A Ratio Between Paise and Rupees Must Be Fixed First

1 : 4 — but only after both are written in the same unit. Converting ₹2 to 200 paise gives $50:200$, which reduces to $1:4$.

Writing $50:2$ instead would be nonsense, and that mistake is the commonest in the chapter. This page covers everything in the ICSE Class 7 Mathematics chapter on ratio and proportion: simplifying ratios, dividing a quantity in a ratio, testing for proportion, and the fourth, third and mean proportionals.

A ratio compares two quantities of the same kind by division, written $a:b$ and read a is to b. The first term is the antecedent and the second the consequent.

To simplify, divide both terms by their HCF:

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

A ratio has no unit, because the units cancel — which is why the two quantities must be in the same unit first.

Different units. Convert, then simplify.

$$250g:2kg=250:2000=1:8$$

$$40cm:1.5m=40:150=4:15$$

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

Recipes use ratios constantly — rice to water at $1:2$ means two cups of water for every cup of rice, whatever the cup's size.

The restriction on "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.

Add the ratio terms to get the total number of parts, find the value of one part, then multiply.

Worked example. Divide ₹1200 between two people in the ratio $2:3$.

Total parts $=2+3=5$, so one part $=\frac{1200}{5}=₹240$. Therefore the shares are

$$2\times 240=₹480\text{and}3\times 240=₹720$$

Check: $480+720=₹1200$, and $480:720=2:3$. Both conditions hold.

Three-way. Divide ₹2400 among three in the ratio $3:4:5$. Total parts $=12$, one part $=₹200$, so the shares are ₹600, ₹800 and ₹1000.

Combining ratios. If $A:B=2:3$ and $B:C=4:5$, make B match in both. The LCM of 3 and 4 is 12, so multiply the first ratio by 4 and the second by 3:

$$A:B=8:12,B:C=12:15$$

$$\text{so }A:B:C=8:12:15$$

Always check the total. If your shares do not add back to the original quantity, a part value went wrong — and that check takes one line.

Four quantities are in proportion when the ratio of the first two equals the ratio of the last two: $a:b::c:d$. The test is cross-multiplication:

$$a\times d=b\times c$$

The outer terms $a$ and $d$ are the extremes, and the inner terms $b$ and $c$ are the means. So the product of the extremes equals the product of the means.

Worked example. Are 3, 4, 9 and 12 in proportion?

$$3\times 12=36,4\times 9=36$$

Equal, so yes — $3:4::9:12$.

Are 2, 5, 6 and 14 in proportion? $2\times 14=28$ and $5\times 6=30$. Not equal, so no.

Fourth proportional. The fourth proportional to 4, 6 and 10 is the number $x$ making $4:6::10:x$:

$$4x=60\Rightarrow x=15$$

A shop bill is the everyday case: if 4 pens cost ₹60, then $4:60::7:x$ gives $x=₹105$ for 7 pens.

The order matters, which is where marks are lost. In $a:b::c:d$ the extremes are the first and last, so 3, 4, 9, 12 are in proportion while 3, 9, 4, 12 are not — the same four numbers, rearranged into a false statement.