## Fractions Questions and Problems with Solutions

Questions and problems with solutions on fractions are presented. Detailed solutions to the examples are also included. In order to master the concepts and skills of fractions, you need a thorough understanding (NOT memorizing) of the rules and properties and lot of practice and patience. I hope the examples, questions, problems in the links below will help you.

- Fractions and Mixed Numbers , define fractions and mixed numbers, and introduce important vocabulary.
- Evaluate Fractions of Quantities .
- Fractions Rules including questions with Solutions
- Properties of Fractions
- Complex Fractions with Variables
- Equivalent Fractions examples and questions with solutions.
- Reduce Fractions examples and questions with solutions.
- Reduce Fractions Calculator
- Simplify Fractions , examples and questions including solutions.
- Factor Fractions , examples with questions including solutions.
- Adding Fractions. Add fractions with same denominator or different denominator. Several examples with detailed solutions and exercises.
- Multiply Fractions. Multiply a fraction by another fraction or a number by a fraction. Examples with solutions and exercises.
- Divide Fractions. Divide a fraction by a fraction, a fraction by a number of a number by a fraction. Several examples with solutions and exercises with answers.

## Fractions Per Grade

- Fractions and Mixed Numbers , questions and problems with solutions for grade 7
- Fractions and Mixed Numbers , questions and problems with solutions for grade 6
- Fractions , questions for grade 5 and their solutions
- Fractions , questions for grade 4 their answers.

## Fractions Calculators

- Fractions Calculator including reducing, adding and multiplying fractions.
- Add Mixed Numbers Calculator

## Fraction Worksheets

Conversion :: Addition :: Subtraction :: Multiplication :: Division

## Conversions

Fractions - addition, fractions - subtraction, fractions - multiplication, fractions - division.

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## Fraction Word Problem Worksheets

Featured here is a vast collection of fraction word problems, which require learners to simplify fractions, add like and unlike fractions; subtract like and unlike fractions; multiply and divide fractions. The fraction word problems include proper fraction, improper fraction, and mixed numbers. Solve each word problem and scroll down each printable worksheet to verify your solutions using the answer key provided. Thumb through some of these word problem worksheets for free!

Represent and Simplify the Fractions: Type 1

Presented here are the fraction pdf worksheets based on real-life scenarios. Read the basic fraction word problems, write the correct fraction and reduce your answer to the simplest form.

- Download the set

Represent and Simplify the Fractions: Type 2

Before representing in fraction, children should perform addition or subtraction to solve these fraction word problems. Write your answer in the simplest form.

Adding Fractions Word Problems Worksheets

Conjure up a picture of how adding fractions plays a significant role in our day-to-day lives with the help of the real-life scenarios and circumstances presented as word problems here.

(15 Worksheets)

Subtracting Fractions Word Problems Worksheets

Crank up your skills with this set of printable worksheets on subtracting fractions word problems presenting real-world situations that involve fraction subtraction!

Multiplying Fractions Word Problems Worksheets

This set of printables is for the ardently active children! Explore the application of fraction multiplication and mixed-number multiplication in the real world with this exhilarating practice set.

Fraction Division Word Problems Worksheets

Gift children a broad view of the real-life application of dividing fractions! Let them divide fractions by whole numbers, divide 2 fractions, divide mixed numbers, and solve the word problems here.

Related Worksheets

» Decimal Word Problems

» Ratio Word Problems

» Division Word Problems

» Math Word Problems

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## How to Solve Fraction Questions in Math

Last Updated: April 14, 2024 Fact Checked

This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Sophia Latorre . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,196,885 times.

Fraction questions can look tricky at first, but they become easier with practice and know-how. Start by learning the terminology and fundamentals, then pratice adding, subtracting, multiplying, and dividing fractions. [1] X Research source Once you understand what fractions are and how to manipulate them, you'll be breezing through fraction problems in no time.

## Doing Calculations with Fractions

- For instance, to solve 5/9 + 1/9, just add 5 + 1, which equals 6. The answer, then, is 6/9 which can be reduced to 2/3.

- For instance, to solve 6/8 - 2/8, all you do is take away 2 from 6. The answer is 4/8, which can be reduced to 1/2.

- For example, if you need to add 1/2 and 2/3, start by determining a common multiple. In this case, the common multiple is 6 since both 2 and 3 can be converted to 6. To turn 1/2 into a fraction with a denominator of 6, multiply both the numerator and denominator by 3: 1 x 3 = 3 and 2 x 3 = 6, so the new fraction is 3/6. To turn 2/3 into a fraction with a denominator of 6, multiply both the numerator and denominator by 2: 2 x 2 = 4 and 3 x 2 = 6, so the new fraction is 4/6. Now, you can add the numerators: 3/6 + 4/6 = 7/6. Since this is an improper fraction, you can convert it to the mixed number 1 1/6.
- On the other hand, say you're working on the problem 7/10 - 1/5. The common multiple in this case is 10, since 1/5 can be converted into a fraction with a denominator of 10 by multiplying it by 2: 1 x 2 = 2 and 5 x 2 = 10, so the new fraction is 2/10. You don't need to convert the other fraction at all. Just subtract 2 from 7, which is 5. The answer is 5/10, which can also be reduced to 1/2.

- For instance, to multiply 2/3 and 7/8, find the new numerator by multiplying 2 by 7, which is 14. Then, multiply 3 by 8, which is 24. Therefore, the answer is 14/24, which can be reduced to 7/12 by dividing both the numerator and denominator by 2.

- For example, to solve 1/2 ÷ 1/6, flip 1/6 upside down so it becomes 6/1. Then just multiply 1 x 6 to find the numerator (which is 6) and 2 x 1 to find the denominator (which is 2). So, the answer is 6/2 which is equal to 3.

Joseph Meyer

Think about fractions as portions of a whole. Imagine dividing objects like pizzas or cakes into equal parts. Visualizing fractions this way improves comprehension, compared to relying solely on memorization. This approach can be helpful when adding, subtracting, and comparing fractions.

## Practicing the Basics

- For instance, in 3/5, 3 is the numerator so there are 3 parts and 5 is the denominator so there are 5 total parts. In 7/8, 7 is the numerator and 8 is the denominator.

- If you need to turn 7 into a fraction, for instance, write it as 7/1.

- For example, if you have the fraction 15/45, the greatest common factor is 15, since both 15 and 45 can be divided by 15. Divide 15 by 15, which is 1, so that's your new numerator. Divide 45 by 15, which is 3, so that's your new denominator. This means that 15/45 can be reduced to 1/3.

- Say you have the mixed number 1 2/3. Stary by multiplying 3 by 1, which is 3. Add 3 to 2, the existing numerator. The new numerator is 5, so the mixed fraction is 5/3.

Tip: Typically, you'll need to convert mixed numbers to improper fractions if you're multiplying or dividing them.

- Say that you have the improper fraction 17/4. Set up the problem as 17 ÷ 4. The number 4 goes into 17 a total of 4 times, so the whole number is 4. Then, multiply 4 by 4, which is equal to 16. Subtract 16 from 17, which is equal to 1, so that's the remainder. This means that 17/4 is the same as 4 1/4.

## Fraction Calculator, Practice Problems, and Answers

## Community Q&A

- Check with your teacher to find out if you need to convert improper fractions into mixed numbers and/or reduce fractions to their lowest terms to get full marks. Thanks Helpful 2 Not Helpful 1
- Take the time to carefully read through the problem at least twice so you can be sure you know what it's asking you to do. Thanks Helpful 2 Not Helpful 2
- To take the reciprocal of a whole number, just put a 1 over it. For example, 5 becomes 1/5. Thanks Helpful 1 Not Helpful 1

## You Might Also Like

- ↑ https://www.sparknotes.com/math/prealgebra/fractions/terms/
- ↑ https://www.bbc.co.uk/bitesize/articles/z9n4k7h
- ↑ https://www.mathsisfun.com/fractions_multiplication.html
- ↑ https://www.mathsisfun.com/fractions_division.html
- ↑ https://medium.com/i-math/the-no-nonsense-straightforward-da76a4849ec
- ↑ https://www.youtube.com/watch?v=PcEwj5_v75g
- ↑ https://sciencing.com/solve-math-problems-fractions-7964895.html

## About This Article

To solve a fraction multiplication question in math, line up the 2 fractions next to each other. Multiply the top of the left fraction by the top of the right fraction and write that answer on top, then multiply the bottom of each fraction and write that answer on the bottom. Simplify the new fraction as much as possible. To divide fractions, flip one of the fractions upside-down and multiply them the same way. If you need to add or subtract fractions, keep reading! Did this summary help you? Yes No

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## Fraction Word Problems (Difficult)

Here are some examples of more difficult fraction word problems. We will illustrate how block models (tape diagrams) can be used to help you to visualize the fraction word problems in terms of the information given and the data that needs to be found.

Related Pages Fraction Word Problems Singapore Math Lessons Fraction Problems Using Algebra Algebra Word Problems

Block modeling (also known as tape diagrams or bar models) are widely used in Singapore Math and the Common Core to help students visualize and understand math word problems.

Example: 2/9 of the people on a restaurant are adults. If there are 95 more children than adults, how many children are there in the restaurant?

Solution: Draw a diagram with 9 equal parts: 2 parts to represent the adults and 7 parts to represent the children.

5 units = 95 1 unit = 95 ÷ 5 = 19 7 units = 7 × 19 = 133

Answer: There are 133 children in the restaurant.

Example: Gary and Henry brought an equal amount of money for shopping. Gary spent $95 and Henry spent $350. After that Henry had 4/7 of what Gary had left. How much money did Gary have left after shopping?

350 – 95 = 255 3 units = 255 1 unit = 255 ÷ 3 = 85 7 units = 85 × 7 = 595

Answer: Gary has $595 after shopping.

Example: 1/9 of the shirts sold at Peter’s shop are striped. 5/8 of the remainder are printed. The rest of the shirts are plain colored shirts. If Peter’s shop has 81 plain colored shirts, how many more printed shirts than plain colored shirts does the shop have?

Solution: Draw a diagram with 9 parts. One part represents striped shirts. Out of the remaining 8 parts: 5 parts represent the printed shirts and 3 parts represent plain colored shirts.

3 units = 81 1 unit = 81 ÷ 3 = 27 Printed shirts have 2 parts more than plain shirts. 2 units = 27 × 2 = 54

Answer: Peter’s shop has 54 more printed colored shirts than plain shirts.

Solve a problem involving fractions of fractions and fractions of remaining parts

Example: 1/4 of my trail mix recipe is raisins and the rest is nuts. 3/5 of the nuts are peanuts and the rest are almonds. What fraction of my trail mix is almonds?

How to solve fraction word problem that involves addition, subtraction and multiplication using a tape diagram or block model

Example: Jenny’s mom says she has an hour before it’s bedtime. Jenny spends 3/5 of the hour texting a friend and 3/8 of the remaining time brushing her teeth and putting on her pajamas. She spends the rest of the time reading her book. How long did Jenny read?

How to solve a four step fraction word problem using tape diagrams?

Example: In an auditorium, 1/6 of the students are fifth graders, 1/3 are fourth graders, and 1/4 of the remaining students are second graders. If there are 96 students in the auditorium, how many second graders are there?

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

## Word Problems on Fraction

In word problems on fraction we will solve different types of problems on multiplication of fractional numbers and division of fractional numbers.

1. 4/7 of a number is 84. Find the number. Solution: According to the problem, 4/7 of a number = 84 Number = 84 × 7/4 [Here we need to multiply 84 by the reciprocal of 4/7]

= 21 × 7 = 147 Therefore, the number is 147.

2. Rachel took \(\frac{1}{2}\) hour to paint a table and \(\frac{1}{3}\) hour to paint a chair. How much time did she take in all?

3. If 3\(\frac{1}{2}\) m of wire is cut from a piece of 10 m long wire, how much of wire is left?

Total length of the wire = 10 m

Fraction of the wire cut out = 3\(\frac{1}{2}\) m = \(\frac{7}{2}\) m

Length of the wire left = 10 m – 3\(\frac{1}{2}\) m

= [\(\frac{10}{1}\) - \(\frac{7}{2}\)] m, [L.C.M. of 1, 2 is 2]

= [\(\frac{20}{2}\) - \(\frac{7}{2}\)] m, [\(\frac{10}{1}\) × \(\frac{2}{2}\)]

= [\(\frac{20 - 7}{2}\)] m

= \(\frac{13}{2}\) m

= 6\(\frac{1}{2}\) m

4. One half of the students in a school are girls, 3/5 of these girls are studying in lower classes. What fraction of girls are studying in lower classes?

Fraction of girls studying in school = 1/2

Fraction of girls studying in lower classes = 3/5 of 1/2

= 3/5 × 1/2

= (3 × 1)/(5 × 2)

= 3/10

Therefore, 3/10 of girls studying in lower classes.

5. Maddy reads three-fifth of 75 pages of his lesson. How many more pages he need to complete the lesson? Solution: Maddy reads = 3/5 of 75 = 3/5 × 75

= 45 pages. Maddy has to read = 75 – 45. = 30 pages. Therefore, Maddy has to read 30 more pages. 6. A herd of cows gives 4 litres of milk each day. But each cow gives one-third of total milk each day. They give 24 litres milk in six days. How many cows are there in the herd? Solution: A herd of cows gives 4 litres of milk each day. Each cow gives one-third of total milk each day = 1/3 of 4 Therefore, each cow gives 4/3 of milk each day. Total no. of cows = 4 ÷ 4/3 = 4 × ¾ = 3 Therefore there are 3 cows in the herd.

Questions and Answers on Word problems on Fractions:

1. Shelly walked \(\frac{1}{3}\) km. Kelly walked \(\frac{4}{15}\) km. Who walked farther? How much farther did one walk than the other?

2. A frog took three jumps. The first jump was \(\frac{2}{3}\) m long, the second was \(\frac{5}{6}\) m long and the third was \(\frac{1}{3}\) m long. How far did the frog jump in all?

3. A vessel contains 1\(\frac{1}{2}\) l of milk. John drinks \(\frac{1}{4}\) l of milk; Joe drinks \(\frac{1}{2}\) l of milk. How much of milk is left in the vessel?

● Multiplication is Repeated Addition.

● Multiplication of Fractional Number by a Whole Number.

● Multiplication of a Fraction by Fraction.

● Properties of Multiplication of Fractional Numbers.

● Multiplicative Inverse.

● Worksheet on Multiplication on Fraction.

● Division of a Fraction by a Whole Number.

● Division of a Fractional Number.

● Division of a Whole Number by a Fraction.

● Properties of Fractional Division.

● Worksheet on Division of Fractions.

● Simplification of Fractions.

● Worksheet on Simplification of Fractions.

● Word Problems on Fraction.

● Worksheet on Word Problems on Fractions.

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5th Grade Math Problems

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## Exploring Fractions

- The first group gives you some starting points to explore with your class, which are applicable to a wide range of ages. The tasks in this first group will build on children's current understanding of fractions and will help them get to grips with the concept of the part-whole relationship.
- The second group of tasks focuses on the progression of ideas associated with fractions, through a problem-solving lens. So, the tasks in this second group are curriculum-linked but crucially also offer opportunities for learners to develop their problem-solving and reasoning skills.

- are applicable to a range of ages;
- provide contexts in which to explore the part-whole relationship in depth;
- offer opportunities to develop conceptual understanding through talk.

## Multiplying Fractions Questions

Practising multiplying fractions questions with solutions is essential to improve children’s skills in fractions. Multiplication of fractions is quite simple; to multiply two fractions, we just have to multiply numerator to numerator and denominator to the denominator. Unlike addition and subtraction of fractions, while multiplying fractions, we do not have to make the denominator the same.

Learn more about fractions .

## Video Lesson on Like and Unlike Fractions

## Multiplying Fractions Questions with Solutions

Now let us solve questions on the multiplication of fractions.

Question 1:

Solve the following:

(ii) 9/7 × ⅜

(iii) ⅘ × ⅚

= (2 × 3)/(3 × 5)

= 6/15 reducing the fraction to the lowest form

= (9 × 3)/(7 × 8)

\(\begin{array}{l}=\frac{4}{\not{5}}\times \frac{\not{{5}}}{6}\end{array} \)

= 4/6 reducing the fraction to the lowest form

Question 2:

Simplify the following:

(i) 2½ × 3⅓

(ii) 3¼ × 5 2 / 9

(iii) 4 1 / 7 × 3 1 / 8

Converting the mixed fraction into the improper fraction

= 5/2 × 10/3

= 25/3 = 8⅓.

(ii) 3 1 / 4 × 5 2 / 9

= 13/4 × 47/9

= (13 × 47)/(4 × 9)

\(\begin{array}{l}=16\frac{35}{36}\end{array} \)

= 29/7 × 25/8

= (29 × 25)/(7 × 8)

\(\begin{array}{l}=12\frac{53}{56}\end{array} \)

Also, Refer:

- Mixed to Improper Fractions
- Mixed to Improper Fraction Calculator
- Improper to Mixed Fraction Calculator

Question 3:

Work out the following and express them in the simplest form:

(i) 22/3 ÷ 11/5

(ii) 34/35 ÷ 6/7

(iii) 56/3 ÷ 9/17

Convert the division into multiplication by taking the reciprocal of the divisor fraction

= 22/3 × 5/11

= 34/35 × 7/6

\(\begin{array}{l}=1\frac{2}{15}\end{array} \)

= 56/3 × 17/9

\(\begin{array}{l}=35\frac{7}{27}\end{array} \)

Question 4:

(i) 26 × 1/13 ÷ 5/169

(ii) 6.4 × ⅘ ÷ ⅔

(iii) 2.98 ÷ ¾ × ⅖

According to the BODMAS rule, we first perform division followed by multiplication.

= 26 × (1/13 ÷ 5/169)

= 26 × (1/13 × 169/5)

= 26 × 13/5

\(\begin{array}{l}=67\frac{3}{5}\end{array} \)

= 64/10 × (⅘ ÷ ⅔)

= 64/10 × (⅘ × 3/2)

= 64/10 × 6/5

\(\begin{array}{l}=7\frac{17}{25}\end{array} \)

= (298/100 ÷ 3/4) × ⅖

= (298/100 × 4/3) × ⅖

= 298/(25 × 3) × ⅖

= 298/75 × ⅖

Question 5:

The length and the width of a rectangular park are 39/4 m and 25/3 m, respectively. Find the area of the park.

Length of the park = 39/4

Width of the park = 25/3

Area of the rectangular park = 39/4 × 25/3

= (13 × 25)/4 = 325/4 = 81.25 m 2 .

Question 6:

A household has an overhead water tank of 1000 litres. Every day 4/5th of the tank is used for household purposes. Find the amount of water needed for a week.

The capacity of the tank = 1000 litres

Amount of water used everyday = 4/5th of 1000 litres

= ⅘ × 1000 = 4 × 200 = 800 litres

Amount of water required for a week = 7 × 800 = 5600 litres.

Question 7:

A farmer plants the sapling of a plant at a uniform distance of 5/3 cm. If he plants 27 such saplings in a row, find the total distance between the first and the last sapling.

Distance between each sapling = 5/3 cm

Number of saplings in a row = 27

Distance between first and the last sapling = 5/3 × 27 = 5 × 9 = 45 cm.

Question 8:

The area of the triangle is 145/3 cm 2 . If the base length of the triangle is ⅔ cm, find the height of the triangle.

The base of the triangle = ⅔ cm

Area of the triangle = ½ × base × height = 145/3 cm 2

⇒ ½ × ⅔ × height = 145/3

⇒ height = 145/3 ÷ ⅓

⇒ height = 145/3 × 3 = 145 cm

Question 9:

Find the following:

(i) 2/5th of a day.

(ii) ¼th of a kilometre.

(iii) 3/4th of a year.

(i) 2/5th of a day

Now, 1 day = 24 hours

⅖ × 24 hours = 48/5

= 9.6 hours

= 9 hours 36 minutes.

(ii) ¼th of a kilometre

1 km = 1000 m

1 year = 12 months

3/4 × 12 = 9 months.

Question 10:

In a class of 60 students, two-thirds are boys. How many girls are there in the class?

Number of students = 60

Number of boys = 60 × ⅔ = 40

Number of girls = 60 – 40 = 20.

## Practice Questions on Multiplying Fractions

1. Evaluate the following:

(ii) 4/7 × 9/21

(iii) 45/7 × 3½

(iv) 19 6 / 7 × 13½

(vi) 34/5 ÷ 6/7

(vii) 2 ÷ ⅘

(viii) ⅞ ÷ 13

(ix) ⅚ ÷ 9 6 / 5

(x) 12 × ¾ ÷ 1/2

2. Find the base of the triangle whose height is 4⅗ cm and the area is 28/35 cm 2 .

3. Three times more flour is needed to make a large cake than a small one. If 3⅕ kg flour is needed to make a large size cake, how much flour is needed to make a small size cake?

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## Arithmetic (all content)

Course: arithmetic (all content) > unit 5.

- Equivalent fractions
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## Simplify fractions

- Your answer should be
- a simplified proper fraction, like 3 / 5
- a simplified improper fraction, like 7 / 4

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## 24 Fraction Questions And Answers To Test Fractions Knowledge And Skills From KS2 to GCSE

Beki Christian

In this blog, we take a look at the sorts of skills pupils need to tackle fraction questions and fraction word problems, and then provide examples of the different sorts of fraction questions children are likely to encounter in KS2, KS3, and KS4.

## Fractions key skills

Fraction questions, lower ks2 fraction questions, upper ks2 fraction questions, manipulating fractions questions, finding a fraction of an amount questions, calculating with fractions and mixed numbers questions, fractions, decimals and percentages questions, gcse fraction questions (foundation), gcse fraction questions (higher).

Some of the key fractions skills that pupils will learn are:

- How to find a fraction of an amount
- How to add fractions and how to subtract fractions
- Fraction, decimal, and percentage conversions

Below we will touch on each of these skills and provide links to more detail and examples.

Fractions Intervention Pack

Try these lessons to help your pupils develop their knowledge and understanding of fractions.

## How to find a fraction of an amount

To find a fraction of a value, we need to know how to multiply fractions and how to divide fractions . The process involves dividing the value by the denominator and then multiplying that answer by the numerator.

For example, find \cfrac{2}{5} \, of 30

First, we find \cfrac{1}{5} \, of 30, by dividing 30 by 5.

\cfrac{1}{5} \, of 30 = 30 \div 5 = 6

Next, we find \cfrac{2}{5} \, of 30 by multiplying the asnwer by 2.

\cfrac{2}{5} \, of 30 = 6 \times 2 = 12

## How to add or subtract fractions with different denominators

To add and subtract fractions with different denominators, we need to rewrite the fractions with a common denominator. To do this, we look for the lowest common multiple (lcm) of the denominators.

Let’s look at an example: \cfrac{2}{5} + \cfrac{1}{4}

The lcm of 5 and 4 is 20, so we rewrite the fractions, using our knowledge of equivalent fractions , with a denominator of 20.

We can rewrite the first fraction as \cfrac{8}{20} \, and the second fraction as \cfrac{5}{20} \, .

Therefore \cfrac{2}{5} + \cfrac{1}{4}=\cfrac{8}{20}+\cfrac{5}{20}

Once we have the two fractions written with the same denominator, we can add the numerators. The denominator remains the same.

\cfrac{8}{20}+\cfrac{5}{20}=\cfrac{13}{20}

## How to compare and convert fractions, decimals and percentages

Fractions are one way of measuring parts of a whole. Percentages and decimals are other ways of measuring parts of a whole. Pupils will first learn about key equivalences between the most common fractions, decimals and percentages, such as halves, quarters, and tenths. Later, in KS3, they will then learn methods for converting between any fractions, decimals, and percentages.

Here are some helpful visuals for the most common fractions, decimals, and percentages:

There are a variety of fraction questions that might be asked in KS2, KS3, and KS4. Here we focus on fraction word problems and problem-solving questions which often provide the greatest challenge to pupils at primary school and secondary school.

At KS2, using real-world problems together with concrete resources or maths manipulatives is one of the best ways to help pupils visualise and understand what they are being asked to do.

In KS3 and KS4, word problems and problem solving questions can encourage students to think more deeply about about the processes and steps involved in a question.

## Fractions in KS2

At the beginning of KS2, pupils will have an understanding of basic fractions, such as \cfrac{1}{2}, \cfrac{1}{4}, and \cfrac{3}{4}.

They will be able to write a fraction and find a fraction of a shape or a quantity. Over the course of KS2, they will spend a significant amount of time developing their knowledge of fractions. By the end of KS2, pupils will have covered:

- Finding a fraction of a quantity
- Equivalent fractions
- Ordering and comparing fractions
- Adding and subtracting fractions with the same denominator
- Adding and subtracting fractions with different denominators
- Multiplying fractions and dividing fractions by whole numbers and fractions
- Improper fractions and mixed numbers (sometimes called mixed fractions)
- Equivalences between fractions, decimal and percentages

Find out more: KS2 fractions : a teacher’s guide

- What Is A Unit Fraction: Explained For Primary School
- What Is An Improper Fraction: Explained For Primary School
- Fractions for Kids: A Comprehensive Guide
- How To Teach Fractions Key Stage 2: Maths Bootcamp
- Fraction games for KS1 and KS2

1. Natalie had 20 sweets. She ate \cfrac{1}{4} \, of them.

How many sweets did Natalie eat?

\cfrac{1}{4} \, of 20 = 20 \div 4 = 5

2. Osian had some money. He gave \cfrac{1}{5} of the money to Ethan. He gave \cfrac{2}{5} of the money to Ffion. What fraction of the money did he give away in total?

He gave away \cfrac{1}{5}+\cfrac{2}{5}=\cfrac{3}{5}

3. Choose the correct fraction to go in the box:

4. There are 32 pupils in a class. \cfrac{3}{8} \, of the pupils are girls.

How many of the pupils are boys?

\cfrac{1}{8} \, of 32 = 32 \div 8 = 4

\cfrac{3}{8} \, of 32 = 4 \times 3=12, so there are 12 girls.

32-12=20, so there are 20 boys.

5. Which fraction is the odd one out?

\cfrac{12}{13} \, is not equivalent to \cfrac{2}{3} \, so it is the odd one out.

6. Ben and Jacob both received the same amount of pocket money.

Ben spent \cfrac{3}{4} \, of his pocket money. Jacob spent \cfrac{13}{20} \, of his pocket money.

Choose the correct symbol to make this sentence correct.

Therefore \cfrac{3}{4} > \cfrac{13}{20}

7. In a jug, there is \cfrac{2}{3} \, litre of juice.

Willow pours \cfrac{1}{5} \, litre of juice from the jug. What fraction of a litre is left in the jug?

\cfrac{2}{3}-\cfrac{1}{5} = \cfrac{10}{15}-\cfrac{3}{15} =\cfrac{7}{15}

8. Tim walked \cfrac{3}{7} \, of a mile each day, for 5 days.

How far did Tim walk in total?

5 \times \cfrac{3}{7} = \cfrac{15}{7} = 2 \cfrac{1}{7}

9. April got 12 questions wrong on a test.

This was \cfrac{2}{5} \, of the questions.

How many questions were on the test?

\begin{aligned} \cfrac{2}{5}&=12\\\\ \cfrac{1}{5}&=12 \div 2 = 6\\\\ \cfrac{5}{5}&=6 \times 5 = 30 \end{aligned}

## Looking for more KS2 fraction questions?

These fraction worksheets provide lots more fraction questions, covering both basic elements and more complex problems. Answers are also included:

- Year 1 Fractions Independent Recap Worksheets
- Year 2 Unit Fractions Worksheet
- Year 3 Equivalent Fractions Worksheet
- Year 4 Subtract Fractions Worksheet
- Year 5 Fractions of Amounts Worksheet
- Year 6 Ordering Fractions, Decimals and Percentages Worksheet

## Fractions in KS3

In KS3, pupils develop their confidence in working with fractions. They practise all of the skills learnt at KS2 and learn to apply these to a variety of problems. Fractions will be learnt about as a topic in their own right, but will also be increasingly embedded into other areas of maths, such as algebra and geometry, as pupils progress through KS3.

10. The ratio of men:women working in a company is 3:5. What fraction of the workers are men?

For every 3 men, there are 5 women. Therefore in a total of 8 people, there are 3 men. The fraction of the workers who are men is \cfrac{3}{8} \, .

11. Ellie says that \cfrac{14}{6}=2\cfrac{1}{3} \, .

Is Ellie correct?

Explain how you know.

We can’t tell

2 \cfrac{1}{3} \,= \cfrac{2 \times 3 +1}{3} \, = \cfrac{7}{3}

\cfrac{7}{3} = \cfrac{14}{6}

12. 2400 people attended a concert. \cfrac{3}{8} \, of the people were men.

\cfrac{5}{12} \, of the people were women. The rest of the people were children. How many children were at the concert?

\cfrac{3}{8} \, of 2400

2400 \div 8 =300

300 \times 3=900

\cfrac{5}{12} \, of 2400

2400 \div 12 =200

200 \times 5=1000

2400-900-1000=500

13. In January the value of a house was £280000.

By August, the value of the house had decreased by \cfrac{1}{10} \, .

Find the value of the house in August.

\cfrac{1}{10} \, of 280000

280000 \div 10=28000

280000-28000=252000

14. Richard wants to calculate \cfrac{5}{8} \div \cfrac{4}{5} \, . Select the correct method.

15. Here is a signpost.

What is the distance from Tresaith to Aberporth?

\begin{aligned} 1\cfrac{1}{4} + 3\cfrac{2}{3} &= \cfrac{5}{4}+\cfrac{11}{3}\\\\ &= \cfrac{15}{12} +\frac{44}{12}\\\\ &=\cfrac{59}{12}\\\\ &=4\cfrac{11}{12} \end{aligned}

16. Frank scored 32 out of 40 in a test. What percentage of the questions did Frank answer correctly?

\cfrac{32}{40}=\cfrac{8}{10}=80\%

17. Write 2.35 as an improper fraction.

Give your answer in its simplest form.

2.35= \cfrac{235}{100}

Simplifying \cfrac{235}{100} gives us \cfrac{47}{20}

Ensure you follow the instructions carefully and give your answer as an improper fraction in its simplest form.

## Fractions in KS4 and for GCSE

In KS4, students continue to develop their fluency in using fractions to solve problems and will be required to apply their knowledge of fractions within many different maths topics. GCSE maths papers, from all exam boards (including AQA, Edexcel and OCR) will, almost certainly, contain questions involving fractions.

These may appear as standard procedural questions, such as calculate 2\cfrac{1}{4} \times 3\cfrac{2}{3} \,. It is also likely that there will be questions from other areas of maths, such as algebra, geometry, and probability, which involve fractions.

Take a look at our GCSE maths guides on fractions :

- Adding and subtracting fractions
- Dividing fractions
- Fractions of amounts
- Fractions, decimals, and percentages
- Improper fraction to mixed number

18. Write these fractions in order of size, starting with the smallest.

\cfrac{2}{5} \hspace{.5cm} \cfrac{3}{8} \hspace{.5cm} \cfrac{3}{10} \hspace{.5cm} \cfrac{9}{20}

\cfrac{2}{5} \, =\cfrac{16}{40}

\cfrac{3}{8} \, =\cfrac{15}{40}

\cfrac{3}{10} \, =\cfrac{12}{40}

\cfrac{9}{20} \, =\cfrac{18}{40}

\cfrac{12}{40}, \, \cfrac{15}{40} , \, \cfrac{16}{40}, \, \cfrac{18}{40}

\cfrac{3}{10}, \, \cfrac{3}{8} , \, \cfrac{2}{5}, \, \cfrac{9}{20}

19. Work out the perimeter of this rectangle. Give your answer as a mixed number in its simplest form.

\begin{aligned} 1 \cfrac{1}{3} \, + 1 \cfrac{1}{3} \, + \cfrac{3}{8} \, + \cfrac{3}{8} &= \cfrac{4}{3} + \cfrac{4}{3} \, + \cfrac{3}{8} \, +\cfrac{3}{8}\\\\ &=\cfrac{32}{24} \, + \cfrac{32}{24} \, + \cfrac{9}{24} \, +\cfrac{9}{24}\\\\ &=\cfrac{82}{24}\\\\ &= \cfrac{41}{12}\\\\ &=3 \cfrac{5}{12} \end{aligned}

20. Here are the first three terms of a sequence:

\cfrac{2}{3}, \, \cfrac{4}{9}, \, \cfrac{8}{27}

The rule to find the next term in the sequence is multiply by \cfrac{2}{3} \, .

Find the 5th term in the sequence.

\cfrac{8}{27} \, \times \cfrac{2}{3} = \cfrac{16}{81}

\cfrac{16}{81} \, \times \cfrac{2}{3} = \cfrac{32}{243}

21. On a farm, the ratio of white sheep:black sheep is 3 \, \text{:} \, 1. \cfrac{2}{3} \, of the white sheep are ewes. \cfrac{2}{5} \, of the black sheep are ewes. There are 144 ewes on the farm. How many sheep are there on the farm in total?

\cfrac{3}{4} \, of the sheep are white. \cfrac{2}{3} \, \times \cfrac{3}{4} \, = \cfrac{6}{12} \, of the sheep are white ewes.

\cfrac{1}{4} \, of the sheep are black. \cfrac{2}{5} \, \times \cfrac{1}{4} \, = \cfrac{2}{20} \, of the sheep are black ewes.

\cfrac{6}{12} \, +\cfrac{2}{20} \, =\cfrac{30}{60} \, +\cfrac{6}{60} \, =\cfrac{36}{60}

\cfrac{36}{60} \, of the sheep are ewes.

\cfrac{36}{60}=144

\cfrac{1}{60} \, =144 \div 36=4

\cfrac{60}{60} \, = 4 \times 60=240

22. Write \cfrac{3}{x-2}-\cfrac{5}{x+4} as a single fraction, in its simplest form.

\begin{aligned} \cfrac{3}{x-2}\,-\cfrac{5}{x+4} \, &=\cfrac{3(x+4)}{(x+4)(x-2)}\,-\cfrac{5(x-2)}{(x+4)(x-2)}\\\\ &=\cfrac{3x+12-(5x-10)}{(x+4)(x-2)}\\\\ &=\cfrac{-2x+2}{(x+4)(x-2)} \end{aligned}

23. Find the value of \cfrac{ab^{2}}{c} when a=-2, \, b=\cfrac{3}{8} \, and c=\cfrac{1}{4}.

\begin{aligned} \cfrac{ab^{2}}{c} \, &=\cfrac{-2 \times (\cfrac{3}{8})^{2}}{\cfrac{1}{4}}\\\\ &=\cfrac{-2 \times \cfrac{9}{64}}{\cfrac{1}{4}}\\\\ &=\cfrac{-\cfrac{18}{64}}{\cfrac{1}{4}}\\\\ &=-\cfrac{18}{64} \, \times \cfrac{4}{1}\\\\ &=-\cfrac{72}{64}\\\\ &=-\cfrac{9}{8} \end{aligned}

24. Express 0.2\dot{8}\dot{1} as a fraction.

\begin{aligned} x&=0.281818181…\\\\ 10x&=2.81818181…\\\\ 1000x&=281.818181…\\\\ 990x&=279\\\\ x&=\cfrac{279}{990}\\\\ x&=\cfrac{31}{110} \end{aligned}

## Looking for more KS3 and KS4 fraction questions?

These GCSE maths worksheets provide lots more fraction questions with answers included:

- Subtracting Fractions Worksheet
- Fractions Decimals and Percentages Worksheet
- Equivalent Fractions Worksheet
- Fractions of Amounts Worksheet
- Mixed Numbers to Improper Fractions Worksheet

## More GCSE fraction support

Third Space Learning’s free GCSE maths resource library contains detailed lessons with step-by-step instructions on how to solve fraction problems at secondary, as well as fraction worksheets with practice questions and more GCSE exam questions.

## Fraction questions FAQs

A fraction of an amount at GCSE is when we are asked to find a certain fraction of an amount, for example \cfrac{3}{5} \, of 20. To do this, we divide by the denominator and then multiply by the numerator. 20\div 5 = 4 4 \times 3 =12 So \cfrac{3}{5} of 20=12.

An example of a fraction is \cfrac{3}{10}. This is read as ‘three tenths’ and means thee out of every ten. You may see different types of fractions: Unit fractions have a numerator of 1, for example \cfrac{1}{8} \, . Proper fractions have a numerator that is smaller than the denominator, for example \cfrac{3}{4} \, . Improper fractions have a numerator that is greater than the denominator, for example \cfrac{5}{2} \, . Mixed number fractions (mixed fractions) are made up of a whole number and a fraction, for example 2\cfrac{1}{3} \, .

When pupils learn how to simplify fractions we look for common factors of the numerator and the denominator. If there are common factors, we divide both the numerator and the denominator by the common factor to find an equivalent, simpler fraction. We continue doing this until there are no more common factors. For example, simplify \cfrac{20}{40} \, . 10 is a common factor of 20 and 40. Dividing both 20 and 40 by 10 gives us \cfrac{2}{4} \, . 2 is a common factor of 2 and 4. Dividing both 2 and 4 by 2 gives us \cfrac{1}{2} \, . There are no more common factors of 1 and 2, so we cannot simplify any further.

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## Related articles

How To Multiply Fractions: Step By Step Guide For Primary School Teachers & Pupils

How To Add Fractions: Step By Step Guide For Primary School Teachers & Pupils

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How To Divide Fractions: Step By Step Guide For Primary School Teachers & Pupils

## FREE Percentages & Decimals All Kinds of Word Problems (Years 5 & 6)

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10 questions for each year group of ascending difficulty, and a challenge question to stretch pupils beyond the core understanding of percentages and decimals and on to mastery.

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## Are ‘Forever Chemicals’ a Forever Problem?

The environmental protection agency says “forever chemicals” must be removed from tap water. but they lurk in much more of what we eat, drink and use..

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 “The New York Times,” I’m Sabrina Tavernise. And this is “The Daily.”

[THEME MUSIC]

This month for the first time, the Environmental Protection Agency began to regulate a class of synthetic chemicals, known as forever chemicals, in America’s drinking water. But the chemicals, which have been linked to liver disease and other serious health problems, are in far more than just our water supply. Today, my colleague Kim Tingley explains.

It’s Wednesday, April 17.

So Kim, any time the EPA announces a regulation, I think we all sort of take notice because implicit in it is this idea that we have been exposed to something — something bad, potentially, lead or asbestos. And recently, the EPA is regulating a type of chemical known as PFAS So for those who don’t know, what are PFAS chemicals

Yeah, so PFAS stands for per and polyfluoroalkyl substances. They’re often called forever chemicals just because they persist so long in the environment and they don’t easily break down. And for that reason, we also use them in a ton of consumer products. They’re in makeup. They’re in carpet. They’re in nonstick cookware. They’re in food packaging, all sorts of things.

Yeah, I feel like I’ve been hearing about these chemicals actually for a very long time. I mean, nonstick pans, Teflon — that’s the thing that’s in my mind when I think PFAS.

Absolutely. Yeah, this class of chemicals has been around for decades. And what’s really important about this is that the EPA has decided, for the first time, to regulate them in drinking water. And that’s a ruling that stands to affect tens of millions of people.

So, help me understand where these things came from and how it’s taken so long to get to the point where we’re actually regulating them.

So, they really actually came about a long time ago. In 1938, DuPont, the people who eventually got us to Teflon, they were actually looking for a more stable kind of refrigerant. And they came upon this kind of chemical, PFAS. The thing that all PFAS chemicals have is a really strong bond between carbon atoms and fluorine atoms. This particular pairing is super strong and super durable.

They have water repellent properties. They’re stain resistant. They’re grease resistant. And they found a lot of uses for them initially in World War II. They were using them as part of their uranium enrichment process to do all these kinds of things. And then —

Well, good thing it’s Teflon.

In the 1950s is when they really started to come out as commercial products.

Even burned food won’t stick to Teflon. So it’s always easy to clean.

So, DuPont started using it in Teflon pans.

Cookware never needs scouring if it has DuPont Teflon.

And then another company, 3M also started using a kind of PFAS —

Scotchgard fabric protector. It keeps ordinary spills from becoming extraordinary stains.

— in one of their big products, Scotchgard. So you probably remember spraying that on your shoes if you want to make your shoes waterproof.

Use Scotchgard fabric protector and let your cup runneth over.

Right — miracle product, Scotchgard, Teflon. But of course, we’re talking about these chemicals because they’ve been found to pose health threats. When does that risk start to surface?

Yeah, so it’s pretty early on that DuPont and 3M start finding effects in animals in studies that they’re running in house.

Around the mid ‘60s, they start seeing that PFAS has an effect on rats. It’s increasing the liver and kidney weights of the rats. And so that seems problematic. And they keep running tests over the next decade and a half. And they try different things with different animals.

In one study, they gave monkeys really, really high levels of PFAS. And those monkeys died. And so they have a pretty strong sense that these chemicals could be dangerous. And then in 1979, they start to see that the workers that are in the plants manufacturing, working with these chemicals, that they’re starting to have higher rates of abnormal liver function. And in a Teflon plant, they had some pregnant workers that were working with these chemicals. And one of those workers in 1981 gave birth to a child who had some pretty severe birth defects.

And then by the mid 1980s, DuPont figures out that it’s not just their workers who are being exposed to these chemicals, but communities that are living in areas surrounding their Teflon plant, particularly the one in Parkersburg, West Virginia, that those communities have PFAS in their tap water.

Wow, so based on its own studies, DuPont knows its chemicals are making animals sick. They seem to be making workers sick. And now they found out that the chemicals have made their way into the water supply. What do they do with that information?

As far as we know, they didn’t do much. They certainly didn’t tell the residents of Parkersburg who were drinking that water that there was anything that they needed to be worried about.

How is that possible? I mean, setting aside the fact that DuPont is the one actually studying the health effects of its own chemicals, presumably to make sure they’re safe, we’ve seen these big, regulating agencies like the EPA and the FDA that exist in order to watch out for something exactly like this, a company that is producing something that may be harming Americans. Why weren’t they keeping a closer watch?

Yeah, so it goes kind of back to the way that we regulate chemicals in the US. It goes through an act called the Toxic Substances Control Act that’s administered by the EPA. And basically, it gives companies a lot of room to regulate themselves, in a sense. Under this act they have a responsibility to report to the EPA if they find these kinds of potential issues with a chemical. They have a responsibility to do their due diligence when they’re putting a chemical out into the environment.

But there’s really not a ton of oversight. The enforcement mechanism is that the EPA can find them. But this kind of thing can happen pretty easily where DuPont keeps going with something that they think might really be a problem and then the fine, by the time it plays out, is just a tiny fraction of what DuPont has earned from producing these chemicals. And so really, the incentive is for them to take the punishment at the end, rather than pull it out early.

So it seems like it’s just self-reporting, which is basically self-regulation in a way.

Yeah, I think that is the way a lot of advocacy groups and experts have characterized it to me, is that chemical companies are essentially regulating themselves.

So how did this danger eventually come to light? I mean, if this is in some kind of DuPont vault, what happened?

Well, there’s a couple different things that started to happen in the late ‘90s.

The community around Parkersburg, West Virginia, people had reported seeing really strange symptoms in their animals. Cows were losing their hair. They had lesions. They were behaving strangely. Some of their calves were dying. And a lot of people in the community felt like they were having health problems that just didn’t really have a good answer, mysterious sicknesses, and some cases of cancers.

And so they initiate a class action lawsuit against DuPont. As part of that class action lawsuit, DuPont, at a certain point, is forced to turn over all of their internal documentation. And so what was in the files was all of that research that we mentioned all of the studies about — animals, and workers, the birth defects. It was really the first time that the public saw what DuPont and 3M had already seen, which is the potential health harms of these chemicals.

So that seems pretty damning. I mean, what happened to the company?

So, DuPont and 3M are still able to say these were just a few workers. And they were working with high levels of the chemicals, more than a person would get drinking it in the water. And so there’s still an opportunity for this to be kind of correlation, but not causation. There’s not really a way to use that data to prove for sure that it was PFAS that caused these health problems.

In other words, the company is arguing, look, yes, these two things exist at the same time. But it doesn’t mean that one caused the other.

Exactly. And so one of the things that this class action lawsuit demands in the settlement that they eventually reach with DuPont is they want DuPont to fund a formal independent health study of the communities that are affected by this PFAS in their drinking water. And so they want DuPont to pay to figure out for sure, using the best available science, how many of these health problems are potentially related to their chemicals.

And so they ask them to pay for it. And they get together an independent group of researchers to undertake this study. And it ends up being the first — and it still might be the biggest — epidemiological study of PFAS in a community. They’ve got about 69,000 participants in this study.

Wow, that’s big.

It’s big, yeah. And what they ended up deciding was that they could confidently say that there was what they ended up calling a probable link. And so they were really confident that the chemical exposure that the study participants had experienced was linked to high cholesterol, ulcerative colitis, thyroid disease, testicular cancer, kidney cancer, and pregnancy induced hypertension.

And so those were the conditions that they were able to say, with a good degree of certainty, were related to their chemical exposure. There were others that they just didn’t have the evidence to reach a strong conclusion.

So overall, pretty substantial health effects, and kind of vindicates the communities in West Virginia that were claiming that these chemicals were really affecting their health.

Absolutely. And as the years have gone on, that was sort of just the beginning of researchers starting to understand all the different kinds of health problems that these chemicals could potentially be causing. And so since the big DuPont class action study, there’s really just been like this building and building and building of different researchers coming out with these different pieces of evidence that have accumulated to a pretty alarming picture of what some of the potential health outcomes could be.

OK, so that really kind of brings us to the present moment, when, at last, it seems the EPA is saying enough is enough. We need to regulate these things.

Yeah, it seems like the EPA has been watching this preponderance of evidence accumulate. And they’re sort of deciding that it’s a real health problem, potentially, that they need to regulate.

So the EPA has identified six of these PFAS chemicals that it’s going to regulate. But the concern that I think a lot of experts have is that this particular regulation is not going to keep PFAS out of our bodies.

We’ll be right back.

So, Kim, you just said that these regulations probably won’t keep PFAS chemicals out of our bodies. What did you mean?

Well, the EPA is talking about regulating these six kinds of PFAS. But there are actually more than 10,000 different kinds of PFAS that are already being produced and out there in the environment.

And why those six, exactly? I mean, is it because those are the ones responsible for most of the harm?

Those are the ones that the EPA has seen enough evidence about that they are confident that they are probably causing harm. But it doesn’t mean that the other ones are not also doing something similar. It’s just sort of impossible for researchers to be able to test each individual chemical compound and try to link it to a health outcome.

I talked to a lot of researchers who were involved in this area and they said that they haven’t really seen a PFAS that doesn’t have a harm, but they just don’t have information on the vast majority of these compounds.

So in other words, we just haven’t studied the rest of them enough yet to even know how harmful they actually are, which is kind of alarming.

Yeah, that’s right. And there’s just new ones coming out all the time.

Right. OK, so of the six that the EPA is actually intending to regulate, though, are those new regulations strict enough to keep these chemicals out of our bodies?

So the regulations for those six chemicals really only cover getting them out of the drinking water. And drinking water only really accounts for about 20 percent of a person’s overall PFAS exposure.

So only a fifth of the total exposure.

Yeah. There are lots of other ways that you can come into contact with PFAS. We eat PFAS, we inhale PFAS. We rub it on our skin. It’s in so many different products. And sometimes those products are not ones that you would necessarily think of. They’re in carpets. They’re in furniture. They’re in dental floss, raincoats, vinyl flooring, artificial turf. All kinds of products that you want to be either waterproof or stain resistant or both have these chemicals in them.

So, the cities and towns are going to have to figure out how to test for and monitor for these six kinds of PFAS. And then they’re also going to have to figure out how to filter them out of the water supply. I think a lot of people are concerned that this is going to be just a really expensive endeavor, and it’s also not really going to take care of the entire problem.

Right. And if you step back and really look at the bigger problem, the companies are still making these things, right? I mean, we’re running around trying to regulate this stuff at the end stage. But these things are still being dumped into the environment.

Yeah. I think it’s a huge criticism of our regulatory policy. There’s a lot of onus put on the EPA to prove that a harm has happened once the chemicals are already out there and then to regulate the chemicals. And I think that there’s a criticism that we should do things the other way around, so tougher regulations on the front end before it goes out into the environment.

And that’s what the European Union has been doing. The European Chemicals Agency puts more of the burden on companies to prove that their products and their chemicals are safe. And the European Chemicals Agency is also, right now, considering just a ban on all PFAS products.

So is that a kind of model, perhaps, of what a tough regulation could look like in the US?

There’s two sides to that question. And the first side is that a lot of people feel like it would be better if these chemical companies had to meet a higher standard of proof in terms of demonstrating that their products or their chemicals are going to be safe once they’ve been put out in the environment.

The other side is that doing that kind of upfront research can be really expensive and could potentially limit companies who are trying to innovate in that space. In terms of PFAS, specifically, this is a really important chemical for us. And a lot of the things that we use it in, there’s not necessarily a great placement at the ready that we can just swap in. And so it’s used in all sorts of really important medical devices or renewable energy industries or firefighting foam.

And in some cases, there are alternatives that might be safer that companies can use. But in other cases, they just don’t have that yet. And so PFAS is still really important to our daily lives.

Right. And that kind of leaves us in a pickle because we know these things might be harming us. Yet, we’re kind of stuck with them, at least for now. So, let me just ask you this question, Kim, which I’ve been wanting to ask you since the beginning of this episode, which is, if you’re a person who is concerned about your exposure to PFAS, what do you do?

Yeah. So this is really tricky and I asked everybody this question who I talked to. And everybody has a little bit of a different answer based on their circumstance. For me what I ended up doing was getting rid of the things that I could sort of spot and get rid of. And so I got rid of some carpeting and I checked, when I was buying my son a raincoat, that it was made by a company that didn’t use PFAS.

It’s also expensive. And so if you can afford to get a raincoat from a place that doesn’t manufacture PFAS, it’s going to cost more than if you buy the budget raincoat. And so it’s kind of unfair to put the onus on consumers in that way. And it’s also just not necessarily clear where exactly your exposure is coming from.

So I talk to people who said, well, it’s in dust, so I vacuum a lot. Or it’s in my cleaning products, so I use natural cleaning products. And so I think it’s really sort of a scattershot approach that consumers can take. But I don’t think that there is a magic approach that gets you a PFAS-free life.

So Kim, this is pretty dark, I have to say. And I think what’s frustrating is that it feels like we have these government agencies that are supposed to be protecting our health. But when you drill down here, the guidance is really more like you’re on your own. I mean, it’s hard not to just throw up your hands and say, I give up.

Yeah. I think it’s really tricky to try to know what you do with all of this information as an individual. As much as you can, you can try to limit your individual exposure. But it seems to me as though it’s at a regulatory level that meaningful change would happen, and not so much throwing out your pots and pans and getting new ones.

One thing about PFAS is just that we’re in this stage still of trying to understand exactly what it’s doing inside of us. And so there’s a certain amount of research that has to happen in order to both convince people that there’s a real problem that needs to be solved, and clean up what we’ve put out there. And so I think that we’re sort of in the middle of that arc. And I think that that’s the point at which people start looking for solutions.

Kim, thank you.

Here’s what else you should know today. On Tuesday, in day two of jury selection for the historic hush money case against Donald Trump, lawyers succeeded in selecting 7 jurors out of the 12 that are required for the criminal trial after failing to pick a single juror on Monday.

Lawyers for Trump repeatedly sought to remove potential jurors whom they argued were biased against the president. Among the reasons they cited were social media posts expressing negative views of the former President and, in one case, a video posted by a potential juror of New Yorkers celebrating Trump’s loss in the 2020 election. Once a full jury is seated, which could come as early as Friday, the criminal trial is expected to last about six weeks.

Today’s episode was produced by Clare Toeniskoetter, Shannon Lin, Summer Thomad, Stella Tan, and Jessica Cheung, with help from Sydney Harper. It was edited by Devon Taylor, fact checked by Susan Lee, contains original music by Dan Powell, Elisheba Ittoop, and Marion Lozano, and was engineered by Chris Wood.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for The Daily. I’m Sabrina Tavernise. See you tomorrow.

- April 18, 2024 • 30:07 The Opening Days of Trump’s First Criminal Trial
- April 17, 2024 • 24:52 Are ‘Forever Chemicals’ a Forever Problem?
- April 16, 2024 • 29:29 A.I.’s Original Sin
- April 15, 2024 • 24:07 Iran’s Unprecedented Attack on Israel
- April 14, 2024 • 46:17 The Sunday Read: ‘What I Saw Working at The National Enquirer During Donald Trump’s Rise’
- April 12, 2024 • 34:23 How One Family Lost $900,000 in a Timeshare Scam
- April 11, 2024 • 28:39 The Staggering Success of Trump’s Trial Delay Tactics
- April 10, 2024 • 22:49 Trump’s Abortion Dilemma
- April 9, 2024 • 30:48 How Tesla Planted the Seeds for Its Own Potential Downfall
- April 8, 2024 • 30:28 The Eclipse Chaser
- April 7, 2024 The Sunday Read: ‘What Deathbed Visions Teach Us About Living’
- April 5, 2024 • 29:11 An Engineering Experiment to Cool the Earth

Hosted by Sabrina Tavernise

Featuring Kim Tingley

Produced by Clare Toeniskoetter , Shannon M. Lin , Summer Thomad , Stella Tan and Jessica Cheung

With Sydney Harper

Edited by Devon Taylor

Original music by Dan Powell , Elisheba Ittoop and Marion Lozano

Engineered by Chris Wood

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

The Environmental Protection Agency has begun for the first time to regulate a class of synthetic chemicals known as “forever chemicals” in America’s drinking water.

Kim Tingley, a contributing writer for The New York Times Magazine, explains how these chemicals, which have been linked to liver disease and other serious health problems, came to be in the water supply — and in many more places.

## On today’s episode

Kim Tingley , a contributing writer for The New York Times Magazine.

## Background reading

“Forever chemicals” are everywhere. What are they doing to us?

The E.P.A. issued its rule about “forever chemicals” last week.

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.

Fact-checking by Susan Lee .

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.

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Questions and problems with solutions on fractions are presented. Detailed solutions to the examples are also included. In order to master the concepts and skills of fractions, you need a thorough understanding (NOT memorizing) of the rules and properties and lot of practice and patience. I hope the examples, questions, problems in the links ...

Math explained in easy language, plus puzzles, games, quizzes, videos and worksheets. For K-12 kids, teachers and parents. Fraction Worksheets ... Fractions - Subtraction. Worksheet. Example. Fractions (Same Denominator) 15 − 25. Unit Fractions. 13 − 19. Easy Proper Fractions. 38 − 27. Harder Proper Fractions. 712 − 1525.

Word problems with fractions: involving a fraction and a whole number. Finally, we are going to look at an example of a word problem with a fraction and a whole number. Now we will have to convert all the information into a fraction with the same denominator (as we did in the example above) in order to calculate. This morning Miguel bought 1 ...

While dividing one fraction by another fraction, we multiply the first fraction by the reciprocal of the other. 5. Multiply the following fractions. (i) (⅖) × 5 ¼. (ii) 2 ⅗ × 3. Solution: (i) (⅖) × 5 ¼. Here, 5 ¼ is a mixed fraction. Let us convert this mixed fraction into an improper fraction.

Analysis: To solve this problem, we will add two mixed numbers, with the fractional parts having unlike denominators. Solution: Answer: The warehouse has 21 and one-half meters of tape in all. Example 8: An electrician has three and seven-sixteenths cm of wire. He needs only two and five-eighths cm of wire for a job.

Identify your areas for growth in these lessons: Dividing fractions by fractions. Dividing fractions word problems. Start quiz. Unit test. Test your understanding of Fractions with these NaN questions. Start test. In this topic, we will explore fractions conceptually and add, subtract, multiply, and divide fractions.

Presented here are the fraction pdf worksheets based on real-life scenarios. Read the basic fraction word problems, write the correct fraction and reduce your answer to the simplest form. Download the set. Represent and Simplify the Fractions: Type 2. Before representing in fraction, children should perform addition or subtraction to solve ...

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To add fractions, they must have the same denominator. If they do, simply add the numerators together. [2] For instance, to solve 5/9 + 1/9, just add 5 + 1, which equals 6. The answer, then, is 6/9 which can be reduced to 2/3. 2. Subtract fractions with the same denominator by subtracting the numerators.

Multiply fractions word problems. Priya spent 1 1 2 days hiking. She was lost 5 6 of the time that she was hiking. What fraction of a day was Priya lost? Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a ...

To solve a fraction word problem, you must understand the context of the word problem, what the unknown information is, and what operation is needed to solve it. ... Use this quiz to check your grade 4 to 6 students' understanding of fraction operations. 10+ questions with answers covering a range of 4th to 6th grade fraction operations ...

Fraction Word Problems - using block models (tape diagrams), Solve a problem involving fractions of fractions and fractions of remaining parts, how to solve a four step fraction word problem using tape diagrams, grade 5, grade 6, grade 7, with video lessons, examples and step-by-step solutions.

A mixed number or a mixed fraction is a type of fraction which is a combination of both a whole number and a proper fraction. We express improper fractions as mixed numbers. For example, 5\(\frac{1}{3}\), 1\(\frac{4}{9}\), 13\(\frac{7}{8}\) are mixed fractions. Unit fraction. A unit fraction is a fraction with a numerator equal to one.

In word problems on fraction we will solve different types of problems on multiplication of fractional numbers and division of fractional numbers. 1. 4/7 of a number is 84. Find the number. Solution: According to the problem, 4/7 of a number = 84. Number = 84 × 7/4.

Example 3: multiplying a mixed number by a fraction with the algorithm. Solve 1 \, \cfrac {11} {12} \times \cfrac {3} {4} \, . 1 1211 × 43. Convert whole numbers and mixed numbers to improper fractions. Convert the mixed number to an improper fraction.

2/3 and 6/9. 3/4 and 2/5. Open Question. Find 3 fractions smaller than 1/3. Find 3 fractions equal to 2/3. Find 3 fractions greater than 2/5. Effective tasks are ones that encourage thinking and analysis, enable students to build upon previous knowledge, and reveal misunderstandings.

Exploring Fractions. Introduction. At NRICH, our aim is to offer rich tasks which develop deep understanding of mathematical concepts. Of course, by their very nature, rich tasks will also provide opportunities for children to work like a mathematician and so help them develop their problem-solving skills alongside this conceptual understanding.

Multiplying fractions questions are given here with solutions for practice. Visit BYJU'S to learn multiplying fractions by solving questions with video lessons and many more study resources. ... Now let us solve questions on the multiplication of fractions. Question 1: Solve the following: (i) ⅔ × ⅗ (ii) 9/7 × ⅜ (iii) ⅘ × ⅚ ...

Questions begin with fluency skills and progress to questions involving reasoning and problem solving. Key Year 6 fractions terminology needed for these questions. ... 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 ...

Write 28 63 in simplest form. 5:51. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

Next: Fractions - Finding Original Practice Questions GCSE Revision Cards. 5-a-day Workbooks

Khan Academy's 100,000+ free practice questions give instant feedback, don't need to be graded, and don't require a printer. Math Worksheets. Khan Academy. Math worksheets take forever to hunt down across the internet. Khan Academy is your one-stop-shop for practice from arithmetic to calculus. Math worksheets can vary in quality from ...

Photomath is a math solver that uses AI and optical recognition technologies to analyze math problems. Just take a screenshot of the question, and the app will identify the math symbols and equations. From calculus to trigonometry, Photomath can accurately answer questions thanks to its powerful AI engine and algebra knowledgebase.

FAQs on Word Problems on Fractions. Here are some of most commonly asked questions on word problems on fractions. Q.1: How do you solve word problems with fractions? Ans: To solve word problems with fractions, first, read and write the given data. Write the mathematical form by given data and perform the operations on fractions according to the ...

369 posts. #3 Yesterday at 8:38 PM • 3 Y. I attended HCSSiM last summer and it was honestly one of the best experiences of my life. Mathematically speaking we did a TON of really cool stuff and a lot of the times we were able to "choose" which direction we wanted to go in, which conjectures we wanted to look at, and this was especially ...

In KS3 and KS4, word problems and problem solving questions can encourage students to think more deeply about about the processes and steps involved in a question. Fractions in KS2 At the beginning of KS2, pupils will have an understanding of basic fractions, such as \cfrac{1}{2}, \cfrac{1}{4}, and \cfrac{3}{4}.

Career Paths for People Who Don't Like Math. If you don't like math, chances are you're in the majority. According to the Harvard Business Review, as early as first and second grade, about half of students are "moderately nervous" or "very, very nervous" about math and even within four-year universities, nearly a quarter of students report moderate to high levels of math anxiety.

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It does take a couple seconds for the ai to process and answer the problems but that's because it is telling you step-by-step how to answer the problem and it's honestly teaching me how to the math properly not just giving me the answer but if you don't want to read through all the steps it takes about 15 to 30 seconds for the full answer ...

Luna, a mini cockapoo, has a unique talent for solving math problems with help from her owner. Up Next in living. Dog that never went on walks without pregnant mom now won't leave newborn baby behind. April 18, 2024. Dog that never went on walks without pregnant mom now won't leave newborn baby behind .