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GCSE Maths · EdexcelChapter 15 · Ratio and proportion
Chapter 15 · Foundation tier

GCSE Maths Ratio and Proportion (Foundation): worked examples and questions with answers

By the end of this chapter you will be able to

  • Write one quantity as a fraction of another, including when the fraction is bigger than 1 and when the units are different
  • Write ratios, simplify them (with different units, decimals and fractions) and write them in the form 1 : n
  • Share a quantity in a given ratio, and solve problems where one part or the difference is known
  • Combine ratios such as a : b and b : c, and move between ratios and fractions
  • Use ratio in recipes, mixing, concentrations and conversions
  • Use the unitary method and unit prices to solve proportion problems and find the best buy
  • Use proportion as equality of ratios and connect it to the linear function y = kx

You should already know

  • Simplifying fractions and finding fractions of an amount (Chapter 3)
  • Multiplying and dividing whole numbers and decimals (Chapters 1 and 4)
  • Highest common factor (Chapter 2)
  • Converting between units of length, mass and time

Where this leads

  • Percentages and percentage change (Chapter 5)
  • Direct and inverse proportion, speed, density and pressure (Chapter 16)
  • Straight-line graphs and gradient (Chapter 12)
  • Scale factors and similar shapes (Chapter 21)
  • Probability as a fraction (Chapter 23)

Edexcel GCSE (9–1) Mathematics 1MA1, Foundation tier. Content references: R3, R4, R5, R6, R7, R8, N11.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

Exercise P  Do you remember?

Non-calculator
1
FWrite the fraction 1824 in its simplest form.
2
FWork out 38 of 64.
3
FChange 2.5 metres into centimetres.
4
FFind the highest common factor (HCF) of 24 and 36.
5
FWork out £4.80 ÷ 6.
15.1

One quantity as a fraction of another

Edexcel 1MA1: R3, N11
Grade 1-4

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: One quantity as a fraction of anotherComing soon

Suppose 18 of the 30 students in a class are girls. One way to describe this is with a fraction. A fraction compares a part with the whole: the part goes on the top (the numerator) and the whole goes on the bottom (the denominator). Here the part is 18 and the whole is 30, so the girls are 1830 of the class. Remember from Chapter 3 that we should always simplify: divide the top and the bottom by their highest common factor. Both 18 and 30 divide by 6, so 1830 = 35.

When a question says “write A as a fraction of B”, the quantity after the word “of” is the whole. It goes on the bottom. For example, “write 3 as a fraction of 8” gives 38. The picture shows why: if the bar is cut into 8 equal blocks and 3 of them are blue, then blue is 3 out of 8 blocks.

3 blue blocks5 other blockswhole bar = 8 equal blocks
Blue is 3 out of 8 equal blocks, so blue is 3/8 of the bar.

Units must match. To write 45 minutes as a fraction of 2 hours, we cannot write 452 because minutes and hours are different units. We change both to the same unit first: 2 hours = 120 minutes, and then the fraction is 45120 = 38. Always change to the smaller unit, because that gives whole numbers.

Fractions bigger than 1. The part is allowed to be bigger than the whole. If one plank is 5 units long and another is 4 units long, the first plank is 54 of the second. This is an improper fraction. It is the same as 114, which tells us the first plank is one whole plank and a quarter more.

Plank A: 5 units1 extra unitPlank B: 4 units
Plank A is 5/4 of plank B because A has 5 units where B has 4.

Key ideas

  • A fraction of a quantity is part ÷ whole: the part on top, the whole on the bottom.
  • In “A as a fraction of B”, B (the quantity after “of”) is the whole.
  • Both quantities must be in the same unit before you write the fraction. Change to the smaller unit.
  • Simplify the fraction by dividing the top and bottom by the highest common factor.
  • If the part is bigger than the whole, the fraction is bigger than 1 (an improper fraction).
Worked example 15.1.1
In a class of 30 students, 18 are girls. What fraction of the class are girls? Give your answer in its simplest form.

Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).

Method. Write the part over the whole (in the same units), then simplify the fraction.

  1. Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.
    Part = 18
    Whole = 30
  2. Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).
    1830
  3. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 18: 1, 2, 3, 6, 9, 18
    Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
    The highest common factor is 6.
  4. Divide the top and the bottom by 6. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    18 ÷ 6 = 3 and 30 ÷ 6 = 5
    1830 = 35

Check. Take the fraction of the whole: 35 of 30 = 30 ÷ 5 × 3 = 18, which is the part 18. ✓

Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.

Worked example 15.1.2
Write 45 minutes as a fraction of 2 hours.

Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).

Method. Write the part over the whole (in the same units), then simplify the fraction.

  1. Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.
    Part = 45 minutes
    Whole = 2 hours
  2. Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.
    1 hours = 60 minutes
    2 hours = 2 × 60 = 120 minutes
  3. Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).
    45120
  4. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 45: 1, 3, 5, 9, 15, 45
    Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
    The highest common factor is 15.
  5. Divide the top and the bottom by 15. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    45 ÷ 15 = 3 and 120 ÷ 15 = 8
    45120 = 38

Check. Take the fraction of the whole: 38 of 120 = 120 ÷ 8 × 3 = 45, which is the part 45. ✓

Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.

Worked example 15.1.3
A parcel weighs 2.4 kg. A letter weighs 0.6 kg. Write the weight of the letter as a fraction of the weight of the parcel.

Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).

Method. Write the part over the whole (in the same units), then simplify the fraction.

  1. Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.
    Part = 0.6 kg
    Whole = 2.4 kg
  2. Multiply both numbers by 5 to remove the decimals. Multiplying the top and the bottom of a fraction by the same number does not change its value.
    0.6 × 5 = 3
    2.4 × 5 = 12
  3. Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).
    312
  4. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 3: 1, 3
    Factors of 12: 1, 2, 3, 4, 6, 12
    The highest common factor is 3.
  5. Divide the top and the bottom by 3. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    3 ÷ 3 = 1 and 12 ÷ 3 = 4
    312 = 14

Check. Take the fraction of the whole: 14 of 12 = 12 ÷ 4 × 1 = 3, which is the part 3. ✓

Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.

Worked example 15.1.4
A rope is 1.5 m long. A string is 60 cm long. Write the length of the rope as a fraction of the length of the string.

Key words. A fraction of a quantity compares part of it with the whole. The numerator is the top (the part) and the denominator is the bottom (the whole).

Method. Write the part over the whole (in the same units), then simplify the fraction.

  1. Decide which quantity is the part and which is the whole. In “A as a fraction of B”, B is the whole (it comes after “of”) and A is the part. The fraction is part over whole.
    Part = 1.5 m
    Whole = 60 cm
  2. Write both quantities in the same unit. A fraction compares two amounts, so they must be in the same unit. Change to the smaller unit to avoid decimals.
    1 m = 100 cm
    1.5 m = 1.5 × 100 = 150 cm
  3. Write the part over the whole. The part goes on top (numerator) and the whole goes on the bottom (denominator).
    15060
  4. List the factors of the top and the bottom. We want the biggest number that divides into both. This is the highest common factor (HCF).
    Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150
    Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
    The highest common factor is 30.
  5. Divide the top and the bottom by 30. Dividing both by the same number does not change the value of the fraction, but it makes the numbers smaller.
    150 ÷ 30 = 5 and 60 ÷ 30 = 2
    15060 = 52
  6. Notice the size of the answer. The part is bigger than the whole, so the fraction is bigger than 1 (an improper fraction). That is allowed.
    52 = 212

Check. Take the fraction of the whole: 52 of 60 = 60 ÷ 2 × 5 = 150, which is the part 150. ✓

Watch out. Put the quantity that comes after “of” on the bottom, and make sure both quantities use the same unit first.

Watch out

Do not put the numbers the wrong way round. “12 as a fraction of 30” is 1230 = 25, not 3012. And never write a fraction until both quantities are in the same unit: 40p as a fraction of £2 is 40200, not 402.

Exercise 15A  One quantity as a fraction of another

Non-calculator
1
FWrite each as a fraction in its simplest form.
(a)7 as a fraction of 10
(b)8 as a fraction of 20
(c)12 as a fraction of 30
(d)9 as a fraction of 36
(e)14 as a fraction of 35
(f)16 as a fraction of 64
2
FA box holds 10 pens. 4 of the pens are blue. What fraction of the pens are blue? Give your answer in its simplest form.
3
FA bag contains 24 sweets. 9 of the sweets are red. What fraction of the sweets are red?
4
FMia scored 36 marks out of 45 in a test. What fraction of the marks did she score? Give your answer in its simplest form.
5
FThe bar is cut into equal blocks. (a) What fraction of the bar is blue? (b) What fraction of the bar is orange? Simplify where you can.
blueorange
(a)Blue as a fraction of the whole bar
(b)Orange as a fraction of the whole bar
6
FWrite the first quantity as a fraction of the second. Change to the same unit first.
(a)20 cm of 1 m
(b)300 g of 2 kg
(c)45 minutes of 3 hours
(d)40p of £2
(e)250 m of 2 km
(f)500 ml of 3 litres
7
FThese fractions are bigger than 1. Write each as an improper fraction in its simplest form.
(a)9 as a fraction of 6
(b)15 as a fraction of 10
(c)7 as a fraction of 4
(d)90 minutes as a fraction of 1 hour
(e)£5 as a fraction of 80p
8
FWrite the first quantity as a fraction of the second. These involve decimals.
(a)0.6 kg of 2.4 kg
(b)0.9 m of 1.5 m
(c)£3.50 of £5.60
9
PSA football team plays 38 matches. It wins 14, draws 9 and loses the rest. Write as a fraction of the matches played: (a) the matches won, (b) the matches drawn, (c) the matches lost.
(a)Matches won
(b)Matches drawn
(c)Matches lost
10
PSTariq is reading a book with 270 pages. He has read 45 pages. What fraction of the book has he not yet read? Give your answer in its simplest form.
11
PSIn a race of 2 km, Kofi has run 750 m. What fraction of the race is left to run? Give your answer in its simplest form.
12
PSIn a class of 28 students, 7 walk to school, 12 come by bus and the rest come by car. Write as a fraction in its simplest form the students who come (a) by bus, (b) by car.
(a)By bus
(b)By car
13
PSImran has £2.40 pocket money. He spends 90p. What fraction of his pocket money does he spend? Give your answer in its simplest form.
14
RAmy scored 17 out of 20 in a spelling test. Ben scored 21 out of 25 in a different test. Who did better, relative to the number of marks available? Show your working.
15
RSara says: “12 as a fraction of 30 is 3012.” Explain her mistake and write the correct fraction.
16
RIvan says: “If A is 35 of B, then B is 53 of A.” Show that Ivan is right by choosing a number for B.
17
RA cake mixture uses 450 g of flour, 0.3 kg of sugar and 150 g of butter. What fraction of the total mass of these three ingredients is sugar? Give your answer in its simplest form.
15.2

Ratio notation and simplifying ratios

Edexcel 1MA1: R4, R6
Grade 1-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Ratio notation and simplifying ratiosComing soon

A ratio compares the sizes of two or more quantities of the same kind. If a bag has 3 red counters and 2 blue counters, we say the ratio of red to blue is 3 : 2 (read “3 to 2”). It means “for every 3 red counters there are 2 blue counters”. The order matters: the ratio of blue to red is 2 : 3, which is a different ratio.

3 red2 blue
Red : blue = 3 : 2. For every 3 red blocks there are 2 blue blocks.

Equivalent ratios. If we double a recipe, every ingredient doubles, but the recipe tastes the same. Multiplying (or dividing) every part of a ratio by the same number gives an equivalent ratio. So 2 : 3 = 4 : 6 = 6 : 9 = 10 : 15. They all describe the same mix.

redblue2369× 3× 3
Multiply both numbers by 3: 2 : 3 = 6 : 9.

Simplest form. A ratio is in its simplest form when no whole number except 1 divides into every part. To simplify, divide every part by the same number, ideally the highest common factor (HCF, see Chapter 2). For 12 : 18 the HCF is 6, so 12 : 18 = 2 : 3. This is exactly like simplifying a fraction, except that there is no line between the numbers.

Different units. A ratio only makes sense when the quantities are in the same unit. To compare 2 m with 50 cm, change both to centimetres first: 200 cm : 50 cm = 4 : 1. The units then disappear from the ratio, because 4 : 1 compares plain numbers of “lots of the same size”. Decimals and fractions: multiply every part by the same number (10 or 100 for decimals, the LCM of the denominators for fractions) to get whole numbers, then simplify.

The form 1 : n. Sometimes we want the first number to be 1, for example in a map scale. Divide both parts by the first number. 4 : 10 becomes 1 : 2.5. In the same way, to write a ratio as n : 1 divide by the second number. The other number does not have to be a whole number.

Ratio and multiplying. The ratio 3 : 1 says the first quantity is 3 times as big as the second. The ratio 3 : 5 says the first is 35 of the second. So a ratio can describe a multiplicative relationship: one quantity is a fixed multiple (or fraction) of the other. If Zara has £48 and Ali has £16, then 48 ÷ 16 = 3, so Zara has 3 times as much, and the ratio is 3 : 1.

Key ideas

  • A ratio compares quantities of the same kind: a : b. The order matters.
  • To get an equivalent ratio, multiply or divide every part by the same number.
  • To simplify, divide every part by the HCF. The quantities must be in the same unit first, and the units are left out of the answer.
  • Decimals and fractions: multiply through to get whole numbers, then simplify.
  • 1 : n form: divide by the first number. n : 1 form: divide by the second number.
  • a : b with a ÷ b = k means the first quantity is k times the second.
Worked example 15.2.1
A box contains 12 red counters and 18 blue counters. Write the ratio of red counters to blue counters in its simplest form.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. To simplify a ratio, divide every part by the same number (their highest common factor) until nothing else divides into all the parts. Both sides are treated in the same way, like simplifying a fraction.

  1. Write the ratio down. Before changing anything, make sure you know which numbers you are comparing.
    12 : 18
  2. List the factors of each number and pick out the highest common factor (HCF). The HCF is the biggest whole number that divides exactly into every part of the ratio. Dividing by it simplifies the ratio in one go.
    Factors of 12: 1, 2, 3, 4, 6, 12
    Factors of 18: 1, 2, 3, 6, 9, 18
    Common factors: 1, 2, 3, 6
    Highest common factor (HCF) = 6
  3. Divide every part by 6. Dividing each part by the same number keeps the ratio equivalent but with smaller numbers.
    12 ÷ 6 = 2
    18 ÷ 6 = 3
    12 : 18 = 2 : 3

Check. Multiply the answer back: 2 × 6 = 12 , 3 × 6 = 18, which gives 12 : 18 again. ✓

Watch out. Do the same to every part of the ratio. Dividing only one part changes the ratio.

Worked example 15.2.2
Write the ratio 2 m : 50 cm in its simplest form.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. Write all the quantities in the same unit, remove any decimals or fractions, then divide every part by the highest common factor.

  1. Write the ratio down. Before changing anything, make sure you know which numbers you are comparing.
    2 m : 50 cm
  2. The units are different, so change them to the same unit. A ratio only makes sense if the quantities are measured in the same unit. Change to the smaller unit (cm), because that avoids decimals.
    1 m = 100 cm
    2 m = 2 × 100 = 200 cm
    The ratio is now 200 cm : 50 cm.
  3. The numbers are big, so divide by common factors one at a time. Each time, divide every part by the same number. Stop when nothing except 1 divides into all of them. The numbers divided by multiply to give the HCF.
    Every number divides by 2: 200 ÷ 2 = 100 , 50 ÷ 2 = 25
    Every number divides by 5: 100 ÷ 5 = 20 , 25 ÷ 5 = 5
    Every number divides by 5: 20 ÷ 5 = 4 , 5 ÷ 5 = 1
    Multiplying the numbers we divided by gives the HCF: 50
  4. Divide every part by 50. Dividing each part by the same number keeps the ratio equivalent but with smaller numbers.
    200 ÷ 50 = 4
    50 ÷ 50 = 1
    200 : 50 = 4 : 1

Check. Multiply the answer back: 4 × 50 = 200 , 1 × 50 = 50, which gives 200 : 50 again. ✓

Watch out. Never leave units in a simplified ratio, and never simplify while the units are different (1 m : 50 cm is 2 : 1, not 1 : 50).

Worked example 15.2.3
Write the ratio 1.2 : 0.8 : 2 in its simplest form.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. To simplify a ratio, divide every part by the same number (their highest common factor) until nothing else divides into all the parts. Both sides are treated in the same way, like simplifying a fraction.

  1. Write the ratio down. Before changing anything, make sure you know which numbers you are comparing.
    1.2 : 0.8 : 2
  2. Multiply every part by 10. Multiplying every part by the same number does not change the ratio. Multiplying by 10 moves the decimal point 1 place and gives whole numbers.
    1.2 × 10 = 12
    0.8 × 10 = 8
    2 × 10 = 20
    The ratio is now 12 : 8 : 20.
  3. List the factors of each number and pick out the highest common factor (HCF). The HCF is the biggest whole number that divides exactly into every part of the ratio. Dividing by it simplifies the ratio in one go.
    Factors of 12: 1, 2, 3, 4, 6, 12
    Factors of 8: 1, 2, 4, 8
    Factors of 20: 1, 2, 4, 5, 10, 20
    Common factors: 1, 2, 4
    Highest common factor (HCF) = 4
  4. Divide every part by 4. Dividing each part by the same number keeps the ratio equivalent but with smaller numbers.
    12 ÷ 4 = 3
    8 ÷ 4 = 2
    20 ÷ 4 = 5
    12 : 8 : 20 = 3 : 2 : 5

Check. Multiply the answer back: 3 × 4 = 12 , 2 × 4 = 8 , 5 × 4 = 20, which gives 12 : 8 : 20 again. ✓

Watch out. Do the same to every part of the ratio. Dividing only one part changes the ratio.

Worked example 15.2.4
Write the ratio 12 : 34 in its simplest form.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. To simplify a ratio, divide every part by the same number (their highest common factor) until nothing else divides into all the parts. Both sides are treated in the same way, like simplifying a fraction.

  1. Write the ratio down. Before changing anything, make sure you know which numbers you are comparing.
    12 : 34
  2. Multiply every part by 4, the lowest common multiple (LCM) of the denominators 2, 4. Multiplying every part of a ratio by the same number gives an equivalent ratio (like making a recipe bigger). The LCM removes all the fractions.
    12 × 4 = 2
    34 × 4 = 3
    The ratio is now 2 : 3.
  3. Check whether it can be simplified further. We look for a number that divides into every part.
    The only common factor is 1, so 2 : 3 is already in its simplest form.

Check. The numbers 2, 3 have no common factor above 1, so the ratio cannot be made simpler. ✓

Watch out. Do the same to every part of the ratio. Dividing only one part changes the ratio.

Worked example 15.2.5
Write the ratio 4 : 10 in the form 1 : n.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. To write a ratio in the form 1 : n, divide both parts by the first number.

  1. Write the ratio. We want the first number to become 1.
    4 : 10
  2. Divide both parts by 4. Dividing every part by the same number gives an equivalent ratio. We divide by the number that we want to turn into 1.
    4 ÷ 4 = 1
    10 ÷ 4 = 2.5
    4 : 10 = 1 : 2.5
  3. Write the answer in the required form. The number we divided by is now 1 (the first part), and the other number is n = 2.5. It does not have to be a whole number.
    1 : 2.5

Check. Multiply back by 4: 1 × 4 = 4 and 2.5 × 4 = 10. ✓

Watch out. The number that is not 1 does not have to be a whole number. It can be a decimal or a fraction.

Worked example 15.2.6
Zara has £48 and Ali has £16. (a) Write the ratio of Zara's money to Ali's money in its simplest form. (b) How many times as much money does Zara have as Ali?

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. Write the ratio, simplify it by dividing by the HCF, then read the first number of the ratio n : 1 as the number of times.

  1. Write the ratio of Zara's money to Ali's money. Zara is mentioned first, so her amount is the first number.
    48 : 16
  2. Find the HCF of 48 and 16. The factors of 16 are 1, 2, 4, 8, 16 and all of them divide into 48 too. The highest is 16.
    HCF = 16
  3. Divide both parts by 16. This gives the simplest ratio.
    48 ÷ 16 = 3 and 16 ÷ 16 = 1, so the ratio is 3 : 1
  4. Read off how many times as much. The ratio 3 : 1 means that for every £1 that Ali has, Zara has £3. So Zara has 3 times as much.
    3 times

Check. 3 × £16 = £48, which is Zara's amount. ✓

Watch out. The ratio 3 : 1 does not mean that Zara has £3. It compares the two amounts.

Watch out

Do not simplify while the units are different. 1 m : 50 cm is not 1 : 50. Change both to centimetres first: 100 : 50 = 2 : 1. And write the numbers in the order the question asks: “red to blue” means red first.

Exercise 15B  Ratio notation and simplifying

Non-calculator
1
FThe diagram shows red and blue blocks. Write the ratio of red blocks to blue blocks.
redblue
2
FWrite each ratio in its simplest form.
(a)4 : 6
(b)10 : 25
(c)12 : 18
(d)15 : 45
(e)24 : 36
(f)16 : 48
3
FA box has 8 red pens and 12 blue pens. Write each ratio in its simplest form.
(a)The ratio of red pens to blue pens
(b)The ratio of blue pens to red pens
4
FWrite each three-part ratio in its simplest form.
(a)6 : 9 : 12
(b)20 : 30 : 50
(c)8 : 12 : 20
5
FWrite each ratio in its simplest form. Change to the same unit first.
(a)50 cm : 2 m
(b)£2 : 50p
(c)20 minutes : 1 hour
(d)3 kg : 750 g
(e)400 m : 1.2 km
(f)1.5 litres : 500 ml
6
FWrite each ratio in its simplest form. These ratios contain decimals.
(a)0.5 : 2
(b)1.2 : 0.8
(c)0.25 : 1.5
(d)0.6 : 0.9 : 1.5
7
FWrite each ratio in its simplest form. These ratios contain fractions.
(a)12 : 34
(b)23 : 56
(c)14 : 16
(d)35 : 110
8
FWrite each ratio in the form given.
(a)2 : 10 as 1 : n
(b)5 : 20 as 1 : n
(c)4 : 10 as 1 : n
(d)8 : 12 as 1 : n
(e)15 : 3 as n : 1
(f)9 : 6 as n : 1
9
PSJan is 1.2 m tall. Her brother is 90 cm tall. Write the ratio of Jan's height to her brother's height in its simplest form.
10
PSZara has £48 and Ali has £16, as in the worked example. Ben has £60 and Cara has £15. Who has the greater number of times as much as the other: Zara compared with Ali, or Ben compared with Cara?
11
PSA rectangular garden is 18 m long and 12 m wide. (a) Write the ratio length : width in its simplest form. (b) Write the ratio width : length in the form 1 : n.
(a)Length : width
(b)Width : length as 1 : n
12
PSA map has a scale where 4 cm on the map represents 1 km on the ground. Write this scale as a ratio in the form 1 : n.
13
RDev writes “1 m : 50 cm = 1 : 50”. Explain his mistake and write the ratio in its simplest form.
14
RWrite down three different ratios that are equivalent to 2 : 5.
15
RThree of these ratios are equivalent to each other. Which one is the odd one out?   3 : 4    6 : 8    9 : 12    12 : 18
16
RMel makes a drink with 150 ml of cordial and 0.6 litres of water. (a) Write the ratio of cordial to water in the form 1 : n. (b) What fraction of the drink is cordial? Explain how you know.
(a)Cordial : water as 1 : n
(b)Fraction that is cordial
15.3

Sharing a quantity in a ratio

Edexcel 1MA1: R5
Grade 2-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Sharing a quantity in a ratioComing soon

Aisha and Ben are given £48 to share in the ratio 3 : 5. This means that for every £3 Aisha gets, Ben gets £5. Aisha gets 3 equal parts and Ben gets 5 equal parts. Before we can work out how much each person gets, we need to know how big one part is. The method has three steps, and the bar below shows why it works.

Aisha: 3 parts£6£6£6Ben: 5 parts£6£6£6£6£6£48 shared into 8 equal parts
Each block is worth £48 ÷ 8 = £6. Aisha gets 3 × £6 = £18 and Ben gets 5 × £6 = £30.
  1. Add the parts to find the total number of equal parts: 3 + 5 = 8.
  2. Divide the amount by the total number of parts to find the value of one part: £48 ÷ 8 = £6.
  3. Multiply the value of one part by each ratio number: 3 × £6 = £18 and 5 × £6 = £30.

Always check that the shares add up to the total: £18 + £30 = £48.

Three or more shares. The same method works for any number of shares. For the ratio 2 : 3 : 5 there are 2 + 3 + 5 = 10 parts. Money. When the total is in pounds and pence, it can help to work in pence (£7.50 = 750p) or to divide the decimal carefully. Remember to write money answers with two decimal places, for example £4.50.

Part : whole ratios. Sometimes a ratio compares one share with the whole, not with the other share. If the ratio of girls to all the students is 3 : 8, then of every 8 students, 3 are girls and the other 5 are not. So girls : not girls = 3 : (8 − 3) = 3 : 5. Change a part : whole ratio into a part : part ratio by taking the part away from the whole, then share as normal.

girls: 3not girls: 8 − 3 = 5whole = 8 parts
Girls : whole is 3 : 8, which means girls : not girls is 3 : 5.

Key ideas

  • Add the ratio numbers to find the total number of equal parts.
  • Divide the amount by the total number of parts to find the value of one part.
  • Multiply the value of one part by each ratio number.
  • Check: the shares must add up to the total.
  • Part : whole ratio, for example 3 : 8, means the other share is 8 − 3 = 5 parts.
Worked example 15.3.1
Aisha and Ben share £48 in the ratio 3 : 5. How much does each person get?

Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.

Method. Add the parts to find how many equal parts there are, find the value of one part, then multiply.

  1. Add up the parts of the ratio. Each number in the ratio is a number of equal parts. The total number of parts is the sum of the ratio numbers.
    3 + 5 = 8 parts
  2. Find the value of one part. £48 is shared into 8 equal parts, so divide it by 8.
    £48 ÷ 8 = £6 for one part
  3. Multiply to find each share. Each person gets as many parts as their ratio number, so multiply the value of one part by that number.
    Aisha: 3 × £6 = £18
    Ben: 5 × £6 = £30

Check. Add the shares: £18 + £30 = £48, which is the total £48. ✓

Watch out. Do not divide the total by one of the ratio numbers. Divide by the total number of parts.

Worked example 15.3.2
Red, yellow and white paint are mixed in the ratio 2 : 3 : 5. The mixture is 300 ml in total. How much of each colour is used?

Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.

Method. Add the parts to find how many equal parts there are, find the value of one part, then multiply.

  1. Add up the parts of the ratio. Each number in the ratio is a number of equal parts. The total number of parts is the sum of the ratio numbers.
    2 + 3 + 5 = 10 parts
  2. Find the value of one part. 300 ml is shared into 10 equal parts, so divide it by 10.
    300 ml ÷ 10 = 30 ml for one part
  3. Multiply to find each share. Each person gets as many parts as their ratio number, so multiply the value of one part by that number.
    Red: 2 × 30 ml = 60 ml
    Yellow: 3 × 30 ml = 90 ml
    White: 5 × 30 ml = 150 ml

Check. Add the shares: 60 ml + 90 ml + 150 ml = 300 ml, which is the total 300 ml. ✓

Watch out. Do not divide the total by one of the ratio numbers. Divide by the total number of parts.

Worked example 15.3.3
Share £7.50 in the ratio 2 : 3.

Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.

Method. Add the parts to find how many equal parts there are, find the value of one part, then multiply.

  1. Add up the parts of the ratio. Each number in the ratio is a number of equal parts. The total number of parts is the sum of the ratio numbers.
    2 + 3 = 5 parts
  2. Find the value of one part. £7.50 is shared into 5 equal parts, so divide it by 5.
    £7.50 ÷ 5 = £1.50 for one part
  3. Multiply to find each share. Each person gets as many parts as their ratio number, so multiply the value of one part by that number.
    First share: 2 × £1.50 = £3
    Second share: 3 × £1.50 = £4.50

Check. Add the shares: £3 + £4.50 = £7.50, which is the total £7.50. ✓

Watch out. Do not divide the total by one of the ratio numbers. Divide by the total number of parts.

Worked example 15.3.4
In a club the ratio of girls to all the students is 3 : 8. There are 40 students in the club. How many girls and how many other students are there?

Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.

Method. Change the part : whole ratio into a part : part ratio. Then share 40 in that ratio.

  1. Find the share of the students who are not girls. The whole is 8 parts and girls are 3 of them, so the rest is the whole take away the girls.
    8 − 3 = 5 parts
  2. Write the ratio girls : not girls. Now both numbers are parts, so we can share in this ratio.
    3 : 5
  3. Add up the parts of the ratio. Each number in the ratio is a number of equal parts. The total number of parts is the sum of the ratio numbers.
    3 + 5 = 8 parts
  4. Find the value of one part. 40 is shared into 8 equal parts, so divide it by 8.
    40 ÷ 8 = 5 for one part
  5. Multiply to find each share. Each person gets as many parts as their ratio number, so multiply the value of one part by that number.
    Girls: 3 × 5 = 15
    Not girls: 5 × 5 = 25

Check. Girls are 38 of 40 = 40 ÷ 8 × 3 = 15. ✓

Watch out. The ratio 3 : 8 is not girls to boys. 8 is the whole group, so boys are 8 − 3 = 5 parts.

Watch out

Do not divide the amount by one of the ratio numbers. To share £48 in the ratio 3 : 5, divide by 8 (the total number of parts), not by 3 or 5. And check that your shares add up to the original amount.

Exercise 15C  Sharing in a ratio

Non-calculator
1
FThe bar shows £35 shared between Eli and Fran in the ratio 2 : 3. Each block is worth the same amount.
EliFran£35
(a)How many equal blocks are there altogether?
(b)What is the value of one block?
(c)How much does Fran get?
2
FShare each amount in the ratio given.
(a)£20 in the ratio 1 : 4
(b)£35 in the ratio 2 : 5
(c)48 kg in the ratio 3 : 1
(d)72 sweets in the ratio 5 : 7
(e)£90 in the ratio 2 : 3
(f)63 m in the ratio 4 : 5
3
FAmir and Beth share £84 in the ratio 3 : 4. How much does each person get?
4
FAna, Ben and Cat share a prize of £600 in the ratio 1 : 2 : 3. How much does each person get?
5
FShare each amount in the ratio given.
(a)60 in the ratio 1 : 2 : 3
(b)£120 in the ratio 2 : 3 : 5
(c)90 kg in the ratio 3 : 4 : 8
6
FShare £2.40 in the ratio 1 : 5.
7
FShare £7.50 between Dan and Eve in the ratio 2 : 3.
8
PSA 20 kg bag of mixture contains sand and cement in the ratio 4 : 1. Work out the mass of sand and the mass of cement.
9
PSIn a bag, the ratio of red counters to all the counters is 2 : 7. There are 49 counters. How many red counters are there, and how many other counters?
10
PSA school has 840 pupils. The ratio of boys to girls is 7 : 5. How many boys and how many girls are there?
11
PSThree partners share a profit of £2400 in the ratio 5 : 4 : 3. How much does each partner receive?
12
PSThe bar shows £84 shared between Priya and Sam in the ratio 4 : 3. How much does Priya receive?
PriyaSam£84
13
PSCompetition A: £60 is shared between two people in the ratio 2 : 3. Competition B: £80 is shared between two people in the ratio 1 : 3. Which competition has the bigger top prize, and how much is it?
14
RSophie says: “To share £50 in the ratio 2 : 3, I divide 50 by 2 and 50 by 3.” Explain what is wrong, and work out the correct shares.
15
RTwo people share £50 in the ratio 3 : 4. Is it possible for both of them to receive a whole number of pounds? Give a reason.
16
RMum, Dad and Ella share £240 in the ratio 3 : 2 : 1. Dad then gives half of his share to Ella. Write the ratio of the three amounts now, in the order Mum : Dad : Ella, in its simplest form.
15.4

Ratio problems: one part known, differences and combining ratios

Edexcel 1MA1: R5, R6
Grade 3-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Ratio problems: one part known, differences and combining ratiosComing soon

In many problems you are not told the total. Instead you know one share, or the difference between two shares. The trick is always the same: find the value of one part, then use it.

One share known. Red and blue counters are in the ratio 3 : 5 and there are 12 red counters. Red is 3 parts, so 3 parts = 12 and one part = 12 ÷ 3 = 4. Then blue is 5 × 4 = 20. Divide by the ratio number of the share you were given, not by the total number of parts.

red: 12444blue: ?444443 parts = 12, so 1 part = 4
Red is 3 parts and is worth 12, so each part is 4. Blue is 5 parts, which is 20.

Difference known. Ali and Bea share money in the ratio 7 : 4. Ali gets £36 more than Bea. The difference in the number of parts is 7 − 4 = 3 parts, and this is worth £36. So one part = £36 ÷ 3 = £12.

Ali: 7 parts3 parts = £36Bea: 4 parts
The difference between the shares is 3 parts, and it is worth £36.

Combining ratios. Suppose A : B = 2 : 3 and B : C = 4 : 5. B is in both ratios, but it is 3 in one and 4 in the other. To join the ratios, make the B numbers equal. The lowest common multiple of 3 and 4 is 12, so we multiply the first ratio by 4 and the second ratio by 3.

ABC
A : B23× 4
becomes812
B : C45× 3
becomes1215
A : B : C81215

Key ideas

  • One share known: divide that share by its ratio number to get the value of one part.
  • Difference known: subtract the ratio numbers, divide the difference by the answer to get the value of one part.
  • Then multiply the value of one part by any ratio number, or by the total number of parts.
  • Combine ratios by making the number for the shared quantity the same in both ratios (use the LCM).
Worked example 15.4.1
Red and blue counters are in the ratio 3 : 5. There are 12 red counters. How many blue counters are there?

Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.

Method. When one share is known, divide it by its number of parts to find one part. Then multiply to find any other share.

  1. Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.
    Red has 3 parts, worth 12
  2. Find the value of one part. 3 parts are worth 12, so divide by 3.
    12 ÷ 3 = 4 for one part
  3. Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.
    Blue: 5 × 4 = 20

Check. Check the share we were given: 3 × 4 = 12, which matches. ✓

Watch out. The amount you are given belongs to one share, not to the whole. Do not divide by the total number of parts.

Worked example 15.4.2
Ali and Bea share money in the ratio 7 : 4. Ali gets £36 more than Bea. How much does each person get?

Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.

Method. Subtract the ratio numbers to find the difference in parts. The known difference is worth that many parts, so divide to find one part.

  1. Find the difference in parts. The difference between two shares is the number of extra parts that the bigger share has.
    7 − 4 = 3 parts
  2. Find the value of one part. We are told the difference is £36. That is worth 3 parts, so divide by 3.
    £36 ÷ 3 = £12 for one part
  3. Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.
    Ali: 7 × £12 = £84
    Bea: 4 × £12 = £48

Check. Check the difference: 7 × £12 − 4 × £12 = £84 − £48 = £36, which is the difference we were given. ✓

Watch out. Subtract the ratio numbers (e.g. 7 − 4 = 3 parts). Do not add them. The sum gives the total, the subtraction gives the difference.

Worked example 15.4.3
A : B = 2 : 3 and B : C = 4 : 5. Write the ratio A : B : C.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. To combine ratios, make the shared quantity (B) the same number in both ratios, using the LCM, then join them.

  1. Spot the quantity that is in both ratios. B appears in both ratios, so it links them together. To combine, B must stand for the same number of parts in both.
    A : B = 2 : 3
    B : C = 4 : 5
  2. Find the lowest common multiple (LCM) of the two B numbers. We make both B numbers equal to the LCM of 3 and 4.
    Multiples of 3: 3, 6, 9, 12
    Multiples of 4: 4, 8, 12
    LCM = 12
  3. Multiply each ratio so that B becomes 12. Multiplying every part of a ratio by the same number gives an equivalent ratio.
    A : B = 2 : 3  (multiply by 4)  = 8 : 12
    B : C = 4 : 5  (multiply by 3)  = 12 : 15
  4. Write the combined ratio A : B : C. B is now the same in both, so the three numbers can be joined.
    A : B : C = 8 : 12 : 15

Check. Look back at the ratios: in 8 : 12 : 15, the first two numbers (8 and 12) simplify to 2 : 3, which equals 2 : 3. The last two (12 and 15) simplify to 4 : 5, which equals 4 : 5. ✓

Watch out. Match the linking quantity using the LCM. Do not just write the numbers side by side.

Worked example 15.4.4
The ratio of James's money to Kim's money is 3 : 5. The ratio of Kim's money to Lee's money is 2 : 3. Lee has £45. How much does Kim have?

Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.

Method. First combine the two ratios into one three-part ratio. Then use the known share (Lee's £45) to find the value of one part.

  1. Spot the quantity that is in both ratios. Kim appears in both ratios, so it links them together. To combine, Kim must stand for the same number of parts in both.
    James : Kim = 3 : 5
    Kim : Lee = 2 : 3
  2. Find the lowest common multiple (LCM) of the two Kim numbers. We make both Kim numbers equal to the LCM of 5 and 2.
    Multiples of 5: 5, 10
    Multiples of 2: 2, 4, 6, 8, 10
    LCM = 10
  3. Multiply each ratio so that Kim becomes 10. Multiplying every part of a ratio by the same number gives an equivalent ratio.
    James : Kim = 3 : 5  (multiply by 2)  = 6 : 10
    Kim : Lee = 2 : 3  (multiply by 5)  = 10 : 15
  4. Write the combined ratio James : Kim : Lee. Kim is now the same in both, so the three numbers can be joined.
    James : Kim : Lee = 6 : 10 : 15
  5. Write down what we know. We are told the value of one of the shares. That tells us how many equal parts it is worth.
    Lee has 15 parts, worth £45
  6. Find the value of one part. 15 parts are worth £45, so divide by 15.
    £45 ÷ 15 = £3 for one part
  7. Multiply to find what the question asks for. Multiply the value of one part by the number of parts needed.
    Kim: 10 × £3 = £30

Check. James has 6 × £3 = £18. Then James : Kim = 18 : 30, which simplifies to 3 : 5. ✓

Watch out. Kim's number is 10 in the combined ratio, not 5 or 2. Always use the number in the combined ratio.

Watch out

When one share is known, divide by that share's ratio number (3 for red in 3 : 5), not by the total number of parts (8). And when you are given a difference, subtract the ratio numbers; do not add them.

Exercise 15D  Ratio problems

Non-calculator
1
FWork out the missing amount in each case.
(a)The ratio of red to blue is 2 : 3. There are 10 red. How many blue?
(b)The ratio of boys to girls is 3 : 4. There are 20 girls. How many boys?
(c)Sam and Tia share money in the ratio 5 : 2. Sam gets £35. How much does Tia get?
(d)The ratio of fiction to non-fiction books is 1 : 6. There are 42 non-fiction books. How many fiction books?
(e)The ratio of cats to dogs is 4 : 9. There are 28 cats. How many dogs?
(f)The ratio of tea bags to coffee bags is 7 : 5. There are 45 coffee bags. How many tea bags?
2
FRed and blue counters are in the ratio 1 : 2. There are 6 red counters. How many blue counters are there?
3
FA bag has red and blue counters in the ratio 3 : 5. There are 15 red counters. How many counters are in the bag altogether?
4
PSWhite and blue paint are mixed in the ratio 4 : 3. There are 24 litres of white paint. Work out how much blue paint there is, and the total amount of paint.
5
FThe difference between two shares is given. Work out the share asked for.
(a)Shares in the ratio 5 : 2. The bigger share is £12 more than the smaller share. Find the bigger share.
(b)Shares in the ratio 7 : 3. The bigger share is 20 kg more than the smaller. Find the smaller share.
(c)Shares in the ratio 9 : 5. The difference is 28 m. Find the total.
6
PSChris and Dan share money in the ratio 5 : 2. Chris gets £18 more than Dan. How much money is shared altogether?
7
FCombine the ratios to write A : B : C.
(a)A : B = 1 : 2 and B : C = 3 : 4
(b)A : B = 2 : 3 and B : C = 3 : 5
(c)A : B = 4 : 5 and B : C = 10 : 7
(d)A : B = 3 : 4 and B : C = 6 : 5
8
PSIn a fruit bowl the ratio of apples to pears is 2 : 3 and the ratio of pears to bananas is 6 : 5. There are 75 pieces of fruit altogether. How many bananas are there?
9
PSRed, green and blue counters are in the ratio 2 : 3 : 7. There are 21 green counters. How many counters are there altogether?
10
PSThe ratio of Amy's money to Ben's money is 2 : 3. The ratio of Ben's money to Cat's money is 5 : 4. Cat has £48. How much do Amy and Ben have?
11
PSEve and Fay share money in the ratio 8 : 3. Eve gets £35 more than Fay. How much does each person get?
12
PSIn a choir the ratio of sopranos to altos is 5 : 3. There are 6 more sopranos than altos. How many sopranos, altos and singers are there altogether?
13
PSx : y = 3 : 4 and y : z = 2 : 5. When z = 30, work out the value of x.
14
RJo says: “The ratio of cats to dogs is 2 : 3, so there are 5 more dogs than cats.” Explain why Jo cannot be sure that this is right.
15
RBen writes: “A : B = 2 : 3 and B : C = 4 : 5, so A : B : C = 2 : 3 : 5.” Explain his mistake and write the correct ratio A : B : C.
16
RA : B = 2 : 3 and B : C = 4 : 5. The sum of A and C is 46. Work out the value of B.
15.5

Ratios and fractions

Edexcel 1MA1: N11, R8, R6
Grade 3-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Ratios and fractionsComing soon

Ratios and fractions are two ways of describing the same sharing. Think of a class where the ratio of boys to girls is 2 : 3. There are 2 + 3 = 5 equal parts altogether. Boys are 2 of the 5 parts, so boys are 25 of the class. Girls are 3 of the 5 parts, so girls are 35 of the class.

boys: 2/5girls: 3/5whole class = 5 parts
Ratio 2 : 3 means boys are 2/5 and girls are 3/5 of the whole.

From a ratio to a fraction. Add the ratio numbers to get the denominator. Each share is its own ratio number over the total: for a ratio a : b, the first share is a/(a + b) and the second is b/(a + b). The fractions add up to 1 whole. Notice that 2 : 3 does not give the fractions 2/3 and 3/2. The denominator is the total, 5.

From a fraction to a ratio. If 35 of a group are girls, the rest are boys: 1 − 35 = 25. Out of every 5 equal parts, 3 are girls and 2 are boys. So girls : boys = 3 : 2. The numerators become the ratio when the fractions have the same denominator. With several fractions, first write them over a common denominator (Chapter 3). For 12, 13 and the rest: the common denominator is 6, so we have 36, 26 and 16, and the ratio is 3 : 2 : 1.

“Is a fraction of” is a ratio. If A is 34 of B, then A has 3 parts for every 4 parts of B, so A : B = 3 : 4. The first quantity is a fraction (or multiple) of the second. This is the multiplicative relationship from Section 15.2. Using fractions to solve ratio problems: if boys : girls = 3 : 4 and there are 28 students, boys are 37 of 28 = 28 ÷ 7 × 3 = 12.

Key ideas

  • Ratio to fraction: add the parts. Each share is its ratio number over the total. The fractions add to 1.
  • Fraction to ratio: find the rest by subtracting from 1, use a common denominator, then the numerators are the ratio.
  • “A is 34 of B” means A : B = 3 : 4.
  • A fraction of a total: use the fraction as an operator, for example 37 of 28.
Worked example 15.5.1
The ratio of red counters to blue counters is 2 : 5. What fraction of the counters are blue?

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. The total number of parts is the denominator. Each ratio number is the numerator of that share.

  1. Add the parts to find the total. The whole group is made of all the parts together.
    2 + 5 = 7 parts
  2. Write each share as a fraction of the total. A share of 3 parts out of 8 equal parts is the fraction 3 over 8: the part goes on top and the total on the bottom.
    red: 27
    blue: 57
  3. Write the fraction(s) asked for in simplest form. Divide the top and the bottom by any common factor so that the fraction cannot be simplified further.
    blue: 57 (already in simplest form)

Check. The fractions should add up to 1: 27 + 57 = 77 = 1. ✓

Watch out. The denominator is the total number of parts (the sum), not the other ratio number. 2 : 3 means 2/5 and 3/5, not 2/3.

Worked example 15.5.2
25 of the students in a club are boys and the rest are girls. Write the ratio of boys to girls in its simplest form.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. Turn the fractions into pieces of the same size. The numbers of pieces are the parts of the ratio.

  1. Find the fraction that is left over. The whole is 1, so the rest is 1 minus the fraction we know.
    1 − 25 = 35
  2. Use the numerators as the parts of the ratio. If the group is split into 5 equal pieces, each category has that many pieces.
    boys : girls = 2 : 3
  3. Write the ratio in the order of the question. Keep the order of the names the same as in the question.
    boys : girls = 2 : 3

Check. Change back: 2 : 3 has 5 parts, so the first share is 25 = 25, which matches. ✓

Watch out. The rest is not the same as the fraction you were given. Subtract from 1 to find it.

Worked example 15.5.3
12 of a bag of counters are red, 13 are blue and the rest are green. Write the ratio red : blue : green.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. Turn the fractions into pieces of the same size. The numbers of pieces are the parts of the ratio.

  1. Find the fraction that is left over. The whole is 1, so the rest is 1 minus all the fractions we know.
    1 − (12 + 13) = 16
  2. Write all the fractions with the same denominator, 6. Fractions with the same denominator are made of equal-sized pieces, so the numerators are the numbers of pieces.
    red: 12 = 36
    blue: 13 = 26
    green: 16 = 16
  3. Use the numerators as the parts of the ratio. If the group is split into 6 equal pieces, each category has that many pieces.
    red : blue : green = 3 : 2 : 1
  4. Write the ratio in the order of the question. Keep the order of the names the same as in the question.
    red : blue : green = 3 : 2 : 1

Check. Change back: 3 : 2 : 1 has 6 parts, so the first share is 36 = 12, which matches. ✓

Watch out. The rest is not the same as the fraction you were given. Subtract from 1 to find it.

Worked example 15.5.4
Write the ratio A : B when A is 34 of B.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. “A is 34 of B” means that if B is split into 4 equal parts, A is 3 of those parts.

  1. Write the statement as a sum. “Is a fraction of” means multiply.
    A = 34 × B
  2. Think of B as 4 equal parts. The denominator 4 tells us how many equal parts make the whole of B.
    B = 4 parts
  3. Count the parts that A has. The numerator 3 tells us how many of those parts A has.
    A = 3 parts
  4. Write the ratio A : B. The parts are in the same order as in the question.
    3 : 4

Check. Take B = 4. Then A = 34 of 4 = 3, and 3 : 4 agrees. ✓

Watch out. The ratio compares the two quantities, so the denominator is the number of parts of the second quantity, not the total.

Worked example 15.5.5
In a class the ratio of boys to girls is 3 : 4. There are 28 students. How many boys are there?

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. Turn the ratio into a fraction, then take that fraction of 28.

  1. Add the parts. The class is 3 + 4 parts altogether.
    3 + 4 = 7 parts
  2. Write the boys as a fraction. Boys are 3 of the 7 parts.
    37
  3. Find the fraction of 28. Divide 28 by the denominator 7 to find one part, then multiply by the numerator 3.
    28 ÷ 7 = 4
    4 × 3 = 12

Check. Girls = 28 − 12 = 16, and 12 : 16 = 3 : 4 (divide both by 4). ✓

Watch out. Divide by 7 (the total), not by 3.

Watch out

The ratio 2 : 3 gives the fractions 25 and 35, not 23. The denominator is the total number of parts. And when you are given one fraction, remember there is a second share: the rest is 1 minus the fraction.

Exercise 15E  Ratios and fractions

Non-calculator
1
FWrite the first share as a fraction of the total.
(a)Ratio 1 : 3
(b)Ratio 2 : 3
(c)Ratio 3 : 5
(d)Ratio 4 : 5
(e)Ratio 5 : 7
(f)Ratio 7 : 3
2
FThe ratio of red counters to blue counters is 2 : 3. What fraction of the counters are blue?
3
FThree friends share a prize in the ratio 1 : 2 : 3. What fraction of the prize does each friend receive?
4
FWrite each as a ratio in its simplest form.
(a)14 of the members are juniors and the rest are seniors. Juniors : seniors
(b)35 of the cars are red and the rest are blue. Red : blue
(c)29 of the sweets are mints and the rest are toffees. Mints : toffees
(d)58 of the pupils walk and the rest cycle. Walk : cycle
5
FWrite each as a ratio in its simplest form.
(a)12 of the beads are red, 13 are blue and the rest are green. Red : blue : green
(b)14 of the snacks are apples, 16 are bananas and the rest are oranges. Apples : bananas : oranges
6
FWrite A : B as a ratio in its simplest form.
(a)A is 34 of B
(b)A is 56 of B
(c)A is 310 of B
(d)A is 47 of B
7
PSIn a class the ratio of boys to girls is 3 : 5. The class has 32 students. Use fractions to work out how many boys there are.
8
PS25 of the students in a club walk to school and the rest cycle. (a) Write the ratio walk : cycle. (b) There are 35 students. How many cycle?
(a)Ratio walk : cycle
(b)Number who cycle
9
PSSam spends 25 of his money on food and 14 on travel. He saves the rest. Write the ratio food : travel : saved.
10
PS37 of the sweets in a box are red and the rest are blue. There are 12 more blue sweets than red sweets. How many sweets are there altogether?
11
PSAisha gets 25 of a sum of money, Ben gets 310 and Cara gets the rest. Write the ratio Aisha : Ben : Cara in its simplest form.
12
PSAt a show the ratio of adults to children is 2 : 5. There are 350 people altogether. How many adults and how many children are there?
13
RZoe says: “If the ratio of x to y is 3 : 7, then x is 37 of y and also 310 of the total.” Is Zoe right? Explain.
14
RThe ratio of boys to girls in a choir is 1 : 2. Dev says: “So the boys are 12 of the choir.” Explain his mistake. What fraction of the choir are boys?
15
R13 of the counters in a bag are red. The rest are blue or green in the ratio 3 : 5. Work out the ratio red : blue : green.
15.6

Ratio in context: recipes, mixing, concentration and conversion

Edexcel 1MA1: R5, R6
Grade 3-6

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Ratio in context: recipes, mixing, concentration and conversionComing soon

Ratios appear all around us. In each case the idea is the same: multiply or divide every quantity by the same number so that the ratios stay the same.

Recipes (scaling). A recipe for 4 people uses 200 g flour, 80 g sugar and 120 g butter. For 6 people, every amount is multiplied by the same number, 6 ÷ 4 = 1.5. Where the multiplier is awkward, use the unitary idea: work out the amounts for 1 person (divide by 4), then multiply by 6. The ratio flour : sugar : butter stays 5 : 2 : 3, so the recipe tastes the same.

Mixing. Squash is made by mixing cordial and water in the ratio 1 : 4. That is 1 part cordial and 4 parts water, so there are 1 + 4 = 5 parts in the drink. The drink is 15 cordial. It is not 14: the ratio compares cordial with the water, not with the whole drink.

1 cordial4 water5 parts: the drink is 1/5 cordial
Cordial : water = 1 : 4 means the drink is 1/5 cordial and 4/5 water.

Concentration. The strength or concentration of a mixture is the fraction of it that is the active ingredient. A larger fraction means a stronger mixture. A drink mixed 1 : 4 is 15 cordial and one mixed 1 : 5 is only 16 cordial, so the 1 : 4 drink is stronger. Conversion is also a ratio. If £1 = €1.15 then the ratio of pounds to euros is 1 : 1.15, so we multiply pounds by 1.15 to get euros and divide euros by 1.15 to get pounds.

Map scales. A scale of 1 : 25 000 means that 1 cm on the map is 25 000 cm in real life. Since 25 000 cm = 250 m, the real distance is 250 m for every 1 cm. (Scale drawings are studied again in Chapter 22.)

Key ideas

  • To scale a recipe, multiply every ingredient by the same multiplier (new amount ÷ old amount).
  • A mixing ratio a : b has a + b parts in the mixture. The fraction of the mixture that is a is a/(a + b).
  • A larger fraction of the active ingredient means a stronger mixture.
  • Conversion rates are ratios: multiply to change into the other unit, divide to change back.
  • To join two mixtures, work out the actual amounts of each liquid, add them, then form the ratio.
Worked example 15.6.1
A recipe for 4 people uses 200 g of flour, 80 g of sugar, 120 g of butter and 2 eggs. Work out the amounts needed for 6 people.

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. Find the amounts for one (the unitary method), then multiply by the number needed.

  1. Find the amounts for 1 people. The unitary method: divide every amount by the number it serves now.
    Flour: 200 g ÷ 4 = 50 g
    Sugar: 80 g ÷ 4 = 20 g
    Butter: 120 g ÷ 4 = 30 g
    Eggs: 2 ÷ 4 = 0.5
  2. Multiply by 6 for 6 people. Multiply the amount for one by the number needed.
    Flour: 50 g × 6 = 300 g
    Sugar: 20 g × 6 = 120 g
    Butter: 30 g × 6 = 180 g
    Eggs: 0.5 × 6 = 3
  3. Write the amounts needed. These are the new amounts, with the units.
    Flour: 300 g
    Sugar: 120 g
    Butter: 180 g
    Eggs: 3

Check. Check the first ingredient in reverse: 300 ÷ 1.5 = 200, the original amount of flour. ✓

Watch out. Scale every ingredient, not just the main one, and keep the units.

Worked example 15.6.2
Squash is made by mixing cordial and water in the ratio 1 : 4. Zara makes 750 ml of squash. How much cordial and how much water does she use?

Key words. To share in a ratio means to split an amount into equal-sized parts, then give each person the number of parts shown in the ratio. The ratio 3 : 5 means 3 parts for the first person and 5 parts for the second.

Method. Add the parts to find how many equal parts there are, find the value of one part, then multiply.

  1. Add up the parts of the ratio. Each number in the ratio is a number of equal parts. The total number of parts is the sum of the ratio numbers.
    1 + 4 = 5 parts
  2. Find the value of one part. 750 ml is shared into 5 equal parts, so divide it by 5.
    750 ml ÷ 5 = 150 ml for one part
  3. Multiply to find each share. Each person gets as many parts as their ratio number, so multiply the value of one part by that number.
    Cordial: 1 × 150 ml = 150 ml
    Water: 4 × 150 ml = 600 ml

Check. Add the shares: 150 ml + 600 ml = 750 ml, which is the total 750 ml. ✓

Watch out. Do not divide the total by one of the ratio numbers. Divide by the total number of parts.

Worked example 15.6.3
£1 = €1.15. (a) Change £80 to euros. (b) Change €138 to pounds.

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. The rate tells us that each £1 is worth €1.15. To get euros, multiply the pounds by 1.15. To get pounds, divide the euros by 1.15.

  1. Write the rate. The ratio £ : € is 1 : 1.15.
    £1 = €1.15
  2. (a) Multiply the pounds by 1.15. There are 80 lots of £1, each worth €1.15.
    80 × 1.15 = €92
  3. (b) Divide the euros by 1.15. We want to know how many lots of €1.15 are in €138. Each lot is £1.
    138 ÷ 1.15 = £120

Check. £120 × 1.15 = €138, which takes us back to €138. ✓

Watch out. Multiply to go to the bigger-numbered currency here (£ to €) and divide to come back. Check that your answer is sensible: €92 is more than £80, as it should be.

Worked example 15.6.4
The scale of a map is 1 : 25 000. A path measures 4 cm on the map. How long is the path in real life? Give your answer in kilometres.

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. A scale of 1 : 25 000 means every 1 cm on the map is 25 000 cm in real life. Multiply, then change the units.

  1. Multiply the map length by the scale. Each 1 cm is 25 000 cm in real life.
    4 × 25 000 = 100 000 cm
  2. Change centimetres to metres. 100 cm = 1 m, so divide by 100.
    100 000 ÷ 100 = 1000 m
  3. Change metres to kilometres. 1000 m = 1 km, so divide by 1000.
    1000 ÷ 1000 = 1 km

Check. 1 km = 100 000 cm and 100 000 ÷ 25 000 = 4 cm. ✓

Watch out. The answer is in the unit of the map measurement (cm) until you convert it.

Watch out

A mixing ratio such as 1 : 4 means 1 part cordial and 4 parts water, so the cordial is 15 of the drink, not 14. And when you scale a recipe, scale every ingredient by the same number.

Exercise 15F  Ratio in context

Calculator allowed
1
FA recipe for 4 people uses 200 g of flour, 80 g of sugar, 120 g of butter and 2 eggs. Work out the amounts needed for each number of people.
(a)For 8 people
(b)For 12 people
(c)For 2 people
(d)For 6 people
2
FRed and white paint are mixed in the ratio 2 : 5 to make 14 litres of pink paint. How many litres of each colour are used?
3
FSquash is made from squash concentrate and water in the ratio 1 : 4. Jo uses 250 ml of concentrate. How much water does she use and how much squash does she make?
4
PSConcrete is made from cement, sand and gravel in the ratio 1 : 2 : 4. A builder makes 56 kg of concrete. How much of each material does he use?
5
FThe exchange rate is £1 = €1.15. Convert each amount.
(a)£80 to euros
(b)£250 to euros
(c)€138 to pounds
(d)€46 to pounds
6
FA map has a scale of 1 : 50 000.
(a)A road is 6 cm long on the map. How long is it in real life, in km?
(b)A lake is 2.5 km long in real life. How long is it on the map, in cm?
7
FA cleaning solution is made by mixing cleaner and water in the ratio 1 : 9. Work out how much cleaner is needed to make 500 ml of solution.
8
PSDrink A is made by mixing squash and water in the ratio 1 : 4. Drink B is made by mixing squash and water in the ratio 1 : 5. Which drink is stronger? Show how you decide.
9
PSA recipe for 12 cupcakes uses 150 g of butter. Mia has 450 g of butter and plenty of the other ingredients. How many cupcakes can she make?
10
PSA gardener mixes plant food and water in the ratio 1 : 24. She uses 100 ml of plant food. How much water does she use, and how much mixture does she make?
11
PSA T-shirt costs $30 in the USA and £22 in the UK. The exchange rate is £1 = $1.25. In which country is the T-shirt cheaper, and by how much (in pounds)?
12
ROrange paint is made from red and yellow paint in the ratio 3 : 2. Ella has 600 ml of red paint and 500 ml of yellow paint. What is the largest amount of orange paint she can make?
13
PSA school canteen makes 24 glasses of squash, each of 250 ml. The squash is made from concentrate and water in the ratio 1 : 4. How many litres of concentrate are needed?
14
RA jug holds 1.2 litres of squash made with concentrate and water in the ratio 1 : 5. How much water must be added so that the ratio becomes 1 : 8?
15.7

Proportion: the unitary method and best buy

Edexcel 1MA1: R7, R5
Grade 2-5

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Proportion: the unitary method and best buyComing soon

Two quantities are in direct proportion when they increase or decrease at the same rate. If you buy twice as many pens you pay twice as much; if you buy half as many, you pay half as much. The cost of each pen stays the same. Many everyday problems can be solved by finding the value of one first. This is called the unitary method.

Suppose 5 pens cost £3.50 and we want the cost of 8 pens. Step 1: divide to find the cost of 1 pen: £3.50 ÷ 5 = £0.70. Step 2: multiply to find the cost of 8 pens: 8 × £0.70 = £5.60. A table helps you keep track of what you have done.

PensCost
5£3.50
÷ 5 ↓
1		↓ ÷ 5
£0.70
× 8 ↓
8		↓ × 8
£5.60

Best buy and value for money. A bigger pack is not always cheaper per gram, and a lower price does not always give better value. To compare, find the cost of one unit (for example the price of 1 g) for each option, in the same units. The lowest price per unit is the best buy. Another way is to find how much you get for the same money (for example per £1): the most is the best buy.

Offers. Offers such as “3 for the price of 2” are compared in the same way. Work out the total price with the offer, divide by the number of items, and compare the price per item with the normal price. Always check the units: change pounds to pence when that avoids decimals, and make sure both quantities use the same unit (g or kg, ml or litres).

penscost5£3.501£0.70÷ 5÷ 5
Unitary method, step 1: divide both by 5 to find the cost of 1 pen.

Key ideas

  • Direct proportion: if one quantity is multiplied by a number, the other is multiplied by the same number.
  • Unitary method: divide to find the value of 1, then multiply to find the value of the amount you want.
  • Best buy: find the price of 1 unit (or the amount for the same money) for every option, using the same units. Lowest price per unit (or most per £1) wins.
  • Offers: find the cost per item with the offer, then compare.
Worked example 15.7.1
5 pens cost £3.50. Work out the cost of 8 pens.

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.

  1. Write down what we know. We know the value for one amount, and want it for a different amount.
    5 pens cost £3.50
  2. Divide to find the value of one. This is the unitary method: find the value for 1 pen first. Divide the total by the amount it was for.
    £3.50 ÷ 5 = £0.70 for 1 pen
  3. Multiply to find the value for 8. 8 is 8 times 1 pen, so multiply the value for 1 pen by 8.
    8 × £0.70 = £5.60

Check. A second way (the multiplier): 8 ÷ 5 = 1.6, so the value is £3.50 × 1.6 = £5.60. ✓

Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.

Worked example 15.7.2
A car uses 6 litres of petrol to travel 90 km. How many litres does it need to travel 240 km?

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. The unitary method: divide to find the value of one, then multiply to find the value of the number you want.

  1. Write down what we know. We know the value for one amount, and want it for a different amount.
    90 km needs 6 litres
  2. Divide to find the value of one. This is the unitary method: find the value for 1 km first. Divide the total by the amount it was for.
    6 litres ÷ 90 = 115 litres for 1 km
  3. Multiply to find the value for 240 km. 240 km is 240 times 1 km, so multiply the value for 1 km by 240.
    240 × 115 litres = 16 litres

Check. A second way (the multiplier): 240 ÷ 90 = 223, so the value is 6 litres × 223 = 16 litres. ✓

Watch out. Divide first to find the value of one, then multiply. Multiplying the original numbers together gives nonsense.

Worked example 15.7.3
Cereal A has 500 g and costs £1.60. Cereal B has 750 g and costs £2.25. Which is the better buy?

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.

  1. Write both prices in pence. Comparing is easier when both prices are in the same unit.
    £1.60 = 160p
    £2.25 = 225p
  2. Work out the cost of 1 g for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.
    Cereal A: 160p ÷ 500 = 0.32p per 1 g
    Cereal B: 225p ÷ 750 = 0.3p per 1 g
  3. Compare and choose. Cost per unit: the lower the cost, the better the value.
    0.32p , 0.3p
    The lowest is Cereal B at 0.3p per 1 g, so Cereal B is the best buy.

Check. A second way: how much do you get for £1 (100p)? Cereal A: 500 ÷ 160 × 100 = 312.5 g , Cereal B: 750 ÷ 225 × 100 = 33313 g. Cereal B gives the most for £1. ✓

Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.

Worked example 15.7.4
A shop sells shampoo in three sizes: Small 250 ml for £1.80, Medium 400 ml for £2.80 and Large 750 ml for £4.95. Which size is the best value for money?

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. Work out the price of one unit (or the same quantity) for each option. The lowest price per unit is the best value.

  1. Write both prices in pence. Comparing is easier when both prices are in the same unit.
    £1.80 = 180p
    £2.80 = 280p
    £4.95 = 495p
  2. Work out the cost of 1 ml for each option. Dividing the price by the quantity gives the cost per single unit. Then they can be compared fairly.
    Small: 180p ÷ 250 = 0.72p per 1 ml
    Medium: 280p ÷ 400 = 0.7p per 1 ml
    Large: 495p ÷ 750 = 0.66p per 1 ml
  3. Compare and choose. Cost per unit: the lower the cost, the better the value.
    0.72p , 0.7p , 0.66p
    The lowest is Large at 0.66p per 1 ml, so Large is the best buy.

Check. A second way: how much do you get for £1 (100p)? Small: 250 ÷ 180 × 100 = 13889 ml , Medium: 400 ÷ 280 × 100 = 14267 ml , Large: 750 ÷ 495 × 100 = 1511733 ml. Large gives the most for £1. ✓

Watch out. The biggest pack is not always the best value, and the lowest price is not always the best value. Compare the price per unit.

Worked example 15.7.5
Shampoo costs £2.40 per bottle. A shop has an offer: “3 bottles for the price of 2”. How much does each bottle cost with the offer, and how much is saved on each bottle?

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. With the offer, 3 bottles cost the same as 2 bottles. Work out that price, divide by 3, and compare.

  1. Find the offer price for 3 bottles. You pay for 2 bottles only.
    2 × £2.40 = £4.80
  2. Find the price of 1 bottle with the offer. Divide by the 3 bottles that you get.
    £4.80 ÷ 3 = £1.60
  3. Find the saving on each bottle. Subtract the offer price from the normal price.
    £2.40 − £1.60 = £0.80

Check. Without the offer 3 bottles cost 3 × £2.40 = £7.20. With the offer they cost £4.80. The saving £2.40 ÷ 3 = £0.80 per bottle. ✓

Watch out. “3 for the price of 2” does not mean the bottles are 2/3 of the price each; work out the total first.

Watch out

Do not compare prices when the quantities are different. A pack that costs less may hold much less. Also, check that both quantities are in the same unit: 1 kg and 500 g are not “1 and 500”.

Exercise 15G  Unitary method and best buy

Calculator allowed
1
FWork out the cost.
(a)4 pens cost £2.80. Find the cost of 7 pens.
(b)6 buns cost £4.50. Find the cost of 10 buns.
(c)5 kg of apples cost £12. Find the cost of 8 kg.
(d)12 eggs cost £3. Find the cost of 18 eggs.
(e)3 books cost £14.25. Find the cost of 5 books.
(f)8 metres of cloth cost £34. Find the cost of 5 metres.
2
F3 pencils cost 90p. Work out the cost of 5 pencils.
3
FA car uses 6 litres of petrol to travel 90 km. How many litres does it need to travel 240 km?
4
FMaya types 150 words in 3 minutes. How many words can she type in 8 minutes at the same speed?
5
PSA printer prints 120 pages in 8 minutes. How long will it take to print 300 pages?
6
FWhich is the better buy, A or B? Show your working.
(a)A: 500 g for £1.60. B: 750 g for £2.25.
(b)A: 2 kg of rice for £3.20. B: 5 kg of rice for £7.50.
(c)A: 1 litre of juice for £1.20. B: 1.5 litres for £1.65.
(d)A: 80 tea bags for £2.40. B: 240 tea bags for £6.60.
7
PSCola is sold in a 6-pack of cans for £2.70 and in a 10-pack for £4.20. Which pack is better value for money?
8
PSCoffee is sold in three sizes: A is 250 g for £1.20, B is 400 g for £1.80 and C is 1 kg for £4.20. Which size is the best value for money?
9
PSA shop sells a 200 g bar of chocolate for 90p and a 500 g bar for £2.50. Which bar gives you more chocolate for your money? How many grams of each do you get for £1?
10
PSZain earns £17.50 for 7 hours of work. At the same rate of pay, how much does he earn in 12 hours?
11
PSA recipe for 12 biscuits uses 90 g of butter. Butter costs £1.80 for a 250 g block. How much does the butter cost for 40 biscuits? (Work out the cost for the exact amount of butter.)
12
PSTickets for a concert are the same price each. 2 tickets cost £7. Complete the table: (a) the cost of 5 tickets, (b) the cost of 8 tickets, (c) the number of tickets that can be bought for £42.
Tickets258?
Cost£7??£42
(a)Cost of 5 tickets
(b)Cost of 8 tickets
(c)Number of tickets for £42
13
RLarge packs are always better value than small packs. Is this true? Use the prices in the previous question (200 g for 90p and 500 g for £2.50) to explain.
14
RRaj says: “5 notebooks cost £7.50, so 15 notebooks cost £7.50 + £10 = £17.50, because you buy 10 more.” Explain his mistake and work out the correct cost.
15
RA shop sells pens in packs. Pack A has 6 pens for £2.10. Pack B has 10 pens for £3.20. Sara needs exactly 30 pens. What is the cheapest way to buy exactly 30 pens, and how much does it cost? Show how you decide.
15.8

Proportion as equality of ratios; ratio and linear functions

Edexcel 1MA1: R7, R8
Grade 4-7

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Proportion as equality of ratios; ratio and linear functionsComing soon

Two ratios are equal (or “in proportion”) when they simplify to the same ratio. 2 : 3 and 6 : 9 are equal because 6 : 9 = 2 : 3 after dividing both parts by 3. We write 2 : 3 = 6 : 9. The second ratio is the first multiplied by 3. To find a missing number in equal ratios, find this multiplier from the pair where both numbers are known, and use it on the other pair.

firstsecond351220× 4× 4
3 : 5 = 12 : 20. The multiplier is 12 ÷ 3 = 4, so the missing number is 5 × 4 = 20.

Checking whether ratios are equal. Simplify both ratios and compare, or cross-multiply: the ratios a : b and c : d are equal if a × d = b × c. For 6 : 9 and 10 : 15 we get 6 × 15 = 90 and 9 × 10 = 90, so they are equal. Remember that equal ratios come from multiplying or dividing both parts, never from adding the same number to both.

Direct proportion and tables. If y is in direct proportion to x, then y ÷ x is the same number every time. Call this number k. For example, if cost is directly proportional to tickets and 2 tickets cost £7, then k = 7 ÷ 2 = 3.5 and every pair has y ÷ x = 3.5. The ratio y : x is always k : 1.

Ratio and linear functions. Since y ÷ x = k, we have y = kx. This is a linear function. Its graph is a straight line through the origin (0, 0), and the gradient of the line is k (Chapter 12). For a conversion graph from miles to kilometres, k = 1.6 and the line is y = 1.6x. A line such as y = 2x + 3 is also straight but does not go through the origin, so it does not show direct proportion: doubling x does not double y.

mileskm12345678910246810121416(5, 8)(10, 16)
Direct proportion: the graph of y = 1.6x is a straight line through the origin. k = 1.6 is the gradient.
xy1234563691215
Blue: y = 2x is direct proportion (through the origin). Red: y = 2x + 3 is straight, but it is not direct proportion because it does not pass through (0, 0).

Key ideas

  • Equal ratios: multiply or divide both parts by the same number. Find the multiplier from the known pair.
  • Test equality by simplifying both ratios, or by cross-multiplying: a : b = c : d if ad = bc.
  • Direct proportion: y ÷ x is always the same number k. y = kx.
  • The graph of y = kx is a straight line through the origin with gradient k.
  • A straight line that does not pass through the origin is not direct proportion.
Worked example 15.8.1
Find the missing number in the equal ratios 3 : 5 = 12 : ?

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.

  1. Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.
    3 : 5 = 12 : ?
  2. Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.
    12 ÷ 3 = 4
  3. Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.
    5 × 4 = 20

Check. Divide the new ratio by 4: 12 ÷ 4 = 3 and 20 ÷ 4 = 5, giving 3 : 5 again. ✓

Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.

Worked example 15.8.2
Are the ratios 6 : 9 and 10 : 15 equal?

Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.

Method. Simplify both ratios. If they are the same, the ratios are equal (the quantities are in proportion).

  1. Simplify the first ratio. Two ratios are equal when they simplify to the same ratio.
    6 : 9 = 2 : 3
  2. Simplify the second ratio. Use the same method: divide by the highest common factor.
    10 : 15 = 2 : 3
  3. Compare. Look at the two simplest forms.
    2 : 3 and 2 : 3 are the same, so the ratios are equal.

Check. Cross-multiply as a check: 6 × 15 = 90 and 9 × 10 = 90. They are equal. ✓

Watch out. Adding the same number to both parts changes a ratio. Equal ratios come from multiplying or dividing both parts, not adding.

Tickets (x)258
Cost in £ (y)717.528
Worked example 15.8.3
Show that the cost of tickets is directly proportional to the number of tickets, using this table, and write the rule connecting cost y (£) and tickets x.

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. For direct proportion, y ÷ x must be the same for every pair of values. Work out y ÷ x for each column.

  1. Divide y by x for each pair. If they are all the same number, the quantities are in direct proportion.
    7 ÷ 2 = 3.5
    17.5 ÷ 5 = 3.5
    28 ÷ 8 = 3.5
  2. State the constant k. All three results are the same, so the cost is directly proportional to the number of tickets, and k = 3.5.
    k = 3.5
  3. Write the rule. Use y = kx.
    y = 3.5x

Check. Substitute x = 8: y = 3.5 × 8 = 28, which matches the table. ✓

Watch out. A table with a constant difference in y is not necessarily proportional. It must have a constant ratio y ÷ x.

Worked example 15.8.4
y is directly proportional to x. When x = 4, y = 10. Find y when x = 14.

Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.

Method. For direct proportion, write y = kx, use the known pair to find k, then use the equation.

  1. Write down the rule for direct proportion. If y is directly proportional to x, then y is always the same multiple of x. That multiple is called k, the constant of proportionality.
    y = kx
  2. Use the pair of values we know to find k. Substitute x and y into y = kx, then divide to get k on its own.
    10 = k × 4
    k = 10 ÷ 4 = 2.5
  3. Write the rule with the value of k. Now we have an equation we can use for any pair of values.
    y = 2.5x
  4. Substitute x = 14. Replace x by the number and multiply.
    y = 2.5 × 14 = 35

Check. 14 is 3.5 times 4, so y should be 3.5 times 10 = 35. ✓

Watch out. k is found by dividing y by x (y ÷ x), not x by y.

Watch out

A straight-line graph is not enough to show direct proportion. It must pass through the origin (0, 0). Also, y = kx means y ÷ x = k. Do not work out x ÷ y by mistake.

Exercise 15H  Equal ratios and direct proportion

Non-calculator
1
FFind the missing number in each pair of equal ratios.
(a)2 : 3 = 8 : ?
(b)5 : 4 = 20 : ?
(c)3 : 7 = ? : 21
(d)4 : 9 = ? : 36
(e)6 : 4 = 9 : ?
(f)10 : 6 = 4 : ?
2
FAre the two ratios equal? Write yes or no and show how you know.
(a)4 : 6 and 10 : 15
(b)3 : 5 and 9 : 20
(c)8 : 12 and 6 : 9
(d)5 : 7 and 15 : 22
3
PSA recipe uses 3 cups of rice for every 5 cups of water. How many cups of water are needed for 12 cups of rice?
4
PSFlour and sugar are in the ratio 5 : 3 in a recipe. Sam uses 450 g of flour. How much sugar does he need?
5
RThe table shows the cost y (£) of x tickets. Show that y is directly proportional to x, and write a rule connecting y and x.
Tickets (x)258
Cost in £ (y)717.528
6
Fy is directly proportional to x. Use the information to find the missing value.
(a)x = 4, y = 10. Find y when x = 14
(b)x = 6, y = 15. Find y when x = 10
(c)x = 5, y = 2. Find x when y = 8
(d)x = 8, y = 36. Find x when y = 54
7
PSThe mass M grams of a piece of steel is directly proportional to its volume V cm³. A piece with volume 5 cm³ has mass 39 g. Work out the mass of a piece with volume 8 cm³.
8
PSThe graph shows the cost y (£) of hiring a bike for x hours. The line is straight and passes through the origin.
xy123456785101520(4, 10)(8, 20)
(a)Work out the gradient of the line (the hourly cost).
(b)Write the equation of the line in the form y = kx.
(c)Find the cost of hiring the bike for 10 hours.
(d)For how many hours can the bike be hired for £35?
9
RWhich of these equations show that y is directly proportional to x?   y = 3x    y = x + 3    y = x/2    y = 5x − 1
10
RTwo tables show values of x and y. In which table is y directly proportional to x? Table P: x = 5, 10, 15 and y = 12, 24, 36. Table Q: x = 5, 10, 15 and y = 12, 22, 32.
11
PSThe exchange rate is £1 = $1.25. (a) Change £40 to dollars. (b) Change $100 to pounds.
(a)£40 to dollars
(b)$100 to pounds
12
PSA train fare depends on the distance travelled. The fare for 8 km is £5.60 and the fare for 20 km is £14. (a) Show that the fare is directly proportional to the distance. (b) Work out the fare for 35 km.
(a)Show that the fare is directly proportional to the distance
(b)Fare for 35 km
13
RA taxi charges £3 plus £2 for each kilometre. Is the cost directly proportional to the distance? Give a reason.
14
RExplain how you can tell from a graph that two quantities are in direct proportion.
15
RA photograph is 15 cm wide and 10 cm high. It is enlarged so that its width is 24 cm and it keeps the same shape. (a) Find the height of the enlargement. (b) Another enlargement has a height of 25 cm. Find its width. (c) Show that a copy that is 24 cm by 20 cm is not in proportion to the original.
(a)Height when the width is 24 cm
(b)Width when the height is 25 cm
(c)Show that 24 cm by 20 cm is not in proportion

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