Chapter 5.2 — Exercise 5.1 — Basic Applications

Question 1 — Converting Units Before Comparing

A ratio only makes sense once both quantities share the same unit. Comparing 6 hours to 8 hours gives 6 : 8 = 3 : 4 directly. But comparing 8 litres of milk to 750 millilitres needs a conversion first: 8 litres = 8000 millilitres, so the ratio is 8000 : 750 = 32 : 3. Speeds compare the same way once units already match: 15 km/h to 30 km/h simplifies straight to 1 : 2.

Questions 2 to 4 — Working Backward Through a Compound Ratio

These questions give you the compound ratio's final result and ask you to recover an unknown term.

• Compound ratio of 5:8 and 3:7 is 45:x. Since 5×3 : 8×7 = 15:56 must equal 45:x, cross-multiplying gives 15x = 56×45, so x = 168.
• Compound ratio of 7:5 and 8:x is 84:60. Since 7×8 : 5×x = 56:5x must equal 84:60, cross-multiplying gives 56×60 = 5x×84, so x = 8.
• Compound ratio of 3:4 and the inverse ratio of 4:5 is 45:x. The inverse of 4:5 is 5:4, so the compound ratio is 3×5 : 4×4 = 15:16, matched against 45:x gives x = 48.

Question 5 — Scaling a Ratio to a New Total

A school has 3 teachers for every 60 students, simplifying to 1 : 20. For 400 students in the same ratio: 1 : 20 = x : 400, so x = 400/20 = 20 teachers. Scaling a ratio up or down to match a new total, while keeping the underlying comparison fixed, is one of the most common real-world uses of proportion.

Question 6 — Every Pairwise Ratio of a Triangle

For a triangle with sides AB = 8 cm, BC = 6 cm and AC = 10 cm, there are six possible ordered ratios between pairs of sides — each side compared to each other side, in both directions:

• AB : BC = 8 : 6 = 4 : 3  |  BC : AB = 6 : 8 = 3 : 4
• BC : AC = 6 : 10 = 3 : 5  |  AC : BC = 10 : 6 = 5 : 3
• AB : AC = 8 : 10 = 4 : 5  |  AC : AB = 10 : 8 = 5 : 4

Notice each pair produces two different ratios depending on the order — AB:BC and BC:AB are reciprocals of each other, not the same comparison.

Questions 7 and 8 — Ratios from Percentages and Letter Counts

If 9 of 24 students scored below 75%, the remaining 24 − 9 = 15 scored 75% or above, giving a ratio of 9 : 15 = 3 : 5. A different kind of counting problem: in the word "MISSISSIPPI," the 4 vowels (I, I, I, I) compare to the 7 consonants (M, S, S, S, S, P, P) as 4 : 7.

Question 9 — A Profit Share from a Percentage

If Rehana receives 25% of a month's profit and that amounted to ₹2080, the total profit x satisfies (25/100) × x = 2080, so x/4 = 2080, giving x = ₹8320. Turning a known percentage share into the whole is the reverse of the more familiar "find x% of y" calculation.

Question 14 — A Point Between Two Points on a Line

Points P and Q lie on segment AB, with AP:PB = 2:3 and AQ:QB = 3:4, and PQ = 2 units. Adding 1 to both sides of each ratio converts it into a fraction of the whole segment: AP:PB = 2:3 gives AB:PB = 5:3, so PB = (3/5)AB; similarly AB:QB = 7:4 gives QB = (4/7)AB. Since PQ = PB − QB:

Since PQ = 2, AB/35 = 2, so AB = 70 units. This technique — converting a part-to-part ratio into a part-to-whole fraction by adding 1 — is worth remembering any time a problem gives ratios of two segments that share an endpoint.

Reading a Ratio Question Before Simplifying Anything

A useful habit across every question in this exercise is identifying, before any arithmetic, exactly which two quantities are being compared and in which order the question asks for them. Question 6's triangle makes this concrete: "AB:BC" and "BC:AB" use the same two numbers but answer different questions, and a problem that specifically asks for "the ratio of AC to AB" is not answered correctly by writing AB:AC instead, even though the two ratios are closely related. Slowing down to name the two quantities in the order the question actually specifies, before simplifying, avoids a whole category of otherwise-easy questions being answered with the correct numbers in the wrong arrangement.

Why Question 14's Trick Is Worth Remembering on Its Own

Adding 1 to a part-to-part ratio to convert it into a part-to-whole fraction is a genuinely reusable technique well beyond this one question. Any time a problem gives AP:PB (a part compared to another part) but the more useful quantity is AP:AB or PB:AB (a part compared to the whole), adding 1 to the ratio AP/PB gives (AP+PB)/PB = AB/PB directly — no need to introduce a variable for AB and solve for it separately. This shortcut turns up again in later geometry, whenever a point divides a segment or an angle in a given ratio.

Order and Units — Where Ratios Trip People Up

• Forgetting to match units before forming a ratio. A ratio between quantities in different units is meaningless until they're converted to match.
• Reversing a ratio's order by mistake. AB:BC and BC:AB are reciprocal, not identical — always check which quantity was named first in the question.
• Adding compound ratios instead of multiplying. The compound ratio of a:b and c:d is ac:bd — a product of the two ratios, not a sum.

Two Different Skills Hiding in One Exercise

Looking back across all fourteen questions, they split into two genuinely different skills wearing the same "ratio" label. Questions 1, 6, 7 and 8 are about forming a ratio correctly from a description — getting the units right, the order right, the counting right. Questions 2 through 5, 9 and 14 are about using an already-formed ratio or proportion to find a missing piece — cross-multiplying, scaling, or rearranging. Recognising which of the two a question is asking for is often the harder half of the problem; the arithmetic that follows, once that's clear, is usually the easy part — cross-multiplication and simplification are the same two or three steps no matter which question they're attached to.

Ratios Turning Into Percentages

The percentage-based reasoning in Questions 7 and 9 is exactly what Exercise 5.2 builds on for discounts, profit and loss, and GST calculations. For the underlying ratio and proportion definitions this exercise assumes, revisit the Introduction to Ratios and Proportion.