Proportion

Proportion is simply a statement that two ratios match. Four quantities $a, b, c, d$ are said to be in proportion, written $a:b :: c:d$, if

where: $a$ and $d$ are the extremes (the outer terms) and $b$ and $c$ are the means (the inner terms). A fast way to check proportion without doing any division is the cross-product rule:

where: the product of the extremes must equal the product of the means. If you are handed four numbers — say 2, 3, 8, 12 — you do not need to simplify any fraction; just check $2 \times 12 = 24$ against $3 \times 8 = 24$. They match, so the numbers are in proportion.

Direct proportion is what happens when two varying quantities, not four fixed numbers, keep the same ratio as they change together. If $x$ and $y$ are in direct proportion,

where: $k$ is the constant of proportionality — for example, the fixed cost of one notebook. The practical meaning is powerful: if $x$ is scaled by any factor, $y$ is scaled by exactly the same factor. Double the notebooks, double the cost; triple the recipe, triple every ingredient. This "same scaling factor" idea is often the quickest route to an answer, because you can skip setting up fractions entirely and just ask "what did the known quantity get multiplied by?" Not every relationship between two changing quantities is a direct proportion — only ones that scale together and pass through zero (zero notebooks cost zero rupees). The figure below shows what direct proportion looks like as a picture: a straight line through the origin, unlike a relationship that shifts or bends.

The goal is to solve any direct-proportion word problem in which three of the four related quantities are known and one is unknown, using nothing beyond ratio equality and multiplication or division — no algebra beyond isolating one letter.

Step 1. Confirm the situation really is direct proportion — check that the two quantities increase together and that doubling one would double the other (zero of one thing goes with zero of the other). Word problems about cost of goods, distance covered at constant speed, recipe scaling, and map distances are the classic direct-proportion settings for this chapter.

Step 2. Set up the two ratios in matching order — the same kind of quantity must sit in the same position in both fractions (both numerators are "cost," both denominators are "number of items," for instance). Write $\dfrac{a}{b} = \dfrac{c}{d}$ with the unknown as one of the four letters.

Step 3. Solve by the equal-ratios method — cross-multiply, $a \times d = b \times c$, and divide to isolate the unknown. Alternatively, solve by the unitary method — divide to find the value for a single unit of the varying quantity, then multiply by however many units the question asks for.

Step 4. Cross-check the two methods against each other. Because a genuine direct-proportion relationship has one fixed constant $k = y/x$, both methods must land on the same number; if they disagree, the ratio was set up "upside down" somewhere in Step 2. This four-step routine works for cost problems, distance-speed-time at constant speed, recipe scaling, and map or scale-drawing enlargements alike, because all of them are really the same underlying idea in different clothing: one fixed rate connecting two quantities that grow together.

Problem: The cost of 8 notebooks is ₹280. Find the cost of 13 notebooks, assuming cost is directly proportional to the number of notebooks. Then use the result to find the cost of 20 notebooks.

Step 1 — Unitary method for 13 notebooks. Find the cost of one notebook first: $\dfrac{280}{8} = ₹35$ per notebook. Scale up: $35 \times 13 = ₹455$.

Step 2 — Check with the equal-ratios method. Let the cost of 13 notebooks be $x$.

Both methods agree: ₹455, confirming the constant cost per notebook really is fixed at ₹35.

Step 3 — Extend to 20 notebooks using the "factor of change" idea. Since 20 notebooks is $20/8 = 2.5$ times as many as 8 notebooks, the cost must also scale by $2.5$: $280 \times 2.5 = ₹700$. Check against the unit cost: $35 \times 20 = ₹700$ — same answer, obtained a third way.

Competitive stretch

Problem: On a map, 3 cm represents 45 km. Two towns are 7.5 cm apart on the map. Find the real distance.

Step 1. Set up matching ratios: $\dfrac{3}{45} = \dfrac{7.5}{x}$.

Step 2. Cross-multiply: $3x = 45 \times 7.5 = 337.5$, so $x = \dfrac{337.5}{3} = 112.5$.

Answer: 112.5 km. Unitary check: 1 cm represents $45 \div 3 = 15$ km, and $7.5 \times 15 = 112.5$ km — the two methods agree once again.

• Ratio set up upside down — writing $\dfrac{3}{45} = \dfrac{x}{7.5}$ instead of $\dfrac{3}{45}=\dfrac{7.5}{x}$ mixes which quantity sits where; always keep the same kind of quantity in the same position in both fractions.
• Assuming every word problem is a direct proportion — direct proportion needs $y = kx$, a straight scaling through zero; some relationships (worker-day type problems, certain unit conversions) do not behave this way and are outside this chapter's scope.
• Mismatched units — mixing cm with m, or minutes with hours, before cross-multiplying gives a wrong answer even though the method itself was correct; convert units first.
• Confusing "in proportion" with "directly proportional" — testing four fixed numbers with the cross-product rule answers a different question from recognising that two changing quantities scale together.
• Stopping after one method — if you only solve by the unitary method or only by the equal-ratios method, an arithmetic slip can slide through unnoticed; solving with one and checking with the other is what actually catches mistakes before they cost marks.

Ask the tutor: If you are stuck on which quantity goes in the numerator and which in the denominator, ask the AI tutor to set up the ratio with you step by step and check your cross-multiplication.