1. What a ratio is
Reading a ratio
- A ratio compares the sizes of two or more quantities, often the parts that together make up a whole
- The quantities are separated by a colon, and two or three parts are usual
- 4 : 5 and 3 : 4 : 5 are both ratios
- Order follows the question: whichever quantity is named first belongs to the first number
- Cement and sand mixed 1 : 6 means one part cement to six parts sand
- Labelling each part in the working prevents the two quantities being swapped later
Ratios and fractions
- Each number counts parts, and the whole is the sum of the parts
- In 3 : 5 there are 8 parts altogether
- A fraction compares a part against the whole, whereas a ratio compares parts against each other
- Where red and blue counters are in the ratio 3 : 5, the fraction that is red is 3/8
- The total never appears in the ratio itself, which is the usual source of error
- 1 : 4 is not another way of writing 1/4, because 1 : 4 means one part in five
2. Equivalent and simplified ratios
Equivalent ratios
- Multiplying or dividing every part by the same number leaves the comparison unchanged
- 2 : 3 is equivalent to 8 : 12 and to 200 : 300
- Scaling exists to make the numbers meaningful: a recipe in the ratio 3 : 2 is useful as 300 g : 200 g
Simplest form
- A ratio is in its simplest form once no whole number above 1 divides every part
- Divide through by the highest common factor of all the parts
- 6 : 9 has highest common factor 3, giving 2 : 3
- 18 : 24 : 30 has highest common factor 6, giving 3 : 4 : 5
- With three parts the factor must divide all three, not merely two of them
Worked example
Simplifying a ratio, including one with units
Write 45 : 60 : 75 in its simplest form, and write 400 m : 1.2 km in its simplest form.
Solution:
- For the first, find a factor that divides all three parts
- 15 divides 45, 60 and 75
- 45 ÷ 15 = 3, 60 ÷ 15 = 4, 75 ÷ 15 = 5, giving 3 : 4 : 5
- Check nothing more divides all three: 3, 4 and 5 share no factor above 1
- For the second, the units differ, so convert to the same unit first
- 1.2 km = 1200 m, so the ratio is 400 : 1200
- Divide both parts by 400: 1 : 3
- Ignoring the units would have given 400 : 1.2, which compares metres with kilometres and is meaningless
3. Dividing a quantity in a given ratio
The method
- Add the parts so the whole is expressed as a number of parts
- One part is then the quantity divided by that total
- Each share follows by multiplying a single part by its own number in the ratio
- Check by adding the shares back to the original amount, which catches an arithmetic slip immediately
Exam tip
One part earns the method mark
Sharing in a ratio is set in 12 of the 22 papers, and every scheme gives a method mark for the value of one part: the amount divided by the number of parts it stands for. That value, or any share worked out from it, earns the mark even when the final answer goes wrong. So write one part on its own line before working out what the question asks for.
Worked example
Dividing a quantity in a given ratio
£540 is divided between two people in the ratio 4 : 5. Work out each share.
Solution:
- A ratio of 4 : 5 means the money is cut into 4 + 5 = 9 equal parts, not into 4 or 5
- Finding the value of one part is the step every ratio-sharing question turns on
- One part = 540 ÷ 9 = £60
- The first share takes 4 of those parts: 4 × 60 = £240
- The second takes 5 of them: 5 × 60 = £300
- Check by adding the shares back: 240 + 300 = 540, which is the amount divided
- A second check is the ratio itself: 240 : 300 cancels by 60 to 4 : 5
- Keep the shares in the order the question set them, since giving £300 to the first person is a different answer
Worked example
Dividing into three parts
96 counters are divided in the ratio 3 : 4 : 5. Work out how many counters are in each group.
Solution:
- The method does not change when there are three parts instead of two
- Total parts = 3 + 4 + 5 = 12
- One part = 96 ÷ 12 = 8 counters
- The three groups take 3, 4 and 5 of those parts
- 3 × 8 = 24, 4 × 8 = 32 and 5 × 8 = 40
- The groups hold 24, 32 and 40 counters
- Check by adding: 24 + 32 + 40 = 96
- The division came out exactly, which it must: a whole number of counters cannot be split into fractions, so a total that does not divide by the number of parts is a signal to re-read the question
When the question gives a difference instead of a total
- The difference between two shares is worth the difference between their two numbers of parts
- Divide to find one part, then continue exactly as before
- In the ratio 7 : 4 one share exceeds the other by 33, and 7 − 4 = 3 parts, so one part is 11
- The shares are 77 and 44, and the total is 11 parts, or 121
When the question gives one share instead of a total
- Match the known quantity against the number of parts it stands for, then divide to find one part
- Paint is mixed red to white in the ratio 5 : 2, and 45 litres of red are available
- 5 parts is 45 litres, so one part is 9 litres, giving 18 litres of white and 63 litres of mixture
Worked example
When the amount given is not the total
(a) Some money is shared between three people in the ratio 2 : 5 : 9. The third person receives £117. Work out the total amount shared.
(b) Two people share some money in the ratio 4 : 9. The second person receives £40 more than the first. Work out the total amount they share.
Solution:
- In both parts the amount given is not the total, so first decide how many parts it stands for
- (a) The £117 is the third person's own share, which is 9 parts
- One part = 117 ÷ 9 = £13
- The total is 2 + 5 + 9 = 16 parts, so it is 16 × 13 = £208
- Check: the three shares are £26, £65 and £117, which add to £208
- (b) The £40 is the gap between the two shares, which is 9 − 4 = 5 parts
- One part = 40 ÷ 5 = £8
- The total is 4 + 9 = 13 parts, so it is 13 × 8 = £104
- Check: the shares are £32 and £72, which differ by £40
- Dividing by 16 in (a), or by 13 in (b), would treat the given amount as the total, which it is not
- Both questions ask for the total, so stopping at one person's share leaves the answer unfinished
Combining two ratios
- Two two-part ratios that share a quantity join into a single three-part ratio
- Scale each ratio so that the shared quantity is represented by the same number in both, using the lowest common multiple of its two values
- Write the three parts out in one line once they agree
- With A : B = 3 : 4 and B : C = 6 : 5, the shared B is 4 and 6, whose lowest common multiple is 12
- Scaling gives A : B = 9 : 12 and B : C = 12 : 10, so A : B : C = 9 : 12 : 10
4. Proportional reasoning
Direct proportion
- Two quantities are in direct proportion when multiplying one by a factor multiplies the other by the same factor
- Their ratio stays fixed, so doubling one doubles the other
- Where the factor is not stated, it is recovered by comparing the two known values of the same quantity, then applied to the other
The unitary method
- Finding the value of a single unit first turns most proportion questions into one division followed by one multiplication
- Divide to reach one unit, then multiply up to the number required
Worked example
Scaling with the unitary method
Six identical books have a total mass of 2.7 kg. Work out the mass of eleven of these books.
Solution:
- The books are identical, so mass and number are in direct proportion: double the books, double the mass
- The unitary method goes down to one first, then up to the number wanted
- Mass of one book = 2.7 ÷ 6 = 0.45 kg
- Write that line down on its own: if the multiplication then slips, the working still shows the correct method
- Mass of eleven books = 0.45 × 11 = 4.95 kg
- Check the direction: eleven books is nearly twice six, and 4.95 kg is nearly twice 2.7 kg
- Multiplying 2.7 by 11 and then dividing by 6 gives the same answer, because the two steps can be done in either order
Inverse proportion
- Two quantities are in inverse proportion when multiplying one by a factor divides the other by that same factor
- More workers means less time, and more people sharing means a smaller share each, which is what makes the relationship inverse rather than direct
- Find the factor by dividing the new quantity by the old one, then divide the other quantity by it
- If 2 machines take 15 hours, then 6 machines is three times as many, so the time is 15 ÷ 3 = 5 hours
- The unitary method still works, provided the scaling is reversed at each step
- 4 taps fill a tank in 90 minutes, so one tap would take 4 × 90 = 360 minutes
- Six taps then take 360 ÷ 6 = 60 minutes
- Check the direction of the answer against the situation: a larger crew finishing in a longer time is a sign the two steps were the wrong way round
Rounding in context
- The context decides the direction of rounding, not the usual half-way rule
- Where a quantity must be sufficient, round up even when the nearer whole number is lower
- A job needing 2.4 tins of paint requires 3 tins
- Where a quantity is limited by what is available, round down instead
5. Ratios in context
Map scales
- A scale written as a ratio has no units, so both sides are measured the same way
- A scale of 1 : 50 000 means one centimetre on the map is 50 000 cm on the ground
- Convert to sensible units at the end: 50 000 cm is 500 m, or 0.5 km
Comparing value
- To decide which of two sizes is better value, reduce both to the same fixed amount
- Either compare the cost of one unit, or compare how much is obtained per unit of currency
- Whichever is chosen, the same comparison must be applied to both options
Worked example
Deciding which pack is better value
A 750 g pack costs £2.40 and a 1.2 kg pack costs £3.72. Determine which is better value.
Solution:
- Two quantities can only be compared once they are measured the same way, so fix the units first
- 1.2 kg = 1200 g, and working in pence avoids decimals: £2.40 is 240p and £3.72 is 372p
- Then reduce both to the same single unit: here, the cost of one gram
- Smaller pack: 240 ÷ 750 = 0.32p per gram
- Larger pack: 372 ÷ 1200 = 0.31p per gram
- The larger pack costs less for the same gram, so the 1.2 kg pack is better value
- Say which one and why, because "0.31 < 0.32" alone does not answer the question asked
- Grams per penny works equally well as the comparison, but then the larger figure is the better value, so decide which way round you are working before comparing