Types of Ratios

Ratio and Proportion · 20 min

Aptitude · Ratio and Proportion · Types of ratios

Six names for four operations on a pair

Duplicate, sub-duplicate, triplicate, sub-triplicate, compound and inverse. Behind the vocabulary there are only squares, cubes, their roots, a multiplication and a flip. Learn which physical question each one answers and the names stop mattering.

Run one ratio through all six operations →
Sides are 2 : 3, so areas are 4 : 9 and volumes are 8 : 27. Duplicate squares a ratio, triplicate cubes it, and the sub- versions take you back.

01 The idea

Why a ratio needs six operations

Two square plots have sides in the ratio 2 : 3. Their areas are not in the ratio 2 : 3. Area is side × side, so the areas are in the ratio 4 : 9 — the duplicate ratio, meaning both terms squared. Cube them instead and you have the triplicate ratio, 8 : 27, which is what you need if the plots are boxes and you want their volumes.

Both operations reverse. Given areas of 4 : 9 for two squares, the sides are √4 : √9 = 2 : 3, called the sub-duplicate ratio. Given volumes of 8 : 27, the edges are the sub-triplicate ratio, 2 : 3 again. Nothing new is happening here — the four names are square, cube, square root and cube root.

One caution the textbooks skip: areas scale as the square of length only for similar figures. Two squares, yes. Two circles compared by radius, yes. Two rectangles whose lengths are 2 : 3 can have any area ratio you like, because their breadths are unconstrained. If a question does not say the shapes are similar, it usually means squares or circles — but check.

The last two operations are not about powers at all. The compound ratio of a : b and c : d is ac : bd, which merges two comparisons into one. The inverse ratio of a : b is b : a, which is what you reach for when the two quantities move in opposite directions — more speed, less time. Compound a ratio with its own inverse and you get 1 : 1, which is the neatest way to remember what “inverse” means.

Duplicate squares the terms, triplicate cubes them, the sub- forms take the roots, compound multiplies two ratios term by term, and inverse flips one.
Duplicate and sub-duplicatea² : b² and √a : √b. Sides to areas, and areas back to sides, for similar figures.
Triplicate and sub-triplicatea³ : b³ and ∛a : ∛b. Edges to volumes, and volumes back to edges.
Compound and inverseac : bd merges two ratios; b : a flips one. A ratio compounded with its inverse is always 1 : 1.

02 Worked example

Sides 2 : 3, taken through all six

One pair of squares runs the whole lesson. The sides of two squares are in the ratio 2 : 3. Find the ratio of their areas, the ratio of the volumes of cubes built on those sides, and show how to get back from each of those to the side ratio.

1
Sides to areas — the duplicate ratioArea of a square is side squared, so squaring both terms answers the question directly. No algebra is needed and no total is needed.2² : 3² = 4 : 9
2
Areas back to sides — the sub-duplicate ratioThe reverse operation. Take the square root of each term, which works cleanly here because 4 and 9 are perfect squares.√4 : √9 = 2 : 3 ✓
3
Sides to volumes — the triplicate ratioVolume of a cube is side cubed, so cube both terms. Compare 8 : 27 with the original 2 : 3: cubing has pushed the gap much wider.2³ : 3³ = 8 : 27, and ∛8 : ∛27 = 2 : 3 ✓
4
Compound the sides with the areasMultiply first terms together and second terms together. Side times area is volume, so this had better give the volume ratio — and it does.(2 × 4) : (3 × 9) = 8 : 27 ✓
5
The inverse, and the check it gives youFlip the terms for the inverse ratio. Compounding a ratio with its own inverse gives 1 : 1, which is a one-line way to confirm you flipped rather than mis-copied.inverse of 2 : 3 is 3 : 2  ⇒  (2×3) : (3×2) = 6 : 6 = 1 : 1

Read the chain 2 : 3, then 4 : 9, then 8 : 27. Each step multiplies the gap rather than adding to it, which is why a modest 2 : 3 difference in length becomes a 1 : 3.375 difference in volume. This is the same effect that makes a 12% length error in a manufactured part a 40% volume error, and it is the reason these operations have names at all.

03 The method

The formula bank, and which question triggers each entry

Six formulas, and one column that matters more than the formulas: the phrase in the question that tells you which one to reach for.

Duplicate a² : b² · Sub-duplicate √a : √b · Triplicate a³ : b³ · Sub-triplicate ∛a : ∛b · Compound (a:b, c:d) = ac : bd · Inverse b : a. Always take the simplest form of the ratio first, then apply the operation.
Decide by direction, not by vocabulary. If the quantity you want grows faster than the one you have, you need a power — duplicate for areas, triplicate for volumes. If it grows slower, you need a root. If it moves the opposite way, you need the inverse. The six names are only labels on those three decisions, and an exam that phrases the question as “the ratio of their surface areas” never uses the word “duplicate” at all.
TypeFormulaOn 2 : 3The phrase that signals it
Duplicatea² : b²4 : 9areas, from sides
Sub-duplicate√a : √b√2 : √3sides, from areas
Triplicatea³ : b³8 : 27volumes, from edges
Sub-triplicate∛a : ∛b∛2 : ∛3edges, from volumes
Compound with 4 : 9ac : bd8 : 27two splits in sequence
Inverseb : a3 : 2speed against time, men against days
Compound with its inversealways 1 : 16 : 6 = 1 : 1a check, not a question

05 Cheat sheet

Six operations on one page

The formulas, the two that get confused with each other, and the geometry fact that makes half of these questions answerable on sight.

CaseRuleOn 2 : 3
Duplicatea² : b²4 : 9
Triplicatea³ : b³8 : 27
Sub-duplicate of 4 : 9√a : √b2 : 3
Sub-triplicate of 8 : 27∛a : ∛b2 : 3
Compoundac : bdwith 4 : 9 → 8 : 27
Inverseb : a3 : 2
Duplicate of a non-similar pairdoes not applytwo rectangles need both dimensions
Simplify before you squareThe duplicate ratio of 18 : 12 is 324 : 144, which is 9 : 4 — the same as squaring 3 : 2. Both are right, but only one is in simplest form, and only one is marked correct.
Powers widen, roots narrow2 : 3 is a 1.5× gap; its duplicate 4 : 9 is 2.25× and its triplicate 8 : 27 is 3.375×. A root moves the other way. Use that as an instant sanity check on any answer.
A ratio times its inverse is 1 : 1The compound of a : b with b : a is ab : ba, which is always 1 : 1. This is the cheapest available check that you flipped the right ratio.

06 Where & why

Where this shows up

These six operations look like vocabulary drill, and in a mock test they are. In a real paper they arrive disguised as geometry, speed or work questions.

SSC CGL · Bank Prelims
“Find the sub-duplicate ratio of 25 : 36”

Direct vocabulary questions, worth a mark each, answerable in seconds. 5 : 6 here. The only risk is confusing sub-duplicate with duplicate under time pressure.

Mensuration questions
Sides, areas and volumes of similar figures

“The radii of two spheres are 2 : 3, find the ratio of their volumes” is a triplicate ratio question wearing a geometry hat. 8 : 27.

Time, speed and distance
The inverse ratio

Over a fixed distance, speeds of 4 : 5 mean times of 5 : 4, so the faster one arrives first. The same flip runs the men-and-days questions in time and work.

Time and work
The compound ratio

Efficiency compounded with days gives work done. Two ratios applied one after the other are one compound ratio, which is why chain-rule questions collapse so neatly.

You will almost never see the word “duplicate” in a real paper. You will constantly see “the ratio of their areas”. Learn the trigger phrases in the section 03 table and the vocabulary becomes optional.

07 Interview questions

What gets asked

Ten. The first six are the definitions an interviewer fires off quickly; the last four are where the marks are.

What is the duplicate ratio of a : b?
a² : b² — both terms squared. The duplicate ratio of 2 : 3 is 4 : 9. It is what you want when the quantity asked about grows as the square of the quantity given, most often areas from sides.
And the sub-duplicate ratio?
√a : √b — the square root of each term, which undoes the duplicate. The sub-duplicate ratio of 25 : 36 is 5 : 6. Exam questions choose perfect squares, so if your roots are not clean, re-read which operation was wanted.
Define the compound ratio.
The compound ratio of a : b and c : d is ac : bd — multiply first terms together and second terms together. Compounding 2 : 3 with 4 : 5 gives 8 : 15. It is how two comparisons applied in sequence collapse into one.
What is the inverse ratio used for?
For quantities that move in opposite directions. Over a fixed distance, speeds in the ratio 4 : 5 give times in the ratio 5 : 4. The same flip handles men against days for a fixed amount of work.
The radii of two spheres are in the ratio 2 : 3. Find the ratio of their volumes.
8 : 27, the triplicate ratio. Volume goes as the cube of the radius, so cube both terms. Their surface areas would be in the ratio 4 : 9, since area goes as the square.
The volumes of two cubes are 64 : 125. Find the ratio of their edges.
4 : 5, the sub-triplicate ratio — take the cube root of each term. If the question then asks for surface areas, square that answer to get 16 : 25.
Two rectangles have lengths in the ratio 3 : 4. Are their areas 9 : 16?
Not necessarily. That step needs the figures to be similar, so that both dimensions scale together. For two squares or two circles it holds; for two rectangles you also need their breadths, because the breadth ratio compounds with the length ratio.
Compound 2 : 3 with 3 : 2. What do you get, and why is it worth knowing?
1 : 1, because (2×3) : (3×2) is 6 : 6. A ratio compounded with its own inverse is always 1 : 1, which makes it the fastest available check that you inverted the ratio you meant to invert.
Duplicate ratio versus the square of a ratio — is there a difference?
No, they are the same operation described two ways: squaring the fraction a/b gives a²/b², which is the duplicate ratio. The vocabulary survives only because exam papers still use it. What matters is recognising that the gap widens, not what it is called.
When would you actually use these outside an exam?
The powers matter constantly in engineering: scale a component up 10% in length and its volume, and therefore its mass and material cost, goes up about 33%. The inverse ratio is everyday work planning. The word “sub-duplicate”, honestly, appears nowhere but in competitive exams — the operation matters, the name does not.

08 Practice problems

Six on the six

The first two are vocabulary speed drills — aim for under fifteen seconds each. The last one goes both ways in one problem.

Two at once

Easy
Find the duplicate ratio of 5 : 7 and the sub-duplicate ratio of 49 : 81.
Follow-up
The two operations are inverses of each other, so doing them back to back stops you confusing which one squares and which one roots.
Show the hint
One answer has three-digit terms and the other has single-digit terms.

Merge three

Easy
Find the compound ratio of 2 : 3, 5 : 4 and 9 : 10, in simplest form.
Follow-up
Three ratios rather than two, so you have to see that compounding extends by multiplying all the first terms and all the second terms. The answer simplifies a long way.
Show the hint
Multiply 2 × 5 × 9 and 3 × 4 × 10, then cancel.

Speeds and times

Medium
Two cars cover the same route. Their speeds are in the ratio 5 : 6. Find the ratio of the times they take, and say which arrives first and by what fraction of the slower car's time.
Follow-up
The flip is the easy half. The second half checks that you can read the inverse ratio as a physical statement rather than just a pair of numbers.
Show the hint
The time ratio is the inverse of the speed ratio; then compare the difference against the larger time.

From volume to surface

Medium
The volumes of two cubes are in the ratio 27 : 64. Find the ratio of their edges and hence the ratio of their total surface areas.
Follow-up
Two operations in sequence, and in opposite directions: a cube root down to edges, then a square back up to areas. Doing it in one jump is the standard error.
Show the hint
Cube root first, then square what you get.

The unsimplified trap

Medium
Find the duplicate ratio of 18 : 12, giving your answer in simplest form. Then find the duplicate ratio of 3 : 2 and explain in one sentence why the two answers agree.
Follow-up
Both routes are legitimate and one is much less work. The explanation is the real question: it tests whether you know that squaring commutes with simplifying.
Show the hint
Simplify 18 : 12 first and the arithmetic collapses from three digits to one.

Scaling a real object

Hard
A cube-shaped water tank is replaced by a larger cube whose edge is 20% longer. (a) Write the ratio of the new edge to the old edge as a ratio of whole numbers. (b) Find the ratio of the surface areas and the ratio of the capacities. (c) The tank is painted, and paint costs scale with surface area while water costs scale with capacity. State which cost rises by the larger percentage, and by how much each rises.
Follow-up
A 20% length change becomes a 44% area change and a 72.8% volume change, and part (c) is where that stops being arithmetic and starts being a decision. This is the same asymmetry that makes large storage cheaper per litre than small storage.
Show the hint
Start by writing 20% longer as the ratio 6 : 5, then apply the duplicate and the triplicate.