did not see itself as winning, and that was the Palestinians.

[MUSIC PLAYING]

First, it’s important to understand that the Palestinians have always expected that the Arab countries in the Middle East would insist that Israel recognize a Palestinian state before those countries were willing to essentially make total peace and have normal relations with Israel.

So when the Abraham Accords happened in the Trump administration, the Palestinians felt like they’d been thrown under the bus because the Abraham Accords gave them virtually nothing. But the Palestinians did still hold out hope that Saudi Arabia would be their savior. And for years, Saudi Arabia has said that Israel must give the Palestinians a state if there’s going to be a normal relationship between Israel and Saudi Arabia.

Now the Palestinians see the Saudis in discussions with the US and Israel about a normalization agreement, and there appears to be very little on offer for the Palestinians. And they are feeling like they’re going to be left out in the cold here.

Right. And in the minds of the Palestinians, having already been essentially sold out by all their other Arab neighbors, the prospect that Saudi Arabia, of all countries, the most important Muslim Arab country in the region, would sell them out, had to be extremely painful.

It was a nightmare scenario for them. And in the minds of many analysts and US officials, this was a factor, one of many, in Hamas’s decision to stage the October 7th attacks.

Hamas, like other Palestinian leaders, was seeing the prospect that the Middle East was moving on and essentially, in their view, giving up on the Palestinian cause, and that Israel would be able to have friendly, normal relations with Arab countries around the region, and that it could continue with hardline policies toward the Palestinians and a refusal, as Prime Minister Benjamin Netanyahu has said publicly, to accept a Palestinian state.

Right. So Michael, once Hamas carries out the October 7th attacks in an effort to destroy a status quo that it thinks is leaving them less and less relevant, more and more hopeless, including potentially this prospect that Saudi Arabia is going to normalize relations with Israel, what happens to these pre-October 7th negotiations between the US, Saudi Arabia, and Israel?

Well, I think there was a snap assumption that these talks were dead and buried. That they couldn’t possibly survive a cataclysm like this.

But then something surprising happened. It became clear that all the parties were still determined to pull-off the normalization.

And most surprisingly of all, perhaps, was the continued eagerness of Saudi Arabia, which publicly was professing outrage over the Israeli response to the Hamas attacks, but privately was still very much engaged in these conversations and trying to move them forward.

And in fact, what has happened is that the scope of this effort has grown substantially. October 7th didn’t kill these talks. It actually made them bigger, more complicated, and some people would argue, more important than ever.

We’ll be right back.

Michael, walk us through what exactly happens to these three-way negotiations after October 7th that ends up making them, as you just said, more complicated and more important than ever?

Well, it’s more important than ever because of the incredible need in Gaza. And it’s going to take a deal like this and the approval of Saudi Arabia to unlock the kind of massive reconstruction project required to essentially rebuild Gaza from the rubble. Saudi Arabia and its Arab friends are also going to be instrumental in figuring out how Gaza is governed, and they might even provide troops to help secure it. None of those things are going to happen without a deal like this.

Fascinating.

But this is all much more complicated now because the price for a deal like this has gone up.

And by price, you mean?

What Israel would have to give up. [MUSIC PLAYING]

From Saudi Arabia’s perspective, you have an Arab population that is furious at Israel. It now feels like a really hard time to do a normalization deal with the Israelis. It was never going to be easy, but this is about as bad a time to do it as there has been in a generation at least. And I think that President Biden and the people around him understand that the status quo between Israel and the Palestinians is intolerable and it is going to lead to chaos and violence indefinitely.

So now you have two of the three parties to this agreement, the Saudis and the Americans, basically asking a new price after October 7th, and saying to the Israelis, if we’re going to do this deal, it has to not only do something for the Palestinians, it has to do something really big. You have to commit to the creation of a Palestinian state. Now, I’ll be specific and say that what you hear the Secretary of State, Antony Blinken, say is that the agreement has to include an irreversible time-bound path to a Palestinian state.

We don’t know exactly what that looks like, but it’s some kind of a firm commitment, the likes of which the world and certainly the Israelis have not made before.

Something that was very much not present in the pre-October 7th vision of this negotiation. So much so that, as we just talked about, the Palestinians were left feeling completely out in the cold and furious at it.

That’s right. There was no sign that people were thinking that ambitiously about the Palestinians in this deal before October 7th. And the Palestinians certainly felt like they weren’t going to get much out of it. And that has completely changed now.

So, Michael, once this big new dimension after October 7th, which is the insistence by Saudi Arabia and the US that there be a Palestinian state or a path to a Palestinian state, what is the reaction specifically from Israel, which is, of course, the third major party to this entire conversation?

Well, Israel, or at least its political leadership, hates it. You know, this is just an extremely tough sell in Israel. It would have been a tough sell before October 7th. It’s even harder now.

Prime Minister Benjamin Netanyahu is completely unrepentantly open in saying that there’s not going to be a Palestinian state on his watch. He won’t accept it. He says that it’s a strategic risk to his country. He says that it would, in effect, reward Hamas.

His argument is that terrorism has forced a conversation about statehood onto the table that wasn’t there before October 7th. Sure, it’s always in the background. It’s a perennial issue in global affairs, but it was not something certainly that the US and Israel’s Arab neighbors were actively pushing. Netanyahu also has — you know, he governs with the support of very right-wing members of a political coalition that he has cobbled together. And that coalition is quite likely to fall apart if he does embrace a Palestinian state or a path to a Palestinian state.

Now, he might be able to cobble together some sort of alternative, but it creates a political crisis for him.

And finally, you know, I think in any conversation about Israel, it’s worth bearing in mind something you hear from senior US officials these days, which is that although there is often finger pointing at Netanyahu and a desire to blame Netanyahu as this obstructionist who won’t agree to deals, what they say is Netanyahu is largely reflecting his population and the political establishment of his country, not just the right-wingers in his coalition who are clearly extremist.

But actually the prevailing views of the Israeli public. And the Israeli public and their political leaders across the spectrum right now with few exceptions, are not interested in talking about a Palestinian state when there are still dozens and dozens of Israeli hostages in tunnels beneath Gaza.

So it very much looks like this giant agreement that once seemed doable before October 7th might be more important to everyone involved than ever, given that it’s a plan for rebuilding Gaza and potentially preventing future October 7th’s from happening, but because of this higher price that Israel would have to pay, which is the acceptance of a Palestinian state, it seems from everything you’re saying, that this is more and more out of reach than ever before and hard to imagine happening in the immediate future. So if the people negotiating it are being honest, Michael, are they ready to acknowledge that it doesn’t look like this is going to happen?

Well, not quite yet. As time goes by, they certainly say it’s getting harder and harder, but they’re still trying, and they still think there’s a chance. But both the Saudis and the Biden administration understand that there’s very little time left to do this.

Well, what do you mean there’s very little time left? It would seem like time might benefit this negotiation in that it might give Israel distance from October 7th to think potentially differently about a Palestinian state?

Potentially. But Saudi Arabia wants to get this deal done in the Biden administration because Mohammed bin Salman has concluded this has to be done under a Democratic president.

Because Democrats in Congress are going to be very reluctant to approve a security agreement between the United States and Saudi Arabia.

It’s important to understand that if there is a security agreement, that’s something Congress is going to have to approve. And you’re just not going to get enough Democrats in Congress to support a deal with Saudi Arabia, who a lot of Democrats don’t like to begin with, because they see them as human rights abusers.

But if a Democratic president is asking them to do it, they’re much more likely to go along.

Right. So Saudi Arabia fears that if Biden loses and Trump is president, that those same Democrats would balk at this deal in a way that they wouldn’t if it were being negotiated under President Biden?

Exactly. Now, from President Biden’s perspective, politically, think about a president who’s running for re-election, who is presiding right now over chaos in the Middle East, who doesn’t seem to have good answers for the Israeli-Palestinian question, this is an opportunity for President Biden to deliver what could be at least what he would present as a diplomatic masterstroke that does multiple things at once, including creating a new pathway for Israel and the Palestinians to coexist, to break through the logjam, even as he is also improving Israel’s relations with Saudi Arabia.

So Biden and the Crown Prince hope that they can somehow persuade Bibi Netanyahu that in spite of all the reasons that he thinks this is a terrible idea, that this is a bet worth taking on Israel’s and the region’s long-term security and future?

That’s right. Now, no one has explained very clearly exactly how this is going to work, and it’s probably going to require artful diplomacy, possibly even a scenario where the Israelis would agree to something that maybe means one thing to them and means something else to other people. But Biden officials refuse to say that it’s hopeless and they refuse to essentially take Netanyahu’s preliminary no’s for an answer. And they still see some way that they can thread this incredibly narrow needle.

Michael, I’m curious about a constituency that we haven’t been talking about because they’re not at the table in these discussions that we are talking about here. And that would be Hamas. How does Hamas feel about the prospect of such a deal like this ever taking shape. Do they see it as any kind of a victory and vindication for what they did on October 7th?

So it’s hard to know exactly what Hamas’s leadership is thinking. I think they can feel two things. I think they can feel on the one hand, that they have established themselves as the champions of the Palestinian people who struck a blow against Israel and against a diplomatic process that was potentially going to leave the Palestinians out in the cold.

At the same time, Hamas has no interest in the kind of two-state solution that the US is trying to promote. They think Israel should be destroyed. They think the Palestinian state should cover the entire geography of what is now Israel, and they want to lead a state like that. And that’s not something that the US, Saudi Arabia, or anyone else is going to tolerate.

So what Hamas wants is to fight, to be the leader of the Palestinian people, and to destroy Israel. And they’re not interested in any sort of a peace process or statehood process.

It seems very clear from everything you’ve said here that neither Israel nor Hamas is ready to have the conversation about a grand bargain diplomatic program. And I wonder if that inevitably has any bearing on the ceasefire negotiations that are going on right now between the two of them that are supposed to bring this conflict to some sort of an end, even if it’s just temporary?

Because if, as you said, Michael, a ceasefire opens the door to this larger diplomatic solution, and these two players don’t necessarily want that larger diplomatic solution, doesn’t that inevitably impact their enthusiasm for even reaching a ceasefire?

Well, it certainly doesn’t help. You know, this is such a hellish problem. And of course, you first have the question of whether Israel and Hamas can make a deal on these immediate issues, including the hostages, Palestinian prisoners, and what the Israeli military is going to do, how long a ceasefire might last.

But on top of that, you have these much bigger diplomatic questions that are looming over them. And it’s not clear that either side is ready to turn and face those bigger questions.

So while for the Biden administration and for Saudi Arabia, this is a way out of this crisis, these larger diplomatic solutions, it’s not clear that it’s a conversation that the two parties that are actually at war here are prepared to start having.

Well, Michael, thank you very much. We appreciate it.

On Tuesday afternoon, under intense pressure from the US, delegations from Israel and Hamas arrived in Cairo to resume negotiations over a potential ceasefire. But in a statement, Israel’s Prime Minister Benjamin Netanyahu made clear that even with the talks underway, his government would, quote, “continue to wage war against Hamas.”

Here’s what else you need to know today. In a dramatic day of testimony, Stormy Daniels offered explicit details about an alleged sexual encounter with Donald Trump that ultimately led to the hush money payment at the center of his trial. Daniels testified that Trump answered the door in pajamas, that he told her not to worry that he was married, and that he did not use a condom when they had sex.

That prompted lawyers for Trump to seek a mistrial based on what they called prejudicial testimony. But the judge in the case rejected that request. And,

We’ve seen a ferocious surge of anti-Semitism in America and around the world.

In a speech on Tuesday honoring victims of the Holocaust, President Biden condemned what he said was the alarming rise of anti-Semitism in the United States after the October 7th attacks on Israel. And he expressed worry that too many Americans were already forgetting the horrors of that attack.

The Jewish community, I want you to know I see your fear, your hurt, and your pain. Let me reassure you, as your president, you’re not alone. You belong. You always have and you always will.

Today’s episode was produced by Nina Feldman, Clare Toeniskoetter, and Rikki Novetsky. It was edited by Liz O. Baylen, contains original music by Marion Lozano, Elisheba Ittoop, and Dan Powell, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for The Daily. I’m Michael Barbaro. See you tomorrow.

$year 6 fraction problem solving$

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Hosted by Michael Barbaro

Featuring Michael Crowley

Produced by Nina Feldman , Clare Toeniskoetter and Rikki Novetsky

Edited by Liz O. Baylen

Original music by Marion Lozano , Elisheba Ittoop and Dan Powell

Engineered by Alyssa Moxley

Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music | YouTube

If and when Israel and Hamas reach a deal for a cease-fire, the United States will immediately turn to a different set of negotiations over a grand diplomatic bargain that it believes could rebuild Gaza and remake the Middle East.

Michael Crowley, who covers the State Department and U.S. foreign policy for The Times, explains why those involved in this plan believe they have so little time left to get it done.

On today’s episode

$year 6 fraction problem solving$

Michael Crowley , a reporter covering the State Department and U.S. foreign policy for The New York Times.

$year 6 fraction problem solving$

Background reading :

Talks on a cease-fire in the Gaza war are once again at an uncertain stage .

Here’s how the push for a deal between Israel and Saudi Arabia looked before Oct. 7 .

From early in the war, President Biden has said that a lasting resolution requires a “real” Palestinian state .

Here’s what Israeli officials are discussing about postwar Gaza.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Michael Crowley covers the State Department and U.S. foreign policy for The Times. He has reported from nearly three dozen countries and often travels with the secretary of state. More about Michael Crowley

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Fraction questions for Year 6: fraction word problems. Fraction question 21. Max has 150 pages in his book. He has read \frac{3}{5} of the book. How many pages does he have left to read? Answer: 60 pages left to read. \frac{1}{5} of 150 = 30 \frac{2}{5} of 150 = 60. Fraction question 22. Sienna has half of a pizza left.
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