Maths · 1 Number

1.10 Ratio and proportion

Before you start

Finding the highest common factor (HCF) of two or three whole numbers, e.g. the HCF of $24$ and $36$ is $12$ .
Simplifying a fraction, e.g. $\frac{12}{30} = \frac{2}{5}$ , and finding a common denominator.
Multiplying and dividing by $10$ , $100$ and $1000$ , and short division with decimals, e.g. $3.40 \div 5 = 0.68$ .
Metric conversions: $100$ cm $= 1$ m, $1000$ m $= 1$ km, $1000$ g $= 1$ kg, $1000$ ml $= 1$ litre, and $60$ minutes $= 1$ hour.

Syllabus objectives covered (1)

Understand and use ratio and proportion to: • give ratios in their simplest form • divide a quantity in a given ratio • use proportional reasoning and ratios in context.

Part 1 of 7

What a ratio says, and its simplest form

A ratio compares the sizes of two or more quantities. A class has 12 boys and 18 girls, so the ratio of boys to girls is $12 : 18$ . You read it as '12 to 18'.
The order of the numbers follows the order of the words. 'Boys to girls' is $12 : 18$ . 'Girls to boys' is $18 : 12$ . Write the ratio in the order the question names the quantities, because the reversed ratio is a different answer.
A ratio compares one part with another part. It does not tell you the whole directly. The class has $12 + 18 = 30$ students, so the fraction of the class that are boys is $\frac{12}{30}$ , not $\frac{12}{18}$ . To change a ratio to a fraction, put one part over the total of the parts.
If you multiply or divide every part by the same number, the ratio still describes the same mix. $12 : 18$ , $6 : 9$ and $2 : 3$ all say 'for every 2 boys there are 3 girls'. This works the same way as equivalent fractions.
$12 : 18 = 6 : 9 = 2 : 3$
A ratio is in its simplest form when every part is a whole number and the parts have no common factor except 1. To get there, divide every part by the HCF of all the parts. For $20 : 30 : 40$ the HCF is 10.
$20 : 30 : 40 = 2 : 3 : 4$
If you cannot see the HCF at once, divide in small steps until no number divides every part. For $84 : 126$ : divide by 2 to get $42 : 63$ , divide by 3 to get $14 : 21$ , divide by 7 to get $2 : 3$ . Now 2 and 3 have no common factor, so you stop.
In a three-part ratio, one number must divide all three parts. The trap is $32 : 48 : 72$ . All three divide by 8, which gives $4 : 6 : 9$ . 4 and 6 share a factor of 2, but 9 does not, so $4 : 6 : 9$ is already the simplest form. Do not simplify two parts and leave the third.
Check by multiplying back. $4 \times 8 = 32$ , $6 \times 8 = 48$ and $9 \times 8 = 72$ . If every part returns to the original, your division was correct.

Part 2 of 7

Units, decimals and fractions inside a ratio

Before you simplify, put every part in the same units. A ratio of 45 minutes to 2 hours is not $45 : 2$ . Change 2 hours to 120 minutes first. Then the HCF of 45 and 120 is 15.
$45 : 120 = 3 : 8$
When the units are the same, they cancel. So the final ratio has no units: write $3 : 8$ , not '3 min : 8 min'.
If a part is a decimal, multiply every part by 10, 100 or 1000 until all the parts are whole numbers. Then simplify as usual. For $0.6 : 1.4$ , multiply by 10 to get $6 : 14$ , then divide by 2.
$0.6 : 1.4 = 6 : 14 = 3 : 7$
If a part is a fraction, multiply every part by a common denominator. For $\frac{2}{3} : \frac{1}{4}$ , the common denominator is 12. $\frac{2}{3} \times 12 = 8$ and $\frac{1}{4} \times 12 = 3$ .
$\frac{2}{3} : \frac{1}{4} = 8 : 3$
Sometimes the question asks for the form $1 : n$ or $n : 1$ . Divide every part by the part that must become 1. For $5 : 8$ in the form $1 : n$ , divide both parts by 5. Here $n$ can be a decimal. This is the one case where a decimal in a ratio is the correct final answer.
$5 : 8 = 1 : 1.6$
The form $1 : n$ is useful for comparing two ratios. $3 : 5 = 1 : 1.67$ (to 3 significant figures) and $4 : 7 = 1 : 1.75$ . In the second ratio, there is more of the second quantity for each 1 of the first quantity.

Part 3 of 7

Dividing a quantity in a given ratio

To divide a quantity in a ratio, think of the ratio as a number of equal parts. In the ratio $3 : 4$ , the first person gets 3 parts and the second person gets 4 parts, so there are $3 + 4 = 7$ parts in total.
A picture helps. Draw a bar of 7 equal boxes. Label 3 boxes for the first person and 4 boxes for the second person. The whole quantity fills all 7 boxes, so one box is the quantity divided by 7.
Then use three steps. Add the parts. Divide the quantity by the total of the parts to get one part. Multiply one part by each number in the ratio. To share $84$ in the ratio $3 : 4$ : there are 7 parts, one part is $84 \div 7 = 12$ , so the shares are $3 \times 12 = 36$ and $4 \times 12 = 48$ .
Check that the shares add back to the quantity: $36 + 48 = 84$ . This check takes a few seconds and finds most arithmetic slips.
A three-part ratio uses the same method. To share $150$ in the ratio $2 : 3 : 5$ , there are $2 + 3 + 5 = 10$ parts, so one part is $15$ .
$2 \times 15 = 30, 3 \times 15 = 45, 5 \times 15 = 75$
Sometimes the question gives you one share, not the total. The ratio is $3 : 5$ and the smaller share is $24$ . The smaller share is 3 parts, so one part is $24 \div 3 = 8$ . The larger share is $5 \times 8 = 40$ and the total is $8 \times 8 = 64$ . The trap is to divide 24 by all 8 parts, but 24 is only 3 of the parts.
Sometimes the question gives you the difference between two shares. The ratio is $2 : 7$ and one share is $35$ more than the other. The difference is $7 - 2 = 5$ parts, so 5 parts are $35$ and one part is $7$ . The shares are $14$ and $49$ , and $49 - 14 = 35$ confirms it.
So read the question carefully to find which amount it gives you: the total, one share, or a difference. That amount tells you how many parts to divide by. When you know one part, you can find every other amount.

Part 4 of 7

Direct proportion and the unitary method

Two quantities are in direct proportion when doubling one doubles the other. The ratio between them stays the same. The cost of identical pens is in direct proportion to the number of pens you buy.
The unitary method finds the value for one first, and then multiplies. 5 pens cost $$ 3.40$ . One pen costs $$ 3.40 \div 5 = $ 0.68$ . So 8 pens cost $8 \times $ 0.68 = $ 5.44$ .
A second method uses a multiplier. Going from 5 pens to 8 pens multiplies the number of pens by $\frac{8}{5} = 1.6$ , so the cost is multiplied by 1.6 too: $$ 3.40 \times 1.6 = $ 5.44$ . Both methods give the same answer. Use the one with easier numbers.
Not every pair of quantities is in direct proportion. Sometimes, when one quantity goes up, the other goes down. 4 workers take 6 days to paint a fence. More workers finish sooner, so you must not multiply the days by the number of workers.
For this kind of question, find the total amount of work. 4 workers for 6 days is $4 \times 6 = 24$ worker-days. If you have 3 workers, the job takes $24 \div 3 = 8$ days. This assumes that all the workers work at the same rate.
$4 \times 6 = 24 = 3 \times 8$
Before you calculate, ask: 'If this quantity goes up, does the other one go up or go down?' If it goes up, the quantities are in direct proportion, so scale both in the same way. If it goes down, multiply to find the total first, then divide.

Part 5 of 7

Adapting a recipe

A recipe is a ratio of ingredients. If you make more or less, every ingredient changes by the same multiplier, so the taste stays the same.
To change a recipe for 4 people into a recipe for 6 people, the multiplier is $\frac{6}{4} = 1.5$ . If the recipe uses 300 g of pasta and 200 ml of cream, then 6 people need $300 \times 1.5 = 450$ g of pasta and $200 \times 1.5 = 300$ ml of cream.
$multiplier = \frac{new number}{old number} = \frac{6}{4} = 1.5$
The trap is to add. Going from 4 people to 6 people is not 'add 2 to every amount'. If you add, you get 302 g of pasta and 202 ml of cream, which is almost the recipe for 4. Proportion always means multiply or divide.
If the multiplier is not easy, use the unitary method. Find the amount for one person, then multiply by the new number of people. For 300 g of pasta for 4 people, one person needs $300 \div 4 = 75$ g, so 7 people need $7 \times 75 = 525$ g.
Some questions ask for the greatest number you can make from the ingredients you have. Work out, for each ingredient, how many people it is enough for. The smallest of those numbers is the answer, because you run out of that ingredient first. This ingredient is called the limiting ingredient.
If the answer must be a whole number of people or items, round down. If one ingredient is enough for 26.7 people, it is enough for 26 people only, because the 27th person does not get a full share.

Part 6 of 7

Map scales

A map scale such as $1 : 50 000$ is a ratio. It means that 1 unit on the map is $50 000$ of the same units on the ground. So 1 cm on the map is $50 000$ cm in real life. It is not $50 000$ km.
To go from the map to real life, multiply by the scale number. To go from real life to the map, divide by the scale number. On a map with a large scale number such as $50 000$ , the real distance is much larger than the map distance, so use that fact to check your answer.
After you multiply, you have an answer in centimetres, so you must convert it. There are 100 cm in 1 m and 1000 m in 1 km, so there are $100 \times 1000 = 100 000$ cm in 1 km.
$100 000 cm = 1000 m = 1 km$
On a $1 : 50 000$ map, two points are 6.2 cm apart. The real distance is $6.2 \times 50 000 = 310 000$ cm. Divide by $100 000$ to change cm to km.
$310 000 cm = 3.1 km$
Now go the other way on the same map. A real road is 4 km long. Change it to centimetres first: $4 \times 100 000 = 400 000$ cm. Then divide by the scale number: $400 000 \div 50 000 = 8$ cm on the map.
Sometimes the scale is given in words, for example '1 cm represents 2 km'. To write this as a ratio, put both parts in the same unit. $2$ km $= 200 000$ cm, so the scale is $1 : 200 000$ .

Part 7 of 7

Deciding which is the best value

A bigger pack is not always cheaper for each gram. To decide which option gives the best value, you must compare like with like. Work out the price for the same amount of each option, for example the price per 100 g or the price per item.
A 350 g jar of honey costs $$ 2.45$ . A 600 g jar costs $$ 4.02$ . Find the price of 100 g in each jar. 350 g is 3.5 lots of 100 g, and 600 g is 6 lots of 100 g.
$2.45 \div 3.5 = 0.70 4.02 \div 6 = 0.67$
The 600 g jar costs $$ 0.67$ for each 100 g and the 350 g jar costs $$ 0.70$ . A lower price for the same amount is better value, so the 600 g jar is the better value.
You can also compare the amount you get for each unit of money. For the 350 g jar, $350 \div 2.45 = 142.9$ g for each dollar. For the 600 g jar, $600 \div 4.02 = 149.3$ g for each dollar. A larger amount for each dollar is better value. Both methods give the same decision, but be careful about which direction is better.
Do not round too soon. Two options can be very close, and rounding to the nearest cent too early can make them look equal or put them in the wrong order. Keep at least 3 significant figures until you decide.
Your answer is a decision supported by numbers. Write the values for every option, then name the best one: 'The 600 g jar, because it costs $$ 0.67$ per 100 g and the 350 g jar costs $$ 0.70$ per 100 g.' A decision with no figures does not show that it is correct.

Worked examples

Example 1. Write each ratio in its simplest form. (a) $24 : 40 : 56$ (b) 75 cm : 2.25 m (c) $1.2 : 0.45$ (d) $\frac{3}{4} : \frac{5}{6}$ . (e) Write $12 : 30$ in the form $1 : n$ .

Show the working

(a) Find one number that divides all three parts. 24, 40 and 56 are all in the 8 times table, and after dividing by 8 the parts 3, 5 and 7 have no common factor.
$24 : 40 : 56 = 3 : 5 : 7$
(b) Make the units the same first. $2.25$ m $= 225$ cm. The HCF of 75 and 225 is 75, because $225 = 3 \times 75$ .
$75 : 225 = 1 : 3$
(c) Two decimal places means multiply both parts by 100 to make whole numbers: $120 : 45$ . The HCF of 120 and 45 is 15.
$1.2 : 0.45 = 120 : 45 = 8 : 3$
(d) Multiply both parts by a common denominator. The lowest common multiple of 4 and 6 is 12. $\frac{3}{4} \times 12 = 9$ and $\frac{5}{6} \times 12 = 10$ . 9 and 10 have no common factor, so you stop.
$\frac{3}{4} : \frac{5}{6} = 9 : 10$
(e) The first part must become 1, so divide both parts by 12. $30 \div 12 = 2.5$ . A decimal is correct here because the question asks for the form $1 : n$ .
$12 : 30 = 1 : 2.5$

Answer: (a) $3 : 5 : 7$ (b) $1 : 3$ (c) $8 : 3$ (d) $9 : 10$ (e) $1 : 2.5$

Example 2. Ana, Ben and Chen share some money in the ratio $2 : 3 : 7$ . Chen receives $$ 45$ more than Ana. Work out how much each person receives, and the total amount of money.

Show the working

The question does not give the total. It gives the difference between Chen and Ana, so use the difference in their parts. Chen has 7 parts and Ana has 2 parts, so the difference is $7 - 2 = 5$ parts.
5 parts are $$ 45$ , so one part is $45 \div 5 = 9$ .
$1 part = $ 9$
Multiply one part by each number in the ratio. Ana gets $2 \times 9 = 18$ , Ben gets $3 \times 9 = 27$ and Chen gets $7 \times 9 = 63$ .
There are $2 + 3 + 7 = 12$ parts, so the total is $12 \times 9 = 108$ . The sum of the shares gives the same result: $18 + 27 + 63 = 108$ .
Check the information you were given: $63 - 18 = 45$ . Chen does receive $$ 45$ more than Ana.

Answer: Ana $$ 18$ , Ben $$ 27$ , Chen $$ 63$ . The total is $$ 108$ .

Example 3. A pancake recipe for 6 people uses 240 g of flour, 3 eggs and 450 ml of milk. (a) Work out the amount of each ingredient for 10 people. (b) Priya has 1 kg of flour, 7 eggs and 2 litres of milk. Work out the greatest number of people she can make pancakes for.

Show the working

(a) The number of people changes from 6 to 10, so the multiplier is $\frac{10}{6} = \frac{5}{3}$ . Multiply every ingredient by $\frac{5}{3}$ : divide by 3, then multiply by 5.
Flour: $240 \div 3 \times 5 = 400$ g. Eggs: $3 \div 3 \times 5 = 5$ . Milk: $450 \div 3 \times 5 = 750$ ml.
(b) Use the unitary method. For one person, the recipe uses $240 \div 6 = 40$ g of flour, $3 \div 6 = 0.5$ of an egg and $450 \div 6 = 75$ ml of milk.
Change Priya's amounts to the units in the recipe: 1 kg $= 1000$ g and 2 litres $= 2000$ ml. Then find how many people each ingredient is enough for.
$flour: 1000 \div 40 = 25 eggs: 7 \div 0.5 = 14 milk: 2000 \div 75 = 26.7$
The smallest number is 14, so the eggs are the limiting ingredient. She has enough flour and milk for more people, but she has no more eggs after 14 people.

Answer: (a) 400 g of flour, 5 eggs and 750 ml of milk. (b) 14 people (the eggs run out first).

Example 4. A map has a scale of $1 : 40 000$ . (a) On the map, a lake is 8.5 cm long. Work out the real length of the lake in kilometres. (b) A real road is 5.2 km long. Work out the length of the road on the map, in centimetres. (c) A different map says '1 cm represents 0.5 km'. Write this scale in the form $1 : n$ .

Show the working

(a) Map to real life, so multiply by the scale number. The answer is in centimetres, because the map length is in centimetres.
$8.5 \times 40 000 = 340 000 cm$
Change to kilometres. There are $100 000$ cm in 1 km, so divide by $100 000$ : $340 000 \div 100 000 = 3.4$ km.
(b) Real life to the map, so first change the real length to centimetres, then divide by the scale number.
$5.2 \times 100 000 = 520 000 cm 520 000 \div 40 000 = 13 cm$
Check if the answer is sensible: a 13 cm line on a map for a 5.2 km road is a reasonable size.
(c) Both parts of a ratio must be in the same unit. $0.5$ km $= 0.5 \times 100 000 = 50 000$ cm.
$1 cm : 50 000 cm = 1 : 50 000$

Answer: (a) 3.4 km (b) 13 cm (c) $1 : 50 000$

Where people go wrong

Mistake: Dividing the total by one of the numbers in the ratio. For example, to share 60 in the ratio $2 : 3$ , writing $60 \div 2 = 30$ and $60 \div 3 = 20$ .

Instead: Add the parts first. $2 + 3 = 5$ parts, so one part is $60 \div 5 = 12$ , and the shares are $24$ and $36$ .

The numbers in a ratio count parts. The whole quantity is spread across all the parts, so you divide by the total number of parts. A quick check shows the error: $30 + 20 = 50$ , not 60.

Mistake: Reading a ratio as a fraction. If boys to girls is $3 : 5$ , saying that $\frac{3}{5}$ of the group are boys.

Instead: There are $3 + 5 = 8$ parts in total, so $\frac{3}{8}$ of the group are boys and $\frac{5}{8}$ are girls.

A ratio compares one part with another part. A fraction compares one part with the whole. The bottom of the fraction must be the total of the parts.

Mistake: Simplifying a ratio while its parts are in different units. For example, writing 50 cm : 2 m as $25 : 1$ .

Instead: Convert first: 2 m $= 200$ cm, so the ratio is $50 : 200 = 1 : 4$ .

A ratio compares the sizes of the quantities. $50 : 2$ says the first quantity is 25 times as large, but 50 cm is actually a quarter of 2 m.

Mistake: Adapting a recipe by adding the same amount to every ingredient. For example, changing a recipe for 4 people to 6 people by adding 2 to each amount.

Instead: Multiply every amount by $\frac{6}{4} = 1.5$ . Alternatively, find the amount for one person and multiply by 6.

Proportion keeps the ratio between the ingredients the same, and only multiplying or dividing does that. With 300 g of flour and 2 eggs, adding 2 gives 302 g of flour, almost no change, but 4 eggs, which is double. The mix is no longer the same.

Mistake: Giving a map distance in the wrong unit. On a $1 : 50 000$ map, saying 3 cm is $150 000$ km, or dividing $150 000$ cm by 1000 to get 150 km.

Instead: $3 \times 50 000 = 150 000$ cm. Divide by $100 000$ to get km: the real distance is 1.5 km.

Both parts of a map scale are in the same unit, so a map length in cm gives a real length in cm. It takes 100 cm to make 1 m and 1000 m to make 1 km, so you divide by $100 000$ , not by 1000.

Check yourself

Write $45 : 60 : 75$ in its simplest form.
Answer
The HCF is 15, so $45 : 60 : 75 = 3 : 4 : 5$ .
Write the ratio 1.5 kg : 600 g in its simplest form.
Answer
1.5 kg $= 1500$ g. $1500 : 600$ , and the HCF is 300, so the ratio is $5 : 2$ .
Write $8 : 5$ in the form $n : 1$ .
Answer
Divide both parts by 5: $8 : 5 = 1.6 : 1$ .
Share $270$ in the ratio $4 : 5$ .
Answer
$4 + 5 = 9$ parts, so one part is $270 \div 9 = 30$ . The shares are $120$ and $150$ (check: $120 + 150 = 270$ ).
A bag contains red and blue counters in the ratio $5 : 3$ . There are 40 red counters. How many blue counters are there? What fraction of the counters are blue?
Answer
5 parts are 40, so one part is 8. There are $3 \times 8 = 24$ blue counters. The total is $40 + 24 = 64$ , so the fraction that are blue is $\frac{24}{64} = \frac{3}{8}$ .
A map has a scale of $1 : 20 000$ . Two churches are 4.5 cm apart on the map. Find the real distance between them in kilometres.
Answer
$4.5 \times 20 000 = 90 000$ cm. $90 000 \div 100 000 = 0.9$ km.
A pack of 6 rolls costs $$ 2.70$ . A pack of 9 rolls costs $$ 3.96$ . Which pack is the better value? Show your working.
Answer
6-pack: $2.70 \div 6 = 0.45$ , so $$ 0.45$ per roll. 9-pack: $3.96 \div 9 = 0.44$ , so $$ 0.44$ per roll. The 9-pack is the better value, because each roll costs less.
3 painters take 8 days to paint a school. How many days would 4 painters take, if they all work at the same rate?
Answer
More painters finish sooner, so do not use direct proportion. Total work is $3 \times 8 = 24$ painter-days, so 4 painters take $24 \div 4 = 6$ days.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1.

(a) Write $32 : 48 : 72$ in its simplest form. 1 mark No calculator

Mark scheme

B1 4 : 6 : 9 cao

(b) Write the ratio 40 cm : 1.2 m in its simplest form. 2 marks No calculator

Mark scheme

M1 40 : 120 or 0.4 : 1.2 seen, i.e. both parts in the same units
A1 1 : 3 cao

(c) Write the ratio $4 : 5$ in the form $1 : n$ . 1 mark No calculator

Mark scheme

B1 1 : 1.25 oe, e.g. 1 : 5/4

Question 2. Mira, Jon and Kai share some prize money in the ratio $3 : 4 : 5$ . Kai receives $$ 56$ more than Mira.

(a) Work out the total amount of prize money. 3 marks No calculator

Mark scheme

M1 56 ÷ (5 − 3) oe, i.e. 28 seen as the value of one part
M1 their 28 × (3 + 4 + 5) oe, or 84 + 112 + 140
A1 336 cao

(b) Write Jon's share as a fraction of the total prize money. Give your fraction in its simplest form. 1 mark No calculator

Mark scheme

B1 1/3 cao

Question 3. A recipe for 12 biscuits uses 150 g of butter, 90 g of sugar and 225 g of flour.

(a) Calculate the amount of flour needed for 20 biscuits. 2 marks

Mark scheme

M1 225 ÷ 12 × 20 oe, e.g. 225 × 5/3 or 18.75 × 20
A1 375 (g) cao

(b) Ama has 500 g of butter, 400 g of sugar and 1 kg of flour. She has plenty of the other ingredients. Work out the greatest number of biscuits she can make. 3 marks

Mark scheme

M1 a correct maximum for one ingredient, e.g. 500 ÷ 150 × 12 = 40, or 3.33… batches from the butter
M1 correct maximums for all three ingredients compared: 40, 53.3… and 53.3… oe, or 3.33…, 4.44… and 4.44… batches
A1 40 cao

Question 4. Leo is planning a walk. His map has a scale of $1 : 25 000$ .

(a) On the map, the distance between two villages is 9.6 cm. Work out the real distance between the villages. Give your answer in kilometres. 2 marks

Mark scheme

M1 9.6 × 25 000 oe, i.e. 240 000 (cm) seen
A1 2.4 (km) cao

(b) A straight path is 1.75 km long. Calculate the length of the path on the map. Give your answer in centimetres. 2 marks

Mark scheme

M1 175 000 ÷ 25 000 oe, or 1.75 × 100 000 ÷ 25 000
A1 7 (cm) cao

(c) Leo buys trail mix for the walk. A 250 g bag costs $$ 2.15$ . A 400 g bag costs $$ 3.36$ . Determine which bag is the better value. You must show your working. 3 marks

Mark scheme

M1 a comparable value for the 250 g bag: 0.86 per 100 g, or 0.0086 per g, or 116 g per dollar (116.27…), oe
M1 the same kind of value for the 400 g bag: 0.84 per 100 g, or 0.0084 per g, or 119 g per dollar (119.0…), oe; any other comparable pair is accepted, e.g. 8.60 and 8.40 per kg, or 17.20 for 8 small bags and 16.80 for 5 large bags (2 kg each)
A1 400 g bag, dep on both M1 with values that are correct and comparable

How this is examined

Sharing an amount in a ratio when you are given the total, one share, or the difference between two shares: A 2- or 3-mark Work out or Find: two or three people share money in a ratio such as $3 : 4 : 5$ , and the stem gives the total, one person's share, or how much more one person gets than another M1 for the value of one part: the amount you are given, divided by the matching number of parts (all the parts for a total, that person's parts for one share, the difference in parts for a difference). M1 for multiplying one part by the ratio number (or the total of the parts) that the question needs. Then A1 for the correct share or total.
Deciding which pack or offer is the best value: A 2- or 3-mark Determine or Show (that) comparing two or three sizes of the same product, each with a price A comparable value for each option, calculated in the same way (for example $$ 0.70$ and $$ 0.67$ per 100 g, or grams per dollar for both). Each value is worth a method mark. Then the decision, naming the option, which earns the final mark only when the values support it. A decision with no figures does not earn the method marks.
Scaling a recipe up or down, and finding the limiting ingredient: A 2-mark Calculate or Work out for one ingredient for a new number of people or items, sometimes followed by a 3-mark Work out for the greatest number you can make from given amounts The multiplier (such as $20 \div 12$ ) or the amount for one, shown in the working, for M1. Then the scaled amount with its unit for A1. For the 'greatest number', show the maximum that each ingredient allows, then choose the smallest one.
Converting between a map distance and a real distance: A 2-mark Calculate or Work out with a scale such as $1 : 25 000$ , in either direction, with the answer in a stated unit M1 for multiplying by the scale number (map to real) or dividing by it (real to map). A1 for the correct conversion to the unit the question asks for, using $100 000$ cm $= 1$ km.
Writing a ratio in its simplest form, including mixed units and the form $1 : n$ : A 1- or 2-mark Write, sometimes with two quantities in different units such as 40 cm and 1.2 m Whole numbers with no common factor: $4 : 6 : 9$ , not $8 : 12 : 18$ . With mixed units, the converted ratio (such as $40 : 120$ ) is the method mark. For the form $1 : n$ , a decimal such as $1 : 1.25$ is the correct answer.

Words to know

Ratio: A comparison of the sizes of two or more quantities, written with colons, e.g. $3 : 4$ . The order follows the order of the words.
Simplest form: A ratio with whole-number parts that have no common factor except 1, e.g. $20 : 30 : 40 = 2 : 3 : 4$ .
Part: One of the equal portions that a ratio divides a quantity into. The ratio $3 : 4$ has $3 + 4 = 7$ parts in total.
The form $1 : n$: A ratio with its first part made equal to 1 by dividing every part by the first part, e.g. $5 : 8 = 1 : 1.6$ . Here $n$ can be a decimal.
Direct proportion: Two quantities are in direct proportion when they increase or decrease by the same multiplier, so their ratio stays the same.
Unitary method: Finding the value for one item or one person first, then multiplying to find the value for the number you need.
Multiplier: The number that changes every amount in proportion, e.g. $\frac{6}{4} = 1.5$ to change a recipe for 4 people into a recipe for 6 people.
Limiting ingredient: The ingredient that runs out first. It sets the greatest number of items or people that a recipe can make.
Map scale: A ratio such as $1 : 50 000$ . It means that 1 unit on the map is $50 000$ of the same units in real life.
Best value: The option that gives the lowest price for the same amount, or the largest amount for the same price.

All Maths lessons Next: 1.11 Rates