GCSE Maths: Ratios and fractions (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)
Ratios and fractions
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.
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.
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.
- Add the parts to find the total. The whole group is made of all the parts together.2 + 5 = 7 parts
- 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 - 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.
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.
- Find the fraction that is left over. The whole is 1, so the rest is 1 minus the fraction we know.1 − 25 = 35
- 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
- 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.
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.
- 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
- 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 - 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
- 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.
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.
- Write the statement as a sum. “Is a fraction of” means multiply.A = 34 × B
- Think of B as 4 equal parts. The denominator 4 tells us how many equal parts make the whole of B.B = 4 parts
- Count the parts that A has. The numerator 3 tells us how many of those parts A has.A = 3 parts
- 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.
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.
- Add the parts. The class is 3 + 4 parts altogether.3 + 4 = 7 parts
- Write the boys as a fraction. Boys are 3 of the 7 parts.37
- 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-calculatorKey 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.
- Add the parts to find the total. The whole group is made of all the parts together.1 + 3 = 4 parts
- 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.first: 14
second: 34 - 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.first: 14 (already in simplest form)
Check. The fractions should add up to 1: 14 + 34 = 44 = 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.
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.
- Add the parts to find the total. The whole group is made of all the parts together.2 + 3 = 5 parts
- 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.first: 25
second: 35 - 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.first: 25 (already in simplest form)
Check. The fractions should add up to 1: 25 + 35 = 55 = 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.
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.
- Add the parts to find the total. The whole group is made of all the parts together.3 + 5 = 8 parts
- 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.first: 38
second: 58 - 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.first: 38 (already in simplest form)
Check. The fractions should add up to 1: 38 + 58 = 88 = 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.
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.
- Add the parts to find the total. The whole group is made of all the parts together.4 + 5 = 9 parts
- 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.first: 49
second: 59 - 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.first: 49 (already in simplest form)
Check. The fractions should add up to 1: 49 + 59 = 99 = 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.
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.
- Add the parts to find the total. The whole group is made of all the parts together.5 + 7 = 12 parts
- 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.first: 512
second: 712 - 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.first: 512 (already in simplest form)
Check. The fractions should add up to 1: 512 + 712 = 1212 = 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.
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.
- Add the parts to find the total. The whole group is made of all the parts together.7 + 3 = 10 parts
- 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.first: 710
second: 310 - 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.first: 710 (already in simplest form)
Check. The fractions should add up to 1: 710 + 310 = 1010 = 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.
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.
- Add the parts to find the total. The whole group is made of all the parts together.2 + 3 = 5 parts
- 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: 25
blue: 35 - 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: 35 (already in simplest form)
Check. The fractions should add up to 1: 25 + 35 = 55 = 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.
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.
- Add the parts to find the total. The whole group is made of all the parts together.1 + 2 + 3 = 6 parts
- 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.first friend: 16
second friend: 26 = 13
third friend: 36 = 12 - 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.first friend: 16 (already in simplest form)
second friend: 26 = 13
third friend: 36 = 12
Check. The fractions should add up to 1: 16 + 13 + 12 = 66 = 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.
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.
- Find the fraction that is left over. The whole is 1, so the rest is 1 minus the fraction we know.1 − 14 = 34
- Use the numerators as the parts of the ratio. If the group is split into 4 equal pieces, each category has that many pieces.juniors : seniors = 1 : 3
- Write the ratio in the order of the question. Keep the order of the names the same as in the question.juniors : seniors = 1 : 3
Check. Change back: 1 : 3 has 4 parts, so the first share is 14 = 14, which matches. ✓
Watch out. The rest is not the same as the fraction you were given. Subtract from 1 to find it.
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.
- Find the fraction that is left over. The whole is 1, so the rest is 1 minus the fraction we know.1 − 35 = 25
- Use the numerators as the parts of the ratio. If the group is split into 5 equal pieces, each category has that many pieces.red : blue = 3 : 2
- Write the ratio in the order of the question. Keep the order of the names the same as in the question.red : blue = 3 : 2
Check. Change back: 3 : 2 has 5 parts, so the first share is 35 = 35, which matches. ✓
Watch out. The rest is not the same as the fraction you were given. Subtract from 1 to find it.
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.
- Find the fraction that is left over. The whole is 1, so the rest is 1 minus the fraction we know.1 − 29 = 79
- Use the numerators as the parts of the ratio. If the group is split into 9 equal pieces, each category has that many pieces.mints : toffees = 2 : 7
- Write the ratio in the order of the question. Keep the order of the names the same as in the question.mints : toffees = 2 : 7
Check. Change back: 2 : 7 has 9 parts, so the first share is 29 = 29, which matches. ✓
Watch out. The rest is not the same as the fraction you were given. Subtract from 1 to find it.
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.
- Find the fraction that is left over. The whole is 1, so the rest is 1 minus the fraction we know.1 − 58 = 38
- Use the numerators as the parts of the ratio. If the group is split into 8 equal pieces, each category has that many pieces.walk : cycle = 5 : 3
- Write the ratio in the order of the question. Keep the order of the names the same as in the question.walk : cycle = 5 : 3
Check. Change back: 5 : 3 has 8 parts, so the first share is 58 = 58, which matches. ✓
Watch out. The rest is not the same as the fraction you were given. Subtract from 1 to find it.
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.
- 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
- 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 - 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
- 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.
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.
- Find the fraction that is left over. The whole is 1, so the rest is 1 minus all the fractions we know.1 − (14 + 16) = 712
- Write all the fractions with the same denominator, 12. Fractions with the same denominator are made of equal-sized pieces, so the numerators are the numbers of pieces.apples: 14 = 312
bananas: 16 = 212
oranges: 712 = 712 - Use the numerators as the parts of the ratio. If the group is split into 12 equal pieces, each category has that many pieces.apples : bananas : oranges = 3 : 2 : 7
- Write the ratio in the order of the question. Keep the order of the names the same as in the question.apples : bananas : oranges = 3 : 2 : 7
Check. Change back: 3 : 2 : 7 has 12 parts, so the first share is 312 = 14, which matches. ✓
Watch out. The rest is not the same as the fraction you were given. Subtract from 1 to find it.
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.
- Write the statement as a sum. “Is a fraction of” means multiply.A = 34 × B
- Think of B as 4 equal parts. The denominator 4 tells us how many equal parts make the whole of B.B = 4 parts
- Count the parts that A has. The numerator 3 tells us how many of those parts A has.A = 3 parts
- 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.
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 56 of B” means that if B is split into 6 equal parts, A is 5 of those parts.
- Write the statement as a sum. “Is a fraction of” means multiply.A = 56 × B
- Think of B as 6 equal parts. The denominator 6 tells us how many equal parts make the whole of B.B = 6 parts
- Count the parts that A has. The numerator 5 tells us how many of those parts A has.A = 5 parts
- Write the ratio A : B. The parts are in the same order as in the question.5 : 6
Check. Take B = 6. Then A = 56 of 6 = 5, and 5 : 6 agrees. ✓
Watch out. The ratio compares the two quantities, so the denominator is the number of parts of the second quantity, not the total.
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 310 of B” means that if B is split into 10 equal parts, A is 3 of those parts.
- Write the statement as a sum. “Is a fraction of” means multiply.A = 310 × B
- Think of B as 10 equal parts. The denominator 10 tells us how many equal parts make the whole of B.B = 10 parts
- Count the parts that A has. The numerator 3 tells us how many of those parts A has.A = 3 parts
- Write the ratio A : B. The parts are in the same order as in the question.3 : 10
Check. Take B = 10. Then A = 310 of 10 = 3, and 3 : 10 agrees. ✓
Watch out. The ratio compares the two quantities, so the denominator is the number of parts of the second quantity, not the total.
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 47 of B” means that if B is split into 7 equal parts, A is 4 of those parts.
- Write the statement as a sum. “Is a fraction of” means multiply.A = 47 × B
- Think of B as 7 equal parts. The denominator 7 tells us how many equal parts make the whole of B.B = 7 parts
- Count the parts that A has. The numerator 4 tells us how many of those parts A has.A = 4 parts
- Write the ratio A : B. The parts are in the same order as in the question.4 : 7
Check. Take B = 7. Then A = 47 of 7 = 4, and 4 : 7 agrees. ✓
Watch out. The ratio compares the two quantities, so the denominator is the number of parts of the second quantity, not the total.
Method. Turn the ratio into a fraction of the whole class, then find that fraction of 32.
- Add the parts. There are 3 + 5 parts altogether.3 + 5 = 8
- Write the boys as a fraction of the class. Boys are 3 of the 8 parts.38
- Find the fraction of 32. Divide by the denominator 8, then multiply by the numerator 3.32 ÷ 8 = 4
4 × 3 = 12
Check. Girls = 32 − 12 = 20 and 12 : 20 = 3 : 5. ✓
Watch out. Divide by the total number of parts (8).
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.
- Find the fraction that is left over. The whole is 1, so the rest is 1 minus the fraction we know.1 − 25 = 35
- Use the numerators as the parts of the ratio. If the group is split into 5 equal pieces, each category has that many pieces.walk : cycle = 2 : 3
- Write the ratio in the order of the question. Keep the order of the names the same as in the question.walk : cycle = 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.
Method. Cyclists are the rest, which is 35 of the students (the ratio is 2 : 3 so the cyclists are 3 of the 5 parts).
- Find the fraction who cycle. The whole is 1, so subtract the walkers.1 − 25 = 35
- Find the fraction of 35. Divide by the denominator, then multiply by the numerator.35 ÷ 5 = 7
7 × 3 = 21
Check. Walkers: 7 × 2 = 14 and 14 + 21 = 35. ✓
Watch out. The cyclists are the rest, not the 2 parts.
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.
- Find the fraction that is left over. The whole is 1, so the rest is 1 minus all the fractions we know.1 − (25 + 14) = 720
- Write all the fractions with the same denominator, 20. Fractions with the same denominator are made of equal-sized pieces, so the numerators are the numbers of pieces.food: 25 = 820
travel: 14 = 520
saved: 720 = 720 - Use the numerators as the parts of the ratio. If the group is split into 20 equal pieces, each category has that many pieces.food : travel : saved = 8 : 5 : 7
- Write the ratio in the order of the question. Keep the order of the names the same as in the question.food : travel : saved = 8 : 5 : 7
Check. Change back: 8 : 5 : 7 has 20 parts, so the first share is 820 = 25, which matches. ✓
Watch out. The rest is not the same as the fraction you were given. Subtract from 1 to find it.
Method. Find the ratio of red to blue, then use the difference in parts.
- Find the fraction that are blue. The rest are blue.1 − 37 = 47
- Write the ratio red : blue. The numerators over the same denominator 7 give the ratio.3 : 4
- Find the difference in parts. Blue has more parts than red.4 − 3 = 1 part
- Find the value of one part. 1 part is worth the 12 more blue sweets.1 part = 12 sweets
- Find the total. There are 3 + 4 = 7 parts altogether.7 × 12 = 84
Check. Red = 3 × 12 = 36 and blue = 4 × 12 = 48. 48 − 36 = 12, and 3684 = 37. ✓
Watch out. One part is worth the difference (12), not the total.
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.
- Find the fraction that is left over. The whole is 1, so the rest is 1 minus all the fractions we know.1 − (25 + 310) = 310
- Write all the fractions with the same denominator, 10. Fractions with the same denominator are made of equal-sized pieces, so the numerators are the numbers of pieces.Aisha: 25 = 410
Ben: 310 = 310
Cara: 310 = 310 - Use the numerators as the parts of the ratio. If the group is split into 10 equal pieces, each category has that many pieces.Aisha : Ben : Cara = 4 : 3 : 3
- Write the ratio in the order of the question. Keep the order of the names the same as in the question.Aisha : Ben : Cara = 4 : 3 : 3
Check. Change back: 4 : 3 : 3 has 10 parts, so the first share is 410 = 25, which matches. ✓
Watch out. The rest is not the same as the fraction you were given. Subtract from 1 to find it.
Method. Write the ratio as fractions of the total, then find each fraction of 350.
- Add the parts. There are 2 + 5 parts altogether.2 + 5 = 7
- Write each share as a fraction. Adults are 2 of the 7 parts and children are 5 of the 7 parts.Adults 27, children 57
- Find one seventh of 350. Divide by 7.350 ÷ 7 = 50
- Multiply to find each number. Adults are 2 parts and children are 5 parts.Adults: 2 × 50 = 100
Children: 5 × 50 = 250
Check. 100 + 250 = 350. ✓
Watch out. Check that your answers add up to the total.
Method. Test both statements with actual numbers in the ratio 3 : 7.
- Choose numbers in the ratio. Take x = 3 and y = 7. Then x : y = 3 : 7.x = 3, y = 7
- Compare x with y. x as a fraction of y is part over y.x ÷ y = 37
- Find the total. The total of x and y is the whole group.x + y = 3 + 7 = 10
- Compare x with the total. x as a fraction of the total.x ÷ total = 310
- Conclude. Both statements are correct, because x is 3 parts, y is 7 parts and the total is 10 parts.Zoe is right.
Check. Try x = 6, y = 14: 6/14 = 3/7 and 6/20 = 3/10. ✓
Watch out. “Of y” means y is the whole. “Of the total” means x + y is the whole.
Method. Add the ratio numbers to find the total number of parts. The fraction of boys has that total as its denominator.
- Add the parts. Boys are 1 part and girls are 2 parts, so the choir is 3 parts.1 + 2 = 3
- Write boys over the total. Boys are 1 of the 3 equal parts.13
- Explain Dev's mistake. Dev has used the number of girls as the denominator. 12 means boys are half the choir, but there are twice as many girls as boys.Boys are 13 of the choir (not 12).
Check. 13 boys + 23 girls = 1. ✓
Watch out. Boys are half as many as girls (1 : 2), but they are one third of the choir.
Method. Find what fraction of the whole school each group of footballers is, then write them as a ratio with a common denominator.
- Find the fraction of the school that are boys. The rest are boys.1 − 35 = 25
- Girls who play football. A fraction of a fraction: multiply.14 × 35 = 320
- Boys who play football. Multiply again.13 × 25 = 215
- Write them with a common denominator. The LCM of 20 and 15 is 60.320 = 960 and 215 = 860
- Write the ratio. The numerators give the ratio. 9 and 8 have no common factor.9 : 8
Check. Take 60 pupils: 36 girls, 24 boys. 9 girls and 8 boys play football (1/4 of 36 = 9, 1/3 of 24 = 8). ✓
Watch out. Multiply the fractions first. Do not add them.
Method. Find the fraction that are not red, split that in the ratio 3 : 5, then put all three fractions over the same denominator.
- Find the fraction that are not red. Subtract from 1.1 − 13 = 23
- Split into blue and green. Blue and green share 23 in the ratio 3 : 5, which is 8 parts.Blue: 38 of 23 = 14
Green: 58 of 23 = 512 - Use a common denominator. The LCM of 3, 4 and 12 is 12.Red 412, blue 312, green 512
- Write the ratio. The numerators give the ratio.4 : 3 : 5
Check. 4 + 3 + 5 = 12, and red is 412 = 13. Blue : green = 3 : 5. ✓
Watch out. Blue and green are fractions of the remaining two thirds, not of the whole.