One Multiplication Tells You Whether Two Ratios Match

Cross-multiply. For $3:4$ and $9:12$, compare $3\times 12=36$ with $4\times 9=36$. They match, so the ratios are equal.

One multiplication each side, and no need to reduce anything. This page covers everything in the CBSE Class 8 Mathematics chapter's first part: writing and simplifying ratios, testing for proportion, finding a missing term, and reading a table of paired values.

A ratio compares two quantities of the same kind by division, written $a:b$ and read a is to b. It can also be written as the fraction $\frac{a}{b}$.

To reduce it to simplest form, divide both terms by their HCF:

$$24:36=\frac{24}{12}:\frac{36}{12}=2:3$$

$$45:60=3:4,18:24=3:4$$

A ratio has no unit, because the units cancel — which is why both quantities must be in the same unit before you start.

Different units. Convert first, then simplify:

$$250\text{g}:2\text{kg}=250:2000=1:8$$

$$40\text{cm}:1.5\text{m}=40:150=4:15$$

$$45\text{minutes}:2\text{hours}=45:120=3:8$$

$$50\text{paise}:₹2=50:200=1:4$$

A recipe calling for rice and water in the ratio $1:2$ means two cups of water for every cup of rice, whatever size the cup.

The requirement of the same kind is real. You can compare 250 g with 2 kg because both are masses, but a ratio of 3 kg to 5 litres is meaningless — there is no unit that makes mass and capacity comparable, so the cancelling that gives a ratio its unit-free nature cannot happen.

Cross-multiply and solve the resulting equation.

Worked example. Find $x$ if $4:6::10:x$.

$$4\times x=6\times 10\Rightarrow 4x=60\Rightarrow x=15$$

Checking: $4:6=2:3$ and $10:15=2:3$. Correct.

Worked example with the unknown first. Find $x$ if $x:8::15:20$.

$$20x=8\times 15=120\Rightarrow x=6$$

Checking: $6:8=3:4$ and $15:20=3:4$. Correct.

Worked example with the unknown in the middle. Find $x$ if $3:x::12:20$.

$$3\times 20=12x\Rightarrow 60=12x\Rightarrow x=5$$

Worked example from a context. If 4 pens cost ₹60, what do 7 pens cost? Setting up $4:60::7:x$:

$$4x=420\Rightarrow x=₹105$$

Checking by the unitary method: one pen costs $\frac{60}{4}=₹15$, so 7 cost $₹105$. The same answer.

When setting up a proportion from words, keep the quantities in matching positions. Writing $4:60::7:x$ puts pens against cost on both sides; writing $4:60::x:7$ would mix them up and give a nonsensical answer — so decide what each position represents before cross-multiplying.

Divide each pair and see whether you always get the same factor.

Worked example. A table of petrol used and distance covered:

- 2 litres, 30 km
- 4 litres, 60 km
- 6 litres, 90 km
- 8 litres, 120 km

Dividing distance by petrol each time:

$$\frac{30}{2}=15,\frac{60}{4}=15,\frac{90}{6}=15,\frac{120}{8}=15$$

The factor is constant at 15 km per litre, so the two quantities are directly proportional — doubling the petrol doubles the distance. The relationship can be written as $\text{distance}=15\times \text{petrol}$.

Worked example that is not proportional:

- 1 hour, ₹5
- 2 hours, ₹9
- 3 hours, ₹13

Dividing:

$$\frac{5}{1}=5,\frac{9}{2}=4.5,\frac{13}{3}\approx 4.33$$

The factor changes, so these are not proportional. What is constant here is the difference — the charge rises by ₹4 each hour, after a fixed ₹1 at the start — so the relationship is additive, not proportional.

The test to apply is the ratio, not the pattern. A table can look beautifully regular and still fail: the second one increases by a steady ₹4 each hour, which feels proportional but is not, because doubling the hours from 1 to 2 does not double the charge from ₹5 to ₹10. Only a constant quotient means proportional.