The classic question is: "Share £$X$ in the ratio $a : b$." There is one reliable method, and it works every time.
- Add the parts of the ratio to find the total number of parts.
- Divide the total amount by the number of parts. This gives the value of one part.
- Multiply the value of one part by each number in the ratio.
- Check that your shares add back up to the original amount.
Share £$450$ between Ali and Ben in the ratio $4 : 5$.
$720$ g of pastry is divided in the ratio $2 : 3 : 7$. Find each amount.
"The ratio of red to blue is $2 : 3$" is part : part — there are $5$ parts altogether.
"$2$ out of every $5$ counters are red" is part : whole — the ratio red : blue is $2 : 3$.
| Statement | Ratio red : blue | Red as a fraction of all |
|---|---|---|
| red : blue $= 3 : 4$ | $3 : 4$ | $\dfrac{3}{7}$ |
| red : total $= 3 : 7$ | $3 : 4$ | $\dfrac{3}{7}$ |
| $\dfrac{3}{7}$ are red | $3 : 4$ | $\dfrac{3}{7}$ |
In a garage, the ratio of cars to all vehicles is $5 : 8$. There are $96$ vehicles. How many are not cars?
Sometimes you are not told the total, but you are told how much bigger one share is. The method is the same, but you divide by the difference in parts.
Money is shared between Cara and Dan in the ratio $7 : 3$. Cara gets £$60$ more than Dan. How much did they share altogether?
Check: Cara $= 7 \times 15 = £105$, Dan $= 3 \times 15 = £45$; difference $= £60$ ✓
If you know the value of just one person's share, work out one part from that, then build up everything else.
A sum is shared between Eve, Fay and Gil in the ratio $2 : 5 : 6$. Fay receives £$85$. Find the total shared.
(Eve gets $2 \times 17 = £34$ and Gil gets $6 \times 17 = £102$; check: $34 + 85 + 102 = 221$ ✓)
Recipe questions are ratio questions in disguise. Find the multiplier and apply it to every ingredient.
A recipe for $6$ scones uses $250$ g flour, $50$ g butter and $150$ ml milk. Rewrite it for $15$ scones.
The same recipe (for $6$ scones: $250$ g flour, $50$ g butter, $150$ ml milk) is to be used. A baker has $2$ kg of flour, $300$ g of butter and $1$ litre of milk. What is the greatest number of scones he can make?
Answer: $6 \times 6 = 36$ scones
These questions combine sharing with comparing. The usual trick is to work out how much of the "active" ingredient there is in each mixture.
Alloy A is $60$ kg of copper and zinc in the ratio $3 : 2$. Alloy B is $40$ kg of copper and zinc in the ratio $1 : 3$. The two alloys are melted together. Find the ratio of copper to zinc in the new alloy.
Check: $46 + 54 = 100$ kg $= 60 + 40$ ✓
Orange squash is mixed with water in the ratio $1 : 6$. How much squash and how much water are needed to make $2.8$ litres of drink?
Harder questions add or remove some of one quantity and ask for the new ratio, or ask you to work backwards. Algebra makes these manageable.
In a box the ratio of red to blue pens is $5 : 3$. When $8$ more blue pens are added, the ratio becomes $5 : 4$. How many red pens are there?
Check: originally $40$ red, $24$ blue ($40:24 = 5:3$ ✓). After adding $8$: $40 : 32 = 5 : 4$ ✓
Standard method
Add the parts → divide the total → multiply each share → check the sum.
Difference given
One part $=$ difference in amount $\div$ difference in parts.
One share given
One part $=$ that share $\div$ its number of parts.
Part : whole
If the second number is the total, do not add — it already includes the first.
Recipes
Multiplier $=$ new amount $\div$ old amount; apply it to every ingredient.
Limited ingredients
Work out batches for each; the smallest one wins; round down.
Mixtures
Split each mixture into its actual amounts, then add the like amounts.
Changing ratios
Let one part be $x$, form an equation and cross-multiply.
Share £$364$ in the ratio $3 : 4$.
▶ Show solution
Total parts $= 3 + 4 = 7$.
One part $= 364 \div 7 = £52$.
Shares: $3 \times 52 = £156$ and $4 \times 52 = £208$.
Check: $156 + 208 = 364$ ✓
$1.5$ kg of sweets is divided in the ratio $2 : 3 : 5$. Find each mass in grams.
▶ Show solution
$1.5$ kg $= 1500$ g. Total parts $= 2 + 3 + 5 = 10$.
One part $= 1500 \div 10 = 150$ g.
Shares: $300$ g, $450$ g, $750$ g.
Check: $300 + 450 + 750 = 1500$ ✓
Two sisters share some money in the ratio $9 : 5$. The older sister gets £$72$ more than the younger. How much did each get?
▶ Show solution
Difference in parts $= 9 - 5 = 4$.
One part $= 72 \div 4 = £18$.
Older $= 9 \times 18 = £162$; younger $= 5 \times 18 = £90$.
Check: $162 - 90 = 72$ ✓
A prize is shared in the ratio $4 : 7 : 9$. The largest share is £$405$. Find the total prize.
▶ Show solution
The largest share has $9$ parts.
One part $= 405 \div 9 = £45$.
Total parts $= 4 + 7 + 9 = 20$.
Total prize $= 20 \times 45 = £900$.
In a bag of $60$ marbles, the ratio of red marbles to all the marbles is $2 : 5$. How many marbles are not red?
▶ Show solution
This is part : whole, so $5$ parts $= 60$ marbles.
One part $= 60 \div 5 = 12$.
Red $= 2 \times 12 = 24$.
Not red $= 60 - 24 = 36$ marbles.
A recipe for $8$ pancakes uses $200$ g flour, $2$ eggs and $300$ ml milk. How much of each is needed for $20$ pancakes?
▶ Show solution
Multiplier $= 20 \div 8 = 2.5$.
Flour $= 200 \times 2.5 = 500$ g.
Eggs $= 2 \times 2.5 = 5$ eggs.
Milk $= 300 \times 2.5 = 750$ ml.
A recipe for $4$ portions of soup needs $600$ g of tomatoes, $2$ onions and $500$ ml of stock. Sam has $2.1$ kg of tomatoes, $9$ onions and $2$ litres of stock. What is the greatest number of portions he can make?
▶ Show solution
Tomatoes: $2100 \div 600 = 3.5$ batches.
Onions: $9 \div 2 = 4.5$ batches.
Stock: $2000 \div 500 = 4$ batches.
Smallest is $3.5$, so $3$ complete batches (tomatoes run out first).
$3 \times 4 = 12$ portions
Note: if half-batches were allowed, $3.5 \times 4 = 14$ portions; the safest exam answer uses whole batches unless the question says otherwise.
A fruit punch is made with juice and lemonade in the ratio $3 : 5$. How much of each is needed to make $4.8$ litres of punch?
▶ Show solution
Total parts $= 3 + 5 = 8$.
One part $= 4.8 \div 8 = 0.6$ litres.
Juice $= 3 \times 0.6 = 1.8$ litres.
Lemonade $= 5 \times 0.6 = 3$ litres.
Alloy P is $50$ kg of tin and lead in the ratio $2 : 3$. Alloy Q is $30$ kg of tin and lead in the ratio $1 : 2$. They are melted together. Find the ratio of tin to lead in the new alloy, in its simplest form.
▶ Show solution
Alloy P: $5$ parts, one part $= 50 \div 5 = 10$ kg. Tin $= 20$ kg, lead $= 30$ kg.
Alloy Q: $3$ parts, one part $= 30 \div 3 = 10$ kg. Tin $= 10$ kg, lead $= 20$ kg.
Total tin $= 30$ kg; total lead $= 50$ kg.
Ratio $= 30 : 50 = 3 : 5$.
The ratio of adults to children at a play scheme is $2 : 7$. After $8$ more adults arrive, the ratio becomes $2 : 3$. How many children are there?
▶ Show solution
Let one original part be $x$: adults $= 2x$, children $= 7x$.
The number of children does not change, so:
$$\frac{2x + 8}{7x} = \frac{2}{3}$$
Cross-multiply: $3(2x + 8) = 2 \times 7x$
$6x + 24 = 14x \Rightarrow 24 = 8x \Rightarrow x = 3$
Children $= 7x = 7 \times 3 = \mathbf{21}$
Check: adults were $2 \times 3 = 6$; after $8$ arrive there are $14$. And $14 : 21 = 2 : 3$ ✓