Ratios

Two classrooms, one 9 metres long and 6 metres long — how do you compare them without just subtracting 3 metres and moving on? A ratio answers that question by division instead of subtraction: it says "how many times as much," not "how much more." That single idea — comparing two quantities of the same kind by dividing one by the other — is the doorway into the whole Proportional Reasoning chapter of Class 8 mathematics. Before a student can handle proportion, direct variation, unitary-method word problems, scale drawings, or splitting a prize among three people, they need to be completely fluent with ratios: writing them correctly, reducing them to lowest terms, building equivalent ratios, and comparing two ratios head to head. This article builds that fluency from the ground up, with one derivation-style worked method, a fully solved example, and the traps that most often cost marks in an exam.

The core insight

A ratio $a:b=\frac{a}{b}$ compares two same-unit quantities by division, stays unchanged when both terms are scaled by the same nonzero number, and must be simplified by dividing both terms by their HCF before it is considered a final answer.

The core idea

A ratio compares two quantities of the same kind — same units — using division rather than subtraction. The ratio of $a$ to $b$ is written

a:b = \frac{a}{b} \qquad (b \neq 0)

where: $a$ is the first term (antecedent), $b$ is the second term (consequent), and the colon notation $a:b$ is just another way of writing the fraction $\frac{a}{b}$ . A ratio carries no unit of its own, because whatever units $a$ and $b$ have cancel out in the division — $a:b$ is a pure number. This is why, before two quantities can be compared as a ratio, they must first be expressed in the same unit: the ratio of 2 kilograms to 500 grams is not $2:500$ ; convert first, $2000\,\text{g}:500\,\text{g} = 4:1$ .

Simplifying a ratio works exactly like reducing a fraction to lowest terms: divide both terms by their highest common factor (HCF). For $18:24$ , since $\text{HCF}(18,24)=6$ , the simplest form is $3:4$ . A ratio that has not been reduced this way is technically correct but is not considered a final answer in an exam.

Equivalent ratios are formed by multiplying — or dividing — both terms by the same nonzero number $k$ :

a:b = ka:kb \qquad (k \neq 0)

where: $k$ is any nonzero scaling factor. So $3:4 = 6:8 = 9:12 = \dots$ — an infinite family of ratios, all describing the same underlying comparison.

Comparing two ratios $a:b$ and $c:d$ means asking which fraction, $\frac{a}{b}$ or $\frac{c}{d}$ , is larger. Two equivalent methods exist: convert both to a common denominator, or cross-multiply — $a:b > c:d$ exactly when $ad > bc$ (assuming $b,d>0$ ). Both must always agree, since cross-multiplication is just a shortcut for comparing fractions with a common denominator of $bd$ . Ratios are not limited to two terms. A three-term ratio $a:b:c$ compares three quantities at once — the tool used later in the chapter for sharing a quantity among three parties.

Figure

Two bars showing length 9 metres and breadth 6 metres of a room reduced by their HCF of 3 to the simplified ratio 3 to 2

Derivation ASimplifying and comparing ratios using the HCF and cross-multiplication

The goal is to show, step by step, why dividing both terms of a ratio by their HCF always produces the same comparison as the original ratio, and why cross-multiplication is a reliable shortcut for deciding which of two ratios is larger — using nothing beyond factors, HCF, and fraction equality, so the method is exam-ready for any pair of ratios a student meets in Class 8.

Step 1. Start from the definition. A ratio $a:b$ means the fraction $\frac{a}{b}$ . If $h$ is the HCF of $a$ and $b$ , write $a = h\,p$ and $b = h\,q$ , where $p$ and $q$ have no common factor left (they are coprime). Then $\frac{a}{b} = \frac{hp}{hq} = \frac{p}{q}$ , since the common factor $h$ cancels top and bottom. This is exactly why simplifying a ratio by its HCF changes nothing about what is being compared — it only removes the shared factor, leaving the same value $\frac{a}{b}$ in its lowest terms $p:q$ .

Step 2. Apply Step 1 to the earlier example. For $18:24$ , $\text{HCF}(18,24) = 6$ , so $18 = 6\times3$ and $24 = 6\times4$ , giving $\frac{18}{24} = \frac{3}{4}$ , i.e. $18:24 = 3:4$ . Any equivalent ratio built by multiplying $3:4$ by some $k$ — such as $6:8$ or $9:12$ — collapses back to $3:4$ once its own HCF is divided out, confirming that "simplest form" is unique for a given ratio.

Step 3. Now derive the cross-multiplication rule for comparing $a:b$ and $c:d$ (with $b,d>0$ ). Comparing $\frac{a}{b}$ and $\frac{c}{d}$ is the same as comparing $\frac{ad}{bd}$ and $\frac{cb}{bd}$ after giving them the common denominator $bd$ . Since $bd>0$ , the fraction with the larger numerator is larger, so $\frac{a}{b} > \frac{c}{d}$ exactly when $ad > cb$ . This is the cross-multiplication shortcut: multiply each numerator by the other fraction's denominator, and compare the two results directly, without ever writing the common denominator explicitly.

Step 4. Verify both methods agree on a fresh pair. Compare $4:5$ and $7:9$ . Common-denominator method: $\frac{4}{5}=\frac{36}{45}$ and $\frac{7}{9}=\frac{35}{45}$ , so $\frac{4}{5}$ is larger since $36>35$ . Cross-multiplication method: compare $4\times9=36$ with $5\times7=35$ ; since $36>35$ , again $4:5 > 7:9$ . Both routes land on the identical inequality, because Step 3 showed they are algebraically the same comparison written two different ways.

Worked example

Problem: A rectangular classroom has length 9 m and breadth 6 m. Find the ratio of length to breadth in simplest form, write one equivalent ratio, and check that this simplified ratio is larger than the ratio $5:4$ .

Step 1. Write the ratio in the given units. Length and breadth are already both in metres, so no unit conversion is needed: Length : Breadth $= 9\,\text{m} : 6\,\text{m}$ , giving the ratio $9:6$ .

Step 2. Simplify using the HCF. $\text{HCF}(9,6) = 3$ , so

9:6 = (9\div3):(6\div3) = 3:2.

Step 3. Write an equivalent ratio by multiplying both terms by the same number, say 5:

3:2 = 15:10.

Step 4. Compare $3:2$ with $5:4$ using cross-multiplication. Compare $3\times4=12$ with $2\times5=10$ . Since $12>10$ , $3:2 > 5:4$ .

Answer: The simplest ratio of length to breadth is $3:2$ , an equivalent ratio is $15:10$ , and $3:2$ is confirmed larger than $5:4$ .

Competitive stretch

A fruit-juice mix uses mango, orange and pineapple pulp in the ratio $2:3:1$ . The total number of "parts" is $2+3+1=6$ , so mango is $\frac{2}{6}=\frac{1}{3}$ of the mix. If the total mix is 1.5 litres, mango pulp is $\frac{1}{3}\times1.5 = 0.5$ litres — the same three-term ratio idea used in Step 1 of the derivation above, just extended from two terms to three, and it is exactly this splitting-by-parts trick that reappears later in the chapter for dividing a quantity among several people.

Common traps

Forgetting to convert units first. Writing the ratio of 1 hour to 20 minutes as $1:20$ is wrong — convert to the same unit first: $60\,\text{min}:20\,\text{min} = 3:1$ .
Treating a ratio like a fraction with a fixed "whole." $a:b$ compares $a$ and $b$ to each other, not each to some total; a ratio of $3:2$ does not mean "3 out of 2 total items" — it means the two quantities stand in that proportion.
Not fully simplifying. Stopping at $6:8$ instead of reducing to $3:4$ loses marks even though the ratio is technically correct — always divide by the HCF.
Reversing the order. The ratio of "boys to girls" is not the same as "girls to boys" — the order in $a:b$ matters and must match the order named in the question.

Ask the tutor: Stuck comparing two ratios or unsure whether you've fully simplified one — ask the AI tutor to walk through it with you step by step.

Why does a ratio have no unit even though the two quantities being compared do have units?

Because a ratio is formed by dividing one quantity by another of the same kind, the units are identical top and bottom and cancel out completely, just like $\frac{5\,\text{kg}}{2\,\text{kg}}=\frac{5}{2}$ . What remains is a pure number describing how many times one quantity is of the other. This is exactly why the two quantities must be converted to the same unit before the ratio is written — if the units do not match, they cannot cancel, and the ratio you write down would be meaningless.

Is 4:6 the same ratio as 6:4, just written the other way around?

No — $4:6$ and $6:4$ are different ratios pointing in opposite directions, even though both simplify using the same HCF (2, giving $2:3$ and $3:2$ ). The order in $a:b$ tells you which quantity is being compared to which: $4:6$ means "4 as a fraction of 6," while $6:4$ means "6 as a fraction of 4." Always match the order to the order named in the question — "ratio of boys to girls" must list boys first.

When comparing two ratios, is it faster to use a common denominator or cross-multiplication?

Both give exactly the same answer, since cross-multiplication is just the common-denominator method with the denominator $bd$ left implicit, as shown in the derivation above. Cross-multiplication is usually faster by hand because you skip writing out the shared denominator — just multiply diagonally and compare the two products. Use whichever method feels more natural, but cross-multiplication is the quicker exam technique once you trust the shortcut.

Ratios

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads