Ratio and proportion

Ratio: comparing like with like. The ratio of $a$ to $b$ (with $b \neq 0$) is written

where: $a$ and $b$ must be quantities of the same kind and the same unit — comparing kilograms to kilograms, not kilograms to litres. A ratio carries no unit of its own, since the units cancel: $6\ \text{kg} : 2\ \text{kg} = 3:1$. Like a fraction, a ratio is reduced to its simplest form by dividing both terms by their HCF, and multiplying or dividing both terms by the same non-zero number never changes its value — $2:3$, $4:6$, and $10:15$ are all equivalent ratios.

Proportion: two ratios agreeing. When $a:b$ and $c:d$ are equal, the four numbers form a proportion, written

where: $a$ and $d$ are the extremes (the outer terms) and $b$ and $c$ are the means (the inner terms). To compare two ratios that are not obviously equal, convert both to fractions with a common denominator or to decimals and compare directly.

Comparing ratios and equivalent ratios. To decide which of two ratios, say $3:5$ and $4:7$, is larger, rewrite both as fractions with a common denominator ($\dfrac{3}{5}=\dfrac{21}{35}$ and $\dfrac{4}{7}=\dfrac{20}{35}$) or convert both to decimals ($0.6$ and $0.571\ldots$) and compare the results directly — here $3:5$ is the larger ratio. This works because a ratio behaves exactly like a fraction under every operation: multiplying or dividing both terms of $a:b$ by the same non-zero number produces an equivalent ratio with the same value, just as $\dfrac{2}{3}$ and $\dfrac{4}{6}$ are equivalent fractions. Every ratio therefore has infinitely many equivalent forms, and only the one obtained by dividing through by the HCF is called its simplest form.

Dividing a quantity in a given ratio. To split a total $Q$ in the ratio $a:b:c$, treat the ratio as $(a+b+c)$ equal parts. One part is worth $\dfrac{Q}{a+b+c}$, so the three shares are

This "value of one part" method is the standard tool for every share-in-a-ratio question, whether it involves two parties or five.

A school stationery box contains pens and pencils in the ratio $5:8$. There are $65$ pens. How many pencils are there, and what is the total number of items?

Step 1. Let the number of pencils be $x$. Since pens : pencils stays fixed at $5:8$, set up the proportion $\dfrac{5}{8} = \dfrac{65}{x}$.

Step 2. Cross-multiply using the rule just derived: $5x = 8 \times 65 = 520$.

Step 3. Solve for $x$: $x = \dfrac{520}{5} = 104$. So there are 104 pencils, and the total number of items is $65 + 104 = 169$.

Step 4. Check by reducing the answer back to a ratio: $65:104$ divides by their HCF, $13$, giving $5:8$ — exactly the ratio we started with.

Competitive stretch

Suppose the same box is restocked so the total items become $220$, keeping the pens-to-pencils ratio at $5:8$. Total parts $= 5+8 = 13$, so one part $= 220/13 \approx 16.9$ — not a whole number, which immediately signals that $220$ is not a valid total for this ratio (a valid total must be a multiple of $13$, such as $169$ or $260$). Spotting this "must divide evenly" check is a fast way to catch an inconsistent problem statement before solving further, and it also explains why ratio questions in exams are almost always built around totals chosen to divide exactly — the setter has done the same divisibility check in reverse while writing the question. The same idea extends naturally to sharing money: an amount of ₹4,400 split among three partners in the ratio $3:4:4$ uses $3+4+4=11$ parts, so one part is $4400/11 = 400$, giving shares of $3\times400=₹1200$, $4\times400=₹1600$, and $4\times400=₹1600$. The check is identical in spirit — the shares add back to $₹4400$, and $1200:1600:1600$ reduces (divide by $400$) back to $3:4:4$. Notice both problems, despite looking different, use the very same "value of one part" logic.

• Forgetting to simplify. A ratio like $20:30$ should always be reduced to $2:3$ before comparing it or stating it as a final answer.
• Mixing units. A ratio must compare the same unit throughout — convert $2$ km and $500$ m to the same unit first: $2000\ \text{m} : 500\ \text{m} = 4:1$, not $2:500$.
• Treating "divide in a ratio" as "divide equally, then adjust." The reliable method is always: total parts, value of one part, then multiply — never guesswork.
• Assuming a ratio fixes the actual quantities, or reversing its order. $5:8$ could mean $5$ and $8$, or $50$ and $80$, or any equivalent pair — one extra fact (a total, or one real value) is needed to pin down the exact numbers. Also, $a:b$ is not the same as $b:a$: "boys to girls $=3:5$" is a different classroom from "girls to boys $=3:5$".

Ask the tutor: Want more practice turning word problems into a proportion, or a walkthrough of dividing an amount among four or five people in a stated ratio?