Ratio & Proportion

1The ratio, and its common multiplier

a : b compares two quantities of the same kind and survives multiplying or dividing both terms by the same non-zero number, so 6 : 8, 3 : 4 and 30 : 40 are one ratio written three ways. A ratio therefore tells you the shape of the answer and nothing about its size.

That is what the common multiplier is for. Writing the two quantities as 3x and 4x keeps the ratio exact and reduces the whole problem to finding one number, x. Almost every ratio question in the paper is solved by that first line, and the ones that look hardest are usually the ones where a candidate assigned two unknowns instead of one.

Figure. The two bars are cut into parts of identical width, and the ratio is just the count of parts in each. Nothing in the picture fixes how wide one part is — that is x, and finding it is the whole of most ratio questions. Halve the part width and label the bars 6 and 8 and you have drawn the same two lengths.

How it works

1. Reduce to lowest termsCancel any common factor first; 42 : 56 is 3 : 4 and much easier to carry.
2. Attach one multiplierWrite the quantities as 3x and 4x — never as a and b, which throws the ratio away.
3. Turn the rest into one equationEvery further condition in the question is now a statement about x alone.

2Splitting a total in a given ratio

To divide a total in the ratio a : b : c, add the parts once, divide the total by that sum to get the value of one part, and then hand out the parts. Each share is (its part / sum of parts) × total, and the sum of parts is the only quantity you should ever compute twice.

This shape covers far more than it looks: partnership profits split by capital, wages split by efficiency, mixtures split by volume, and every "A gets ₹x more than B" question — which is asking for the difference of the parts times the value of one part, not for the shares themselves.

Figure. The whole bar is the total and the two internal cuts are the only decisions in the question. Once the bar is nine equal cells wide, every quantity the examiner can ask for — a share, a difference, what fraction B receives — is a count of cells multiplied by ₹700.

How it works

1. Add the parts2 + 3 + 4 = 9. Write it down; every share and every difference is measured in these units.
2. Find one partTotal ÷ sum of parts. If this is not a whole number the ratio has probably not been reduced.
3. Answer only what was askedA share is parts × one part; a difference is the difference in parts × one part.

3Chaining ratios that share a term

A : B = 3 : 4 and B : C = 6 : 7 cannot simply be written side by side, because B is 4 in one and 6 in the other. Scale each ratio so that B becomes the LCM of the two values it takes — 12 here — and the three quantities line up: A : B : C = 9 : 12 : 14.

Long chains work the same way, one link at a time, and the guard against error is cheap: at the end, read every neighbouring pair back out of the combined ratio and check it reduces to what you were given.

Figure. Left and middle clusters are the two given ratios — B is 4 in one and 6 in the other, so they cannot sit side by side. Scale each until B is the LCM 12 (\times 3 and \times 2), and the right cluster is the joined ratio A:B:C = 9:12:14. Read neighbouring pairs back out to check.

4Proportion, and the three proportionals

a : b :: c : d says the two ratios are equal, and multiplying out gives ad = bc — the product of the extremes equals the product of the means. That one line solves every "find the fourth proportional" question: the fourth proportional to a, b, c is bc/a.

Two special cases have their own names. The mean proportional between a and b is the number that sits in both middle places, so a : m :: m : b and m = √(ab) — the geometric mean, which is also the altitude drawn to the hypotenuse of a right triangle. The third proportional to a and b is the fourth proportional to a, b, b, which is b²/a.

Figure. The apex carries the right angle, and the vertical dropped from it cuts the hypotenuse into 4 and 9. That vertical is the mean proportional between the two pieces: 4 : 6 :: 6 : 9, so h = √36 = 6. This is what "geometric mean" means geometrically, and it is why the mean proportional can never exceed the arithmetic mean — the altitude of a right triangle cannot be longer than half its hypotenuse.

How it works

1. Write the four in orderFirst, second, third, fourth. The outer two are extremes, the inner two are means.
2. Cross-multiplyad = bc. Whichever term is unknown, this is now a one-step division.
3. Name the caseMean proportional repeats the middle; third proportional repeats the second.

5Componendo and dividendo

If a/b = c/d then (a + b)/(a − b) = (c + d)/(c − d). Componendo adds 1 to both sides, dividendo subtracts it, and dividing the two results gives the rule. It is not a new fact — it is a shortcut past two lines of algebra.

It earns its place whenever a question hands you a sum and a difference of the same two quantities, which is exactly the shape of expressions like (3x + 4y)/(3x − 4y). Applied in reverse it converts that shape straight into x : y, with no expansion and no collecting of terms.

Figure. Componendo adds 1 to both sides of a/b=c/d; dividendo subtracts it. Dividing those two results is the identity (a+b)/(a-b)=(c+d)/(c-d) — a shortcut past two lines of algebra whenever a question hands you a sum and a difference of the same pair.

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 ratioBoth sides collapse: the cross terms cancel and only the two quantities are left.

6Direct and inverse variation

In direct variation the quotient is fixed: y = kx, double one and the other doubles. In inverse variation the product is fixed: xy = k, double one and the other halves. Deciding which of the two applies is the entire question, and getting it backwards is the most expensive single error in this chapter.

The test is physical, not algebraic. More men, fewer days — inverse. More days, more wages — direct. More speed, less time for the same distance — inverse. Once the type is fixed, one line finishes it: equate the quotients or equate the products.

Figure. The straight line through the origin is direct variation: the quotient y/x is the same at every point on it. The falling curve is inverse variation: the product xy is the same at every point on that. They are easy to tell apart on a graph and almost impossible to tell apart in a sentence, which is why the question is always posed in words.

How it works

1. Ask which way it movesIf more of the first genuinely means less of the second, it is inverse.
2. Write the invariantDirect: y₁/x₁ = y₂/x₂. Inverse: x₁y₁ = x₂y₂. One line, not a proportion sum.
3. Solve and sanity-check the directionIf more workers came out as more days, you picked the wrong invariant.

7Ratios that change over time

Adding the same number to both terms of a ratio changes it. That is the whole reason age problems exist: 5 : 7 today is not 5 : 7 in four years, because both ages gain 4 and the ratio drifts towards 1 : 1.

The method never varies. Write the present ages as 5x and 7x, add the time to each, set the result equal to the new ratio, and solve the one linear equation that comes out. Two unknowns are never needed — and the number of years is added, never multiplied by anything.

Figure. Both lines rise at exactly one year per year, so they are parallel and the gap between them — eight years — never changes. What does change is the ratio, because a fixed gap is a shrinking fraction of a growing age. Extend the lines far enough right and the ratio approaches 1 : 1 without ever reaching it.

How it works

1. Present ages carry the multiplierWrite 5x and 7x, not a and b. Everything the question says is now about x.
2. Shift the time"Four years hence" adds 4 to each; "four years ago" subtracts 4 from each.
3. Cross-multiply onceOne equation, one unknown. Then answer the age that was actually asked for.

8Duplicate, triplicate and compound ratios

The duplicate ratio of a : b is a² : b², the triplicate is a³ : b³, and the sub-duplicate is √a : √b. The names are old but the fact is current: lengths in the ratio 2 : 3 give areas in 4 : 9 and volumes in 8 : 27, which is where most of the marks attached to this concept actually sit.

Compounding is different and is often confused with it. To compound a : b with c : d you multiply term by term, giving ac : bd — and a ratio of two ratios, (a/b) ÷ (c/d), is ad : bc. Neither is obtained by reading off outer terms.

Figure. The larger square is one and a half times the smaller in each direction, and that factor applies twice — once across and once down — so the area goes up by 1.5² = 2.25, from 36 to 81. The picture makes the doubling of the exponent visible: the ratio of sides is a length seen once, the ratio of areas is that same length seen in both directions at once.

How it works

1. Decide the dimensionLength is power one, area power two, volume power three. Raise the ratio to that power.
2. Go down as well as upFrom areas 9 : 16 back to sides is the sub-duplicate ratio, 3 : 4.
3. Compound by multiplyinga : b with c : d gives ac : bd — never a : d.