Ratio, Proportion, and Mixtures
Simplifying ratios, telling direct proportion apart from inverse proportion, and using alligation to work out mixing ratios.
What you'll learn
- Simplify a ratio of two quantities to its lowest terms
- Distinguish a direct-proportion relationship from an inverse-proportion relationship in a word problem
- Apply the alligation method to find the ratio in which two quantities should be mixed to reach a target average rate
Prerequisites
Explanation
A ratio compares two quantities of the same kind by division rather than subtraction. The ratio 18:24 simplifies by dividing both sides by their HCF (from an earlier lesson): HCF(18, 24) = 6, so 18:24 simplifies to 3:4. A ratio is only meaningful in its simplest form when you're comparing it against another ratio or reading off a "parts" breakdown.
Direct proportion means two quantities increase or decrease together, at a fixed rate: double one, and the other doubles too (think: cost of apples is directly proportional to the number of apples bought). Inverse proportion means one quantity increases exactly as fast as the other decreases, so their product stays fixed (think: if a fixed amount of work needs doing, more workers means proportionally fewer days needed — workers × days = constant work). Mixing these up is a frequent test mistake: more workers doesn't take more time, it takes less, because "number of workers" and "days needed" are inversely, not directly, proportional.
Worked example (inverse proportion). If 6 workers can build a wall in 10 days, how long would 15 workers take, assuming the same total work and everyone works at the same steady rate? Total work = 6 × 10 = 60 "worker-days." With 15 workers: 60 ÷ 15 = 4 days. More workers, fewer days — an inverse relationship.
Alligation is a shortcut for mixture problems: when you blend a "cheap" quantity (rate c) and a "dear" (more expensive) quantity (rate d) to reach some target mean rate m, the ratio in which they must be mixed is:
cheap parts : dear parts = (d − m) : (m − c)
Worked example. A trader mixes milk worth ₹40/L with milk worth ₹60/L to produce a mixture worth ₹52/L. Using alligation: cheap parts = 60 − 52 = 8, dear parts = 52 − 40 = 12. Ratio 8:12 simplifies (divide by their HCF, 4) to 2:3 — for every 2 parts of the ₹40/L milk, use 3 parts of the ₹60/L milt. Sanity check: (2×40 + 3×60)/5 = (80+180)/5 = 260/5 = ₹52/L exactly, matching the target.
The alligation formula makes intuitive sense once you see why it works: the mean rate m must sit between c and d, and the ratio is inversely related to each price's "distance" from the mean — the closer a price is to the target mean, the more of it you need, because it contributes less "pull" away from the mean per unit. That's precisely why the cheap quantity's ratio number comes from (d − m) — the dearer price's distance — and not the other way around.
Alligation cross-diagram
Cheap rate (c) and dear rate (d) are written at the two ends of a cross, with the mean rate (m) at the center. The cheap quantity's ratio number is (d - m); the dear quantity's ratio number is (m - c) -- read diagonally across the cross from the opposite price.
Example
Alligation finds the mixing ratio of a cheaper and a dearer quantity to reach a target mean rate.
function hcfPair(a, b) {
while (b !== 0) {
[a, b] = [b, a % b];
}
return a;
}
function alligationRatio(cheapRate, dearRate, meanRate) {
const cheapParts = dearRate - meanRate;
const dearParts = meanRate - cheapRate;
const divisor = hcfPair(cheapParts, dearParts);
return [cheapParts / divisor, dearParts / divisor];
}
// Example: alligationRatio(40, 60, 52) -> [2, 3]Guided exercise
Guided exercise
Write simplifyRatio(a, b) that returns a two-element array [a divided by their HCF, b divided by their HCF] -- the ratio a:b reduced to its lowest terms. Assume a and b are non-negative integers, not both zero.
Checks: Simplifies a ratio with a common factor of 6 · Simplifies a ratio with a common factor of 15 · plus 1 hidden check
Code editor. Press Escape then Tab to leave the editor if keyboard focus becomes trapped. Press Control+Shift+M inside the editor to toggle Tab-key focus trapping.
Stuck? Get a hint.
Independent exercise
Independent exercise
Write alligationRatio(cheapRate, dearRate, meanRate) that returns the simplified [cheapParts, dearParts] mixing ratio using cheapParts = dearRate - meanRate, dearParts = meanRate - cheapRate, then reduced to lowest terms by their HCF.
Checks: Computes a typical alligation ratio · Handles a mean exactly between the two rates · plus 1 hidden check
Code editor. Press Escape then Tab to leave the editor if keyboard focus becomes trapped. Press Control+Shift+M inside the editor to toggle Tab-key focus trapping.
Stuck? Get a hint.
Common mistakes
- Simplifying a ratio by subtracting the same amount from both sides instead of dividing both sides by their HCF, which does not preserve the ratio.
- Assuming a word problem is direct proportion by default, even when the underlying quantities (like workers and days for fixed total work) are actually inversely related.
- Mixing up the alligation cross and computing cheapParts as (meanRate - cheapRate) instead of (dearRate - meanRate) -- the subtraction is criss-crossed, not matched to the same quantity's own rate.
Knowledge check
Takeaway
Ratios simplify by dividing out the HCF, inverse proportion means one quantity's increase causes another's decrease at a fixed product, and alligation's criss-cross gives the mixing ratio for any target average rate.
Summary
A ratio reduces to lowest terms via HCF. Direct proportion keeps a ratio between two quantities fixed as both grow together; inverse proportion keeps their product fixed as one grows while the other shrinks. Alligation finds a mixing ratio from cheapParts = dearRate - meanRate and dearParts = meanRate - cheapRate, then simplified like any other ratio.
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