A ratio is a recipe, not a total — count the parts before you count anything else.
The problem
A recipe mixes sugar and flour in the ratio 2 : 5. If 350 g of flour is used, how much sugar is needed?
The 5 parts are the 350 g, so one part is 70 g and sugar, at 2 parts, is 140 g. B flips the ratio and scales up. Sugar is the smaller share here, so any answer above 350 g is wrong before you check the arithmetic.
2 : 5 does not mean 2 grams and 5 grams. It means 2 parts to 5 parts.
2 parts sugar, 5 parts flour, 7 parts in total
Every part is the same size. Find the size of one part and the whole problem collapses into a multiplication.
sugarflour = 25 = s350
5s = 2(350)
cross-multiply
5s = 700
tidy the right
s = 140
÷ 5 both sides
Sugar over flour on the left means sugar over flour on the right. Flipping one side gives 875 g, and nobody bakes that.
Divide ₱4,500 among three people in the ratio 2 : 3 : 4.
2 + 3 + 4 = 9 parts
count the parts first
4500 ÷ 9 = 500
one part
2 parts
₱1,000
3 parts
₱1,500
4 parts
₱2,000
1,000 + 1,500 + 2,000 = 4,500 ✓
Always add the shares back up. It catches an arithmetic slip in one line.
The same works for sides. A triangle's sides are in the ratio 5 : 12 : 13 and its perimeter is 90 cm: 30 parts, one part is 3 cm, so the sides are 15, 36 and 39 cm. Since 5² + 12² = 13², it is a right triangle with legs 15 and 36, and its area is (12)(15)(36) = 270 cm².
The commonest wrong answer in this topic comes from mixing these two up.
2 : 5
part to part
2 : 7
part to whole
“What fraction of the mixture is sugar?” is 27, not 25.
“The ratio of boys to girls” fixes the order. Answering with girls to boys inverts everything that follows.
Ask whether more of one thing needs more, or less, of the other.
| Situation | More gives | Then |
|---|---|---|
| more cloth, more cost | more | direct — cross-multiply |
| more workers, less time | less | inverse — multiply across |
6 workers · 12 days = 72
the job is 72 worker-days
8 workers · d = 72
same job, more people
d = 9 days
fewer days, as expected
For an inverse relationship the product stays constant, not the quotient. Cross-multiplying here would give 16 days, which is nonsense.
The one-part method works whatever you are told — a total, a difference, or one share.
Cats to dogs at a shelter are 2 : 5, and there are 18 more dogs than cats. How many animals in all?
difference: 5 − 2 = 3 parts = 18
the gap is counted in parts too
one part = 6
18 ÷ 3
total = 7 parts = 42
12 cats + 30 dogs
If a : b = 2 : 5 and b : c = 3 : 4, write a : b : c.
b is 5 in one ratio and 3 in the other
make them match
a : b = 6 : 15 and b : c = 15 : 20
scale by 3 and by 5 — b becomes 15
a : b : c = 6 : 15 : 20
so a : c = 6 : 20 = 3 : 10
Gluing the ratios together uses two different sizes of “b”. Scale both ratios so the shared letter has the same number first; then any total can be shared with the one-part method.
Terms that are fractions or decimals: simplify 23 : 34 and 0.25 : 1.5.
23 : 34 → × 12 → 8 : 9
multiply both terms by the LCD
0.25 : 1.5 → × 4 → 1 : 6
clear the decimals the same way
What is different: a ratio does not change when both terms are multiplied by the same number, so we multiply until both are whole.
Different units: write 45 cm : 1.2 m in lowest terms.
1.2 m = 120 cm
one unit for both terms
45 : 120 = 3 : 8
÷ 15
What is different: a ratio compares like with like. Leaving the units mixed gives 45 : 1.2 = 75 : 2, which says the shorter length is the longer one.
The unknown on both sides: x + 23 = 2x − 15.
5(x + 2) = 3(2x − 1)
cross-multiply, keep the brackets
5x + 10 = 6x − 3
expand both sides
x = 13
check: 153 = 5 and 255 = 5 ✓
What is different: each side is an expression, so the 5 and the 3 multiply the whole numerator. Writing 5x + 2 drops a term.
Two numbers are in the ratio 3 : 4. When 5 is added to each, the ratio becomes 4 : 5. Find the numbers.
let them be 3x and 4x
x is one part
3x + 54x + 5 = 45
adding changes the ratio, not the parts
15x + 25 = 16x + 20, so x = 5
cross-multiply
the numbers are 15 and 20
check: 20 : 25 = 4 : 5 ✓
10 kg of a mix with sugar to water 1 : 4 is combined with 5 kg of a mix that is 2 : 3. What is the new ratio?
first mix: 10 kg in 5 parts → 2 kg sugar, 8 kg water
turn each ratio into kilograms
second mix: 5 kg in 5 parts → 2 kg sugar, 3 kg water
one part is 1 kg
sugar : water = 4 : 11
add the actual amounts
Answering 20 and 25 gives the numbers after the change. Adding the ratio terms, (1 + 2) : (4 + 3) = 3 : 7, ignores that the batches have different sizes — always convert to real amounts before combining.
Ratio items hand you a total, a difference or one share and ask for another; harder ones chain two ratios or change both terms. Find one part, scale the shared term to match, and check by rebuilding the given fact from your answer.
The problem
The ratio of boys to girls in a class is 3 : 4. There are 28 girls. How many boys?
4 parts make 28 girls, so one part is 7 and the boys, at 3 parts, number 21. A multiplies 3 by 4 rather than by the value of a part. Boys are the smaller share, so the answer has to come in under 28.
