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GCSE Maths · EdexcelChapter 15 · Ratio and proportion
GCSE Maths · Ratio and Proportion · Higher · Lesson 15.2

GCSE Maths: Ratio notation and simplifying ratios (Higher)

Part of the GCSE Maths ratio and proportion chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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15.2

Ratio notation and simplifying ratios

Edexcel 1MA1: R4, R6
Grade 1-5

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
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
2
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
3
FWrite each three-part ratio in its simplest form.
(a)6 : 9 : 12
(b)20 : 30 : 50
(c)8 : 12 : 20
4
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
5
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
6
FWrite each ratio in its simplest form. These ratios contain fractions.
(a)12 : 34
(b)23 : 56
(c)14 : 16
(d)35 : 110
7
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
8
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.
9
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?
10
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
11
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.
12
RDev writes “1 m : 50 cm = 1 : 50”. Explain his mistake and write the ratio in its simplest form.
13
RWrite down three different ratios that are equivalent to 2 : 5.
14
RThree of these ratios are equivalent to each other. Which one is the odd one out?   3 : 4    6 : 8    9 : 12    12 : 18
15
FWrite the ratio 0.125 : 34 in its simplest form.
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

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