8.1 Fractions, Decimals and Percentages

Revision slides, flash cards, quizzes, revision notes and a text version of the slides for this unit. Open to everyone, no password needed.

Slide 1 of 8

8.1 revision slides as a PDF

Flash cards

Click a card to flip it. Say the answer first, then check.

Print the cards (PDF): word on the front, answer on the back, double-sided, flip on the long edge

Quizzes

Pick an answer to see if you are right. Your score shows at the end of each quiz.

Revision notes

Download and print the revision notes (PDF, A4)

The big idea. Equivalence and relationships between fractions, decimals and percentages: write quantities in different forms, convert between them, and calculate percentage increase and decrease.

Key vocabulary

Word	Meaning
Numerator	the number above the line in a fraction: how many parts you have
Denominator	the number below the line in a fraction: how many equal parts the whole is cut into
Equivalent fractions	fractions that name the same amount, e.g. 1/2 and 3/6
Highest common factor (HCF)	the biggest number that divides exactly into two or more numbers
Lowest terms	a fraction written so that the numerator and denominator share no common factor except 1
Improper fraction	a fraction whose numerator is at least as big as its denominator, e.g. 7/4
Mixed numeral	a whole number and a fraction together, e.g. 1 3/4
Common denominator	a denominator shared by two or more fractions, so they can be compared, added or subtracted
Reciprocal	a fraction turned upside down: the reciprocal of 4/5 is 5/4
Per cent	per hundred: 75% means 75/100
Terminating decimal	a decimal that ends, e.g. 1/8 = 0.125
Recurring decimal	a decimal with digits that repeat forever, e.g. 1/3 = 0.333...
Multiplier	the number you multiply by to increase or decrease an amount by a percentage, e.g. × 1.2 for a 20% increase

Forms of fractions

Equivalent: multiply or divide numerator and denominator by the same number: 1/2 = 3/6.
Simplify: divide the numerator and denominator by the HCF: 12/18 = 2/3.
Improper to mixed: 17 ÷ 5 = 3 remainder 2, so 17/5 = 3 2/5. Mixed to improper: 2 1/3 = (3 × 2 + 1)/3 = 7/3.
Compare: common denominator first. 3/4 = 9/12 > 2/3 = 8/12.
Why it works: you cut the pieces the same way everywhere, so the amount does not change.

Four operations with fractions

Add/subtract: common denominator, then add or subtract the numerators. 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Never add the denominators: 1/2 + 1/3 is not 2/5.
Multiply: numerator × numerator, denominator × denominator (cancel first). 2/3 × 3/4 = 6/12 = 1/2.
Divide: multiply by the reciprocal. 3 ÷ 1/4 = 3 × 4 = 12 (how many quarters fit in 3?).
Mixed numerals: change to improper first. 2 1/2 ÷ 5/6 = 5/2 × 6/5 = 3.

Fractions, decimals and percentages

Fraction to decimal: divide numerator by denominator (3/4 = 0.75). Decimal to percentage: × 100. Percentage to decimal: ÷ 100. Percentage to fraction: over 100, then simplify.
Per cent means per hundred: 3/4 = 75/100 = 0.75 = 75%.
Common ones: 1/2 = 0.5 = 50%; 1/4 = 0.25 = 25%; 1/5 = 0.2 = 20%; 1/10 = 0.1 = 10%; 1/8 = 0.125 = 12.5%.
Terminating decimals end (1/8 = 0.125); recurring decimals repeat (1/3 = 0.333...). A fraction in lowest terms terminates only if its denominator has no prime factors except 2 and 5.
To order mixed forms, change them all to percentages.

Percentage increase and decrease

Multiplier = 1 + the rise, or 1 − the fall, as a decimal. +20% is × 1.2; −15% is × 0.85.
Why 1.2 and not 0.2? The original is 100%, and 100% + 20% = 120%.
45 francs up 20%: 45 × 1.2 = 54. 300 francs down 15%: 300 × 0.85 = 255.
Repeated change: multiply the multipliers. Up 20% then down 20% = 1.2 × 0.8 = 0.96, which is 4% less than the start (the second 20% is of a bigger amount).
Check by estimating: a 20% rise on 45 is about 9 more.

Common mistakes

✕ “1/2 + 1/3 = 2/5.” ✓ Find a common denominator first: 3/6 + 2/6 = 5/6.
✕ “3 ÷ 1/4 = 3/4.” ✓ Dividing by 1/4 multiplies by 4: 3 × 4 = 12.
✕ “A 20% rise is × 0.2.” ✓ It is × 1.2, because the original stays (100% + 20%).
✕ “Up 20% then down 20% gets you back to the start.” ✓ It gives 96% of the start: 1.2 × 0.8 = 0.96.
✕ “Leave 12/18 as it is.” ✓ Simplify to lowest terms: 2/3.

Practice booklet

Unit 8.1 Practice Booklet

mathscafe.net
Lorna Brown © 2026

Ms B’s mathscafe

Practice Booklet

MYP 2 · Year 8 Mathematics

Unit 8.1 · Fractions, Decimals and Percentages

Name:

8.1.1: Forms of fractions · Questions 1–17
8.1.2: Four operations with fractions · Questions 18–34
8.1.3: Fractions, decimals and percentages · Questions 35–52
8.1.4: Percentage increase and decrease · Questions 53–69
Homework: Homework page and checking record
Answers: For checking and correcting in green pen

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice BookletMs B’s mathscafe · page 1

How to use this booklet and your homework

Read this first

Choosing your level

Every part of the unit has four levels. Start where you feel comfortable and move between them freely. Most questions ask you to explain as well as answer, because being able to say why is what makes a method stick.

Learning

Simple numbers, with a bar or a grid drawn beside you.

Practising

The standard question, with your reasoning written out.

Challenging

Working backwards, and finding what is wrong with someone else’s thinking.

Mastery

Explaining why a rule must be true, and testing it on new cases.

Working on your homework

Pens and setting out

Answers in blue or black non-erasable pen, or in pencil.
Colour and highlighters are for annotating your work and showing the steps of a method.
Never erase a mistake. Rework it underneath so you can see what changed.

Green pen is for checking. When you finish, use the answers at the back to check your work.

Got it wrong? Circle the question number, then make your correction and write your error analysis in green pen.
Still can’t work out how to solve it? Put a star ★ next to the question number, so we know to go over it in class.

Each sheet

Your name goes on every sheet, because they are handed in one at a time.
Each question is worth up to 3 marks. If a question has parts (a, b, c…), each part is marked out of 3. At the end of each level, write your score out of the total for that level.
Questions are numbered 1, 2, 3… all the way through the booklet, so we can say exactly which ones to do.
If you are stuck after trying, ask for help. Showing me what you tried is a good start.

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 2

8.1.1 Forms of fractions

Learning

Name:

Key vocabulary

Word	Meaning	Example
numerator	the number above the line: how many parts you have	3 in 34
denominator	the number below the line: how many equal parts the whole is cut into	4 in 34
the whole	the complete amount, or 1	44 = 1
equivalent fractions	fractions with the same value, written with different numbers	12 = 24 = 48
highest common factor (HCF)	the biggest number that divides exactly into two numbers	HCF of 12 and 18 is 6
simplest form	no common factor except 1	1218 = 23
improper fraction	numerator bigger than or equal to the denominator	74
mixed numeral	a whole number and a fraction together	134

The value of a fraction does not change when you write it a different way, only how it is written changes. “The big one” is a fraction equal to 1, such as 22, that you multiply or divide by.

Learning

Look at the two bars.
a) What fraction of the first bar is shaded?
b) What fraction of the second bar is shaded?
c) What do you notice about the two fractions?

Use the big one to fill in the missing numbers.
a) 12 = 6 b) 23 = 8 c) 34 = 12

Simplify by dividing the numerator and the denominator by the highest common factor (HCF).
a) 68 b) 1015 c) 1218

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 3

8.1.1 Forms of fractions

Learning · Practising

Name:

Learning (continued)

a) Write 74 as a mixed numeral. Draw 7 quarters to help you.
b) Write 213 as an improper fraction. Work out 2 × 3 + 1 first

Use the words in the vocabulary box to complete the sentences. In 35, 3 is the ________ and 5 is the ________. Equivalent fractions have the same ________ but are written in different ways

Learning score / 15

Practising

Which is larger, 58 or 23? Show how you compared them

Put in order, smallest first: 34, 58, 712. Say which common denominator you used and why

Fill the gaps. a) 35 = 20 b) 1421 = 2 c) 2436 = 3

Simplify fully, in as few steps as you can. a) 4256 b) 4560 c) 1848

a) Write 175 as a mixed numeral.
b) Write 427 as an improper fraction.
c) Write 338 as an improper fraction

Practising score / 27

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 4

8.1.1 Forms of fractions

Challenging · Mastery

Name:

Challenging

Find a fraction that lies between 13 and 12. Explain the method you used

Find and describe the mistake. Ben simplifies 1218 by subtracting 6 from the numerator and the denominator to get 612 = 12

n12 is between 13 and 34. List every whole number n could be

Write 1007 as a mixed numeral. Then check your answer by changing it back into an improper fraction

Challenging score / 12

Mastery

Use a bar model to explain why multiplying the numerator and the denominator by the same number does not change the value of a fraction

A fraction is equivalent to 23. Its numerator and denominator add up to 35. Find the fraction

How many fractions lie between 13 and 12? Justify your answer

Mastery score / 9

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 5

8.1.2 Four operations with fractions

Learning

Name:

Key vocabulary

Word	Meaning	Example
common denominator	a denominator two fractions share, so the pieces are the same size	12 = 36 and 13 = 26
simplify	write a fraction in simplest form, dividing by the big one	612 = 12
reciprocal	a fraction turned upside down; a number times its reciprocal is 1	the reciprocal of 45 is 54
product	the answer to a multiplication	23 × 34 has product 12
quotient	the answer to a division	3 ÷ 14 has quotient 12
mixed numeral	a whole number and a fraction together	212 = 52

Add and subtract with a common denominator. Multiply with an area model. Divide by asking how many fit.

Learning

Look at the denominators. Why do you not need a common denominator?
a) 15 + 25 b) 56 − 16 c) 38 + 18

Cut both bars into quarters, then work out 12 + 14.

The grid shows 12 of 13.
a) How many equal pieces are in the whole grid?
b) How many pieces are shaded in both colours?
c) Write 12 × 13 as one fraction

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 6

8.1.2 Four operations with fractions

Learning · Practising

Name:

Learning (continued)

How many quarters fit into 3 wholes? Draw 3 bars cut into quarters, then write 3 ÷ 14 =

Learning score / 12

Practising

a) 23 + 14 b) 56 − 14 c) 35 + 310

Cancel before you multiply in at least one of these.
a) 34 × 25 b) 56 × 310 c) 23 × 916

Multiply by the reciprocal. a) 34 ÷ 18 b) 4 ÷ 25 c) 23 ÷ 49

Work with mixed numerals.
a) 112 + 234
b) 212 × 25. For b), change the mixed numeral to an improper fraction first

Without calculating, say whether each answer will be bigger or smaller than the starting number.
a) 12 × 14 b) 3 ÷ 14 c) 8 × 32

A recipe needs 34 cup of flour. You make 23 of the recipe. How much flour do you need, and which operation did you use?

Practising score / 30

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 7

8.1.2 Four operations with fractions

Challenging · Mastery

Name:

Challenging

Fill the gap: 23 + = 56. Explain how you worked backwards

Find and describe the mistake. Mia writes 12 + 13 = 25

23 of a number is 12. Find the number and say why you divided rather than multiplied

A class has 24 pupils. 23 play sport and 34 of those play football.
a) How many pupils play sport?
b) How many play football?
c) What fraction of the class plays football?

Challenging score / 12

Mastery

Use an area diagram to explain why you multiply the numerators and multiply the denominators

Multiplying makes numbers bigger. Is that always true? Give an example where it is not, and explain what decides it

Explain why dividing by 14 gives the same answer as multiplying by 4, in words a Year 7 student would follow

Mastery score / 9

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 8

8.1.3 Fractions, decimals and percentages

Learning

Name:

Key vocabulary

Word	Meaning	Example
fraction	parts of a whole: a numerator over a denominator	34
decimal	a number with a decimal point	0.75
per cent (%)	out of 100	75% = 75100
terminating decimal	a decimal that ends	18 = 0.125
recurring decimal	a decimal whose digits repeat forever	13 = 0.333…
equivalent forms	the same value written in different ways	34 = 0.75 = 75%

The fraction bar means divide: numerator ÷ denominator. The value does not change, only how it is written.

Learning

Look at the hundred square. Write the shaded part three ways. a) as a fraction over 100 b) as a decimal c) as a percentage

Divide the numerator by the denominator to write each fraction as a decimal and as a percentage.
a) 12 b) 14 c) 34

Write as percentages. a) 0.3 b) 0.45 c) 0.07. What did you multiply by?

Use the words in the vocabulary box. A percentage means out of ________. A decimal that ends, like 0.125, is called a ________ decimal. A decimal whose digits repeat forever, like 0.333…, is called a ________ decimal

Learning score / 24

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 9

8.1.3 Fractions, decimals and percentages

Practising

Name:

Practising

Write as decimals by dividing. a) 38 b) 720 c) 56

Write as a fraction in simplest form. a) 50% b) 40% c) 75%

Write as a fraction in simplest form. a) 0.35 b) 12.5% c) 0.04

Which is largest: 0.7, 34 or 72%? Say which form you converted everything to, and why

Put in order, smallest first: 25, 0.38, 41%

A test result is 17 out of 25. Convert it to a percentage. Another test is 31 out of 40. Which result is better, and how do you know?

Without a calculator, say whether each decimal terminates or recurs.
a) 14 b) 13 c) 15

Practising score / 39

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 10

8.1.3 Fractions, decimals and percentages

Challenging · Mastery

Name:

Challenging

Put in order, smallest first: 0.65, 58, 63%, 23

Find and describe the mistake. A student says 0.4 is smaller than 0.35 because 4 is smaller than 35

Convert 16 and 19 to decimals. What do you notice? Can you predict 112 before you check?

In a survey 720 of students walk, 0.3 take the bus and 28% cycle. The rest are driven. What percentage are driven?

Challenging score / 12

Mastery

Test the denominators 2, 3, 4, 5, 6, 8, 9, 10, 12 and 20. Which unit fractions terminate and which recur? Describe the pattern, then predict for 16 and 15 before you check

Explain why 34, 0.75 and 75% are all the same number, even though they look different. Use the words “value” and “written”

Explain why every fraction gives either a terminating or a recurring decimal and never anything random

Mastery score / 9

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 11

8.1.4 Percentage increase and decrease

Learning

Name:

Key vocabulary

Word	Meaning	Example
percentage	a number out of 100	20% = 20100
increase	the amount gets bigger	40 up to 50
decrease	the amount gets smaller	40 down to 30
multiplier	the number you multiply by to make the change in one step	× 1.2
original amount	the amount before the change	45 in 45 × 1.2
new amount	the amount after the change	54 in 45 × 1.2

Increase: multiplier = 1 + the rise as a decimal. Decrease: multiplier = 1 − the fall as a decimal.

Learning

Find each amount without a calculator. a) 10% of 80 b) 20% of 80 c) 25% of 60 d) 50% of 44

The bar shows 40 as 100% with a further 10% added on. a) What is 10% of 40? b) What is the new amount?

Complete. Increase by 20%: 100% + 20% = % so multiply by . Decrease by 20%: 100% − 20% = % so multiply by

Decrease 90 by 20% using the multiplier

Use the words in the vocabulary box. A jacket costs CHF 45 and goes up to CHF 54. The CHF 45 is the ________ amount and CHF 54 is the ________ amount. To go from 45 to 54 we multiply by the ________ 1.2

Learning score / 27

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 12

8.1.4 Percentage increase and decrease

Practising

Name:

Practising

Write the multiplier for each change. a) a 15% rise b) an 8% rise c) a 35% fall d) a 4% fall

A jacket costs CHF 45 and its price rises by 20%. Find the new price using one multiplication

A coat costs CHF 120 and is reduced by 35%. Find the sale price. Estimate first to check your answer

a) Increase 60 by 15%. b) Decrease 250 by 12%

A class of 25 grows to 28.
a) What is the increase?
b) What is the increase as a percentage of the original number?

Practising score / 27

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 13

8.1.4 Percentage increase and decrease

Challenging · Mastery

Name:

Challenging

After a 20% rise a price is CHF 54. Find the original price and explain why you divided

Find and describe the mistake. Sam increases 50 by 20% by working out 50 × 0.2 = 10 and says the new amount is 10

A price rises by 20% and then falls by 20%. Is it back where it started? Use CHF 100 to show your working

A shop offers “20% off, then a further 10% off”. Is that the same as 30% off? Justify your answer

Challenging score / 12

Mastery

Explain, using a bar model, why the multiplier for a 20% rise is 1.2 and not 0.2

One newspaper says “crime up 100%” and another says “crime doubled”. Are they saying the same thing? Which is clearer, and why?

What single multiplier undoes a 25% decrease? Explain why it is not 1.25

Mastery score / 9

Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 14

Homework and checking record

Hand this page in with your sheets

Name:

Your teacher will tell you which sheet and questions to do. Choose your level, set out your work in pen or pencil, then check it with the answers at the back using a green pen: circle wrong question numbers, and star ★ any you still cannot solve.

Sheet	Questions	Which ones I did	Checked in green ✓	Teacher
8.1.1 Forms of fractions	Q1–17
8.1.2 Four operations with fractions	Q18–34
8.1.3 Fractions, decimals and percentages	Q35–52
8.1.4 Percentage increase and decrease	Q53–69

Error analysis

Circle the number of any question you got wrong. Star ★ it if you still cannot solve it after correcting. Then write the questions you need to rework, and what you will do differently:

One thing I understand better now

Score: Parent / guardian signature:

MYP 2 · Year 8 · Unit 8.1 Fractions, Decimals and Percentages · Practice Booklet

mathscafe.net
Lorna Brown © 2026page 15

Answers

Check in green pen

8.1.1 Forms of fractions

Learning

1a) 48
b) 24
c) They are equivalent: the shaded part is the same size, so the value does not change, only how it is written changes2a) 3, because 12 × 33 = 36 b) 12, because 23 × 44 = 812 c) 9, because 34 × 33 = 9123a) 34, dividing by 22 b) 23, dividing by 55 c) 23, dividing by 664a) 134: four quarters make 1 whole and 3 quarters are left over
b) 2 × 3 + 1 = 7, so 213 = 735In 35, 3 is the numerator and 5 is the denominator. Equivalent fractions have the same value but are written in different ways

Practising

623, because 58 = 1524 and 23 = 1624 and 1624 is larger7712, 58, 34. 24 is a common denominator of 4, 8 and 12, and the fractions become 1824, 1524 and 14248a) 12, multiplying by 44 b) 3, dividing by 77 c) 2, dividing by 12129a) 34 (HCF 14) b) 34 (HCF 15) c) 38 (HCF 6)10a) 17 ÷ 5 = 3 remainder 2, so 325 b) 4 × 7 + 2 = 30, so 307 c) 3 × 8 + 3 = 27, so 278

Challenging

11Any of 512, 25 or 37. One way: 13 = 412 and 12 = 612, so 512 is between them12Subtracting changes the value of the fraction. To keep the value the same, divide the numerator and denominator by the same number, here the big one 66: 1218 = 231313 = 412 and 34 = 912, so n = 5, 6, 7 or 814100 ÷ 7 = 14 remainder 2, so 1427. Check: 14 × 7 + 2 = 100, so 1007 ✓

Mastery

15Each part is cut into the same number of smaller parts, so the number of parts shaded and the number of parts in the whole both multiply by the same number. The shaded part of the bar is still the same size. Multiplying by the big one, which equals 1, never changes the value1623 = 2k3k, so 5k = 35 and k = 7. The fraction is 142117Infinitely many. Writing the fractions with a larger common denominator always leaves room for more. With denominator 120, 13 = 40120 and 12 = 60120 have 19 fractions between them; with denominator 1200 there are 199, and so on

8.1.2 Four operations with fractions

Learning

18a) 35
b) 46 = 23
c) 48 = 12. The pieces are already the same size, so only the number of pieces changes1912 = 24, so 24 + 14 = 3420a) 6
b) 1
c) 16. The whole grid has 3 × 2 = 6 pieces and 1 × 1 = 1 is shaded twice2112 quarters fit, because 3 × 4 = 12, so 3 ÷ 14 = 12

Practising

22a) 812 + 312 = 1112 b) 1012 − 312 = 712 c) 610 + 310 = 91023a) 620 = 310 b) 1560 = 14 c) 1848 = 3824a) 34 × 8 = 6 b) 4 × 52 = 10 c) 23 × 94 = 1812 = 32 = 11225a) 1 + 2 = 3 and 12 + 34 = 24 + 34 = 54 = 114, so the total is 414
b) 212 = 52, and 52 × 25 = 1010 = 126a) smaller, because 14 is less than 1
b) bigger, because more quarters fit than wholes
c) bigger, because 32 is more than 12734 × 23 = 612 = 12 cup. “Of” means multiply

Challenging

2856 − 23 = 56 − 46 = 16. You undo the addition by subtracting29She added the numerators and added the denominators. Halves and thirds are different sizes of piece, so they must be changed to sixths first: 36 + 26 = 563012 ÷ 23 = 12 × 32 = 18. You are undoing a multiplication, so you divide31a) 16
b) 12
c) 34 × 23 = 612 = 12, which matches 12 out of 24

Mastery

32The denominators set the grid: rows × columns gives the number of pieces in the whole. The numerators pick out a block of rows and columns, and the overlap has numerator × numerator pieces. For example 23 × 34 has 12 pieces in the whole and 6 in the overlap33Not always. Multiplying by a number between 0 and 1 gives a smaller answer, for example 12 × 14 = 3. It depends on whether the multiplier is greater than 134Dividing by 14 asks how many quarters fit. Four quarters fit into each whole, so you multiply the number of wholes by 4

8.1.3 Fractions, decimals and percentages

Learning

35a) 30100 b) 0.3 c) 30%36a) 1 ÷ 2 = 0.5 = 50% b) 1 ÷ 4 = 0.25 = 25% c) 3 ÷ 4 = 0.75 = 75%37a) 30%
b) 45%
c) 7%. You multiply by 100 because a percentage counts hundredths38A percentage means out of 100. A decimal that ends is a terminating decimal. A decimal whose digits repeat forever is a recurring decimal

Practising

39a) 0.375 b) 0.35 c) 0.8333… (recurring)40a) 50100 = 12 b) 40100 = 25 c) 75100 = 3441a) 35100 = 720 b) 12.5% = 0.125 = 1251000 = 18 c) 4100 = 1254234 is largest. As decimals they are 0.7, 0.75 and 0.72. Using one form makes the values easy to compare430.38, 25, 41%. As percentages they are 38%, 40% and 41%441725 = 68%. 3140 = 0.775 = 77.5%. The second is better. Percentages put both results out of the same total45a) terminates, 0.25 b) recurs, 0.333… c) terminates, 0.2

Challenging

4658 = 62.5%, 63%, 0.65 = 65%, 23 = 66.7%, so 58, 63%, 0.65, 2347The digits after the decimal point are tenths and hundredths, not whole numbers. 0.4 = 0.40 = 40 hundredths, which is more than 35 hundredths4816 = 0.1666… and 19 = 0.111… Both recur. 112 = 0.08333… also recurs49720 = 35%, 0.3 = 30%, so 35% + 30% + 28% = 93% and 100% − 93% = 7% are driven

Mastery

50Terminate: 2, 4, 5, 8, 10, 20. Recur: 3, 6, 9, 12. The terminating ones only have 2s and 5s as prime factors. 116 terminates (0.0625) and 115 recurs (0.0666…)51They have the same value because they describe the same part of the whole: 75 out of 100 squares, or 3 out of 4 equal parts. The value does not change, only how it is written changes52When you divide, only a limited number of remainders is possible. Either the remainder becomes 0 and the decimal ends, or a remainder repeats and then the digits repeat from there on

8.1.4 Percentage increase and decrease

Learning

53a) 8 b) 16, which is double 10% c) 15 d) 2254a) 4 b) 40 + 4 = 4455120% so multiply by 1.2. 80% so multiply by 0.85690 × 0.8 = 7257The CHF 45 is the original amount and CHF 54 is the new amount. We multiply by the multiplier 1.2

Practising

58a) 1.15 b) 1.08 c) 0.65 d) 0.965945 × 1.2 = CHF 5460Estimate: 35% is a bit more than a third, so about 40 off, and 120 − 40 = 80. 120 × 0.65 = CHF 7861a) 60 × 1.15 = 69 b) 250 × 0.88 = 22062a) 28 − 25 = 3
b) 325 = 0.12 = 12%. You divide the increase by the original 25

Challenging

6354 ÷ 1.2 = CHF 45. The multiplier 1.2 must be undone, and division undoes multiplication. Check: 45 × 1.2 = 546450 × 0.2 only finds 20% of 50, which is the extra. The new amount keeps the original 100% and adds the 20%, so the multiplier is 1.2 and 50 × 1.2 = 6065No. 100 × 1.2 = 120 and then 120 × 0.8 = 96. The fall is 20% of the larger amount, 120, so it takes off more than the rise added66No. The multiplier is 0.8 × 0.9 = 0.72, which is a 28% reduction, not 30%. The second 10% is taken off the reduced price

Mastery

67The original bar is 100% of itself and you keep all of it. A rise of 20% adds another 20% on the end, so the new bar is 120% of the original. 120% = 1.2. Multiplying by 0.2 would give only the extra piece68Yes, they mean the same: the multiplier is 1 + 1 = 2. “Doubled” is usually clearer because “up 100%” can be misread as no change69Undo a multiplication by 0.75 by dividing by 0.75, which is multiplying by 43 = 1.333… The fall was a quarter of the original amount, but a rise of 25% would be a quarter of the smaller amount, so it is not enough

Revision slides as text (easy to translate)

This is a plain-text version of the 8.1 revision slides. Your browser can translate it into any language.

Key vocabulary

Numerator: the number above the line in a fraction: how many parts you have
Denominator: the number below the line in a fraction: how many equal parts the whole is cut into
Equivalent fractions: fractions that name the same amount, for example 12 and 36
Highest common factor (HCF): the biggest number that divides exactly into two or more numbers
Lowest terms: a fraction written so that the numerator and denominator share no common factor except 1
Improper fraction: a fraction whose numerator is at least as big as its denominator, for example 74
Mixed numeral: a whole number and a fraction together, for example 134
Common denominator: a denominator shared by two or more fractions, so they can be compared, added or subtracted
Reciprocal: a fraction turned upside down: the reciprocal of 45 is 54
Per cent: per hundred: 75% means 75100
Terminating decimal: a decimal that ends, for example 18 = 0.125
Recurring decimal: a decimal with digits that repeat forever, for example 13 = 0.333...
Multiplier: the number you multiply by to increase or decrease an amount by a percentage, for example × 1.2 for a 20% increase

Forms of fractions

Why it works

The value of a fraction does not change when you write it a different way.

You cut the pieces the same way everywhere, so the amount does not change.

Equivalent fractions

Multiply or divide the numerator and denominator by the same number.

12 = 36

Simplify

Divide the numerator and denominator by the HCF.

1218 = 23

Improper fraction to mixed numeral

Divide the numerator by the denominator.

17 ÷ 5 = 3 remainder 2

175 = 325

Mixed numeral to improper fraction

213 = (3 × 2 + 1)3 = 73

Compare fractions

Write them with a common denominator first.

34 = 912 and 23 = 812

so 34 > 23

Four operations with fractions

Add and subtract

Find a common denominator.

Then add or subtract the numerators.

12 + 13 = 36 + 26 = 56

Never add the denominators.

12 + 13 is not 25

Multiply

Multiply numerator by numerator and denominator by denominator.

Cancel first if you can.

23 × 34 = 612 = 12

Divide

Multiply by the reciprocal.

Think: how many quarters fit in 3?

3 ÷ 14 = 3 × 4 = 12

Mixed numerals

Change to improper fractions first.

212 ÷ 56 = 52 × 65 = 3

Fractions, decimals and percentages

Per cent means per hundred

34 = 75100 = 0.75 = 75%

Changing between forms

Fraction to decimal: divide the numerator by the denominator.

34 = 0.75

Decimal to percentage: multiply by 100.

Percentage to decimal: divide by 100.

Percentage to fraction: write it over 100, then simplify.

Common ones to know

12 = 0.5 = 50%

14 = 0.25 = 25%

15 = 0.2 = 20%

110 = 0.1 = 10%

18 = 0.125 = 12.5%

Terminating and recurring decimals

A terminating decimal ends: 18 = 0.125

A recurring decimal repeats: 13 = 0.333...

A fraction in lowest terms terminates only if its denominator has no prime factors except 2 and 5.

Ordering

To order mixed forms, change them all to percentages.

Percentage increase and decrease

Multiplier

For a rise, the multiplier is 1 + the rise, as a decimal.

For a fall, the multiplier is 1 − the fall, as a decimal.

A 20% increase is × 1.2

A 15% decrease is × 0.85

Why 1.2 and not 0.2?

The original is 100%, and 100% + 20% = 120%.

Worked examples

45 francs up 20%: 45 × 1.2 = 54

300 francs down 15%: 300 × 0.85 = 255

Repeated change

Multiply the multipliers.

Up 20% then down 20%: 1.2 × 0.8 = 0.96

This is 4% less than the start, because the second 20% is of a bigger amount.

Check by estimating

A 20% rise on 45 is about 9 more.

Common mistakes

✗ Wrong: 12 + 13 = 25

✓ Correct: Find a common denominator first: 36 + 26 = 56

✗ Wrong: 3 ÷ 14 = 34

✓ Correct: Dividing by 14 multiplies by 4: 3 × 4 = 12

✗ Wrong: A 20% rise is × 0.2

✓ Correct: It is × 1.2, because the original stays (100% + 20%)

✗ Wrong: Up 20% then down 20% gets you back to the start

✓ Correct: It gives 96% of the start: 1.2 × 0.8 = 0.96