Ratio, Proportion, Mixtures and Alligation

1The ratio, and its common multiplier

A ratio a : b compares two quantities of the same kind and is unchanged when both terms are multiplied or divided by the same non-zero number. It fixes the shape of the answer and says nothing about its size.

The size comes from one common multiplier. Writing the quantities as kx and x for a condition a : b = k converts the ratio into a linear equation in a single unknown — never into two free letters that throw the ratio away.

Figure. Each bar is cut into equal parts of identical value, so the ratio is just the count of parts. Nothing in the picture fixes how large one part is — that is the common multiplier x, and finding it is most of the chapter.

How it works

1. Reduce firstCancel any common factor so the parts are small integers.
2. Attach one multiplierFor a : b = k write a = kx and b = x, or write both as parts times the same x.
3. One equationEvery further condition in the question is now a statement about that one unknown.

2Proportion and the cross product

If a : b = c : d, written a : b :: c : d, the cross product of extremes and means is equal: ad = bc. That single line solves every "find the missing proportional" question.

The fourth proportional to a, b, c is the d in a : b :: c : d, namely bc/a. Keep the given three in order — swapping them swaps the answer.

Figure. Both bars are the same shape — each left part is three-fifths of its right. The cross product checks that: 3\times 20 = 5\times 12 = 60. Scale is common across both bars so 20 is four times 5 and 12 is four times 3.

How it works

1. Write the four in orderFirst and fourth are extremes; second and third are means.
2. Cross-multiplyad = bc. Whichever term is unknown is now one division away.
3. Check by reducingThe completed proportion must reduce to the same ratio on both sides.

3Componendo and dividendo

If a/b = c/d, then \frac{a+b}{a-b} = \frac{c+d}{c-d}. Componendo adds 1 to both sides of the ratio, dividendo subtracts it, and dividing those two results gives the rule. It is a shortcut past two lines of algebra, not a new fact.

It earns its place whenever a question hands you a sum and a difference of the same two quantities — expressions shaped like (3x + 4y)/(3x - 4y). Applied in reverse, that shape collapses straight into x : y.

Figure. Componendo–dividendo turns 5 : 3 into 8 : 2, which is 4. The sum segment grows; the difference shrinks — that contrast is the whole identity, and (a+b)/(a-b)=8/2=4.

How it works

1. Spot the shapeNumerator and denominator must be the sum and difference of the same two terms.
2. Add and subtractNew numerator is sum + difference; new denominator is sum − difference.
3. Simplify to a plain ratioCross terms cancel and only the two quantities remain.

4Chaining ratios that share a term

To merge a : b and b : c that share the middle term, scale each ratio so the common value matches, then read off a single chain a : b : c. Writing the two ratios side by side without scaling leaves b meaning two different numbers.

Longer chains work the same way, one shared term at a time. At the end, read every neighbouring pair back out of the combined ratio and check it reduces to what you were given.

Figure. The tall bars are the shared term before and after scaling: B arrives as 4 and as 6, so both links are scaled to LCM 12 (positive). Only then the chain bar 9 : 12 : 14 is honest — the finished parts alone would hide that rescale step.

5Alligation as inverse differences

When ingredients at values c_1 and c_2 mix to a mean m, the quantities are in the inverse ratio of their gaps from the mean: \frac{\text{cheaper qty}}{\text{dearer qty}} = \frac{c_2 - m}{m - c_1}. Place the mean in the middle and take absolute differences crosswise.

The same rule solves any weighted-average problem — mixtures, average speed, average age, blended interest rates — not just liquids. For three or more components, apply alligation to two at a time, or set the total as a variable and equate the weighted sum.

Figure. On the price line the mean sits between the two grades. Scale: 10 rupees of gap maps to 0.80 of width, so the 6-rupee left gap is 0.48 wide and the 4-rupee right gap is 0.32 wide. Cross the gaps for quantities — cheaper : dearer = 4 : 6 = 2 : 3 — so more of the dearer rice appears because m sits nearer c₂. Tick marks are not to the rupee scale of a full 0–40 axis; only the two gaps are to scale with each other.

How it works

1. Name cheaper, dearer, meanc₁ below the mean, c₂ above it; m must sit strictly between them.
2. Take cross differencesCheaper quantity pairs with (c₂ − m); dearer quantity pairs with (m − c₁).
3. Simplify the ratioCancel a common factor; the cheaper side always pairs with the larger gap when m sits nearer the dearer price.

6Repeated replacement

Removing a fixed volume x from a container of volume V and replacing it — done n times — leaves the original component as \left(1 - \frac{x}{V}\right)^n of its start. Each operation multiplies the remaining pure amount by the same fraction.

Track concentration as a power of that ratio. Computing absolute litres round by round works for n = 1 and becomes an arithmetic trap for n ≥ 2.

Figure. Each bar is the fraction of the original milk still present. The same factor 0.8 multiplies every round, so the heights are 1.00, 0.80, 0.64 — a geometric sequence, not an arithmetic drop of 0.20 per step. Bars are to scale with max = 1.

How it works

1. Write the retention factorEach round keeps (1 − x/V) of whatever pure liquid was present at the start of that round.
2. Raise to nAfter n identical rounds the factor is (1 − x/V)^n.
3. Scale by the start amountPure left = V × (1 − x/V)^n when the vessel began full of the pure liquid.

7Partnership: capital times time

Profit is divided in the ratio of each partner's capital multiplied by the time it stays invested. Equal money for unequal months is not an equal split; equal products C × T are.

When every partner invests for the same duration, the time factor cancels and the profit ratio collapses to the capital ratio alone.

Figure. The two bars are the capital × time products, not the capitals. They are equal at 96000, so the profit split is 1 : 1 even though B put in more money. Bars are to scale with each other (identical values draw identical heights).

How it works

1. Weight each partnerCompute Cᵢ × Tᵢ for every partner, in the same money and time units.
2. Form the ratioProfit shares are in the ratio of those products.
3. Scale to the profitIf a total profit is given, multiply each partner's fraction of the sum of products.