Aptitude · Ratio and Proportion · Model 6

Two formulas that look alike and are not

The third proportional squares the second term and divides by the first. The mean proportional multiplies both terms and takes the root. A paper puts them in the same question precisely because the pair is easy to mix up — and there are shortcuts for both that skip most of the arithmetic.

Run both formulas on one combination question →

Third proportional to 12 and 30 is 75; mean proportional between 9 and 25 is 15; their ratio is 5 : 1. One question, two formulas, and the arithmetic is the easy part.

01 The idea

The combination question

“Find the ratio between the third proportional to 12 and 30, and the mean proportional between 9 and 25.” That is one question with two formulas in it, and it is set in exactly this shape because a candidate who half-remembers the pair will apply one of them to the wrong numbers.

The third proportional to 12 and 30 completes 12 : 30 :: 30 : x. The middle term repeats, so x = 30²/12 = 900/12 = 75. There is a faster route: a continued proportion multiplies by the same factor each step, and 12 to 30 is a factor of 2.5, so one more jump gives 30 × 2.5 = 75. That is the multiplier-jump shortcut, and when the factor is a whole number it is dramatically quicker.

The mean proportional between 9 and 25 is the term that sits between them, so 9 : x :: x : 25 and x² = 225, giving x = 15. Note that 15 is not the average of 9 and 25, which is 17. The mean proportional is the geometric mean and it always sits below the arithmetic mean for two unequal numbers.

So the answer is 75 : 15 = 5 : 1. Two formulas, two checks, one simplification. And each check is a single line: 12 × 75 = 900 = 30², and 9 × 25 = 225 = 15². Running them costs a few seconds and is the only thing that catches a formula applied to the wrong pair.

Third proportional to a and b is b²/a. Mean proportional between a and c is √(ac). Both are the same cross-product equation with a different term left blank.

Multiplier jumpA continued proportion is a geometric chain, so the step from a to b repeats. 18 → 36 is ×2, so the third proportional is 36 × 2 = 72 — no squaring and no division.

Split and rootFor an ugly product, factor it into perfect squares before rooting. √(8×18) = √(4×2×9×2) = 2×3×2 = 12, then count the decimal places separately.

The make-it-proportional trap“What must be added to 10, 16, 22 and 32 to put them in proportion?” The x² terms cancel when you cross-multiply, so it is linear. The answer is 8, giving 18 : 24 :: 30 : 40.

02 Worked example

Third proportional to 12 and 30, against the mean proportional between 9 and 25

One combination question runs the whole lesson. Find the ratio between the third proportional to 12 and 30 and the mean proportional between 9 and 25.

Third proportional: repeat the second termThe chain is 12 : 30 :: 30 : x, with 30 written twice. Cross-multiply and divide — and note that 12 divides 900 exactly, which is not an accident.12x = 30 × 30 = 900 ⇒ x = 75

The same answer by multiplier jumpA continued proportion steps by a fixed factor. From 12 to 30 the factor is 2.5, so one more step lands on the third proportional.30/12 = 2.5 ⇒ 30 × 2.5 = 75 ✓

Mean proportional: multiply and rootThe missing term sits in the middle, so x² equals the product of the outer two. This is the geometric mean.x² = 9 × 25 = 225 ⇒ x = 15 (the average would be 17)

Check both chainsEach answer sits inside its own continued proportion, and each test is one multiplication a side.12 × 75 = 900 = 30² ✓ 9 × 25 = 225 = 15² ✓

Take the ratio the question asked forBoth numbers are on the page, so all that remains is a simplification. Fifteen divides 75 exactly.75 : 15 = 5 : 1

Notice what the two formulas have in common: both come from product of extremes = product of means, with a different term left blank. The third proportional leaves the last term blank and repeats the second; the mean proportional leaves the middle blank and gets a square root out of it. If you can write the chain down with the blank in the right place, you never have to remember which formula is which.

03 The method

The shortcuts, and the trap that looks quadratic

Five techniques from the exam sets, and one problem shape that appears difficult and is not.

Third proportional to a and b: b²/a, or the multiplier jump b × (b/a). Mean proportional between a and c: √(ac). Fourth proportional to a, b, c: bc/a, or scale visually — simplify a/b first, then see what multiplies the numerator into c.

The make-it-proportional trap is linear, not quadratic. “What number x must be added to 10, 16, 22 and 32 so the results are in proportion?” Write (10+x)(32+x) = (16+x)(22+x); the x² terms appear on both sides and cancel, leaving 4x = 32 and x = 8. Check: 18 : 24 = 3 : 4 and 30 : 40 = 3 : 4. For the subtraction version there is a faster route still, the ratio of differences: for (23−x)/(39−x) = (32−x)/(56−x), the two gaps are 16 and 24, which is 2 : 3, so 3(23−x) = 2(32−x) and x = 5 in one line.

Technique	Use it for	Example
Repeat the middle term	third proportional	12 : 30 :: 30 : x → 75
Multiplier jump	third proportional, whole factor	18 → 36 is ×2, so 72
Multiply and root	mean proportional	√(9×25) = 15
Split and root	mean proportional with decimals	√(0.08×0.18) = 0.12
Visual scaling	fourth proportional	12/21 = 4/7, and 4→8 doubles, so 7→14
Value assumption	algebraic proportionals	put x = 3, y = 2 and test the options
Ratio of differences	make-it-proportional by subtraction	gaps 16 : 24 = 2 : 3 → x = 5
Treating a mean proportional as an average	`always wrong`	√225 = 15, not 17

05 Cheat sheet

Model 6 on one page

Three formulas, four shortcuts, and the two errors that account for nearly every wrong answer in this model.

Case	Rule	On the lesson numbers
Third proportional to a, b	`b²/a`	30²/12 = 75
Multiplier jump	`b × (b/a)`	30 × 2.5 = 75
Mean proportional between a, c	`√(ac)`	√225 = 15
Fourth proportional to a, b, c	`bc/a`	21×8/12 = 14
Add x to put four numbers in proportion	`the x² terms cancel`	10, 16, 22, 32 → x = 8
Subtract x, by ratio of differences	`gap₁ : gap₂`	23, 39, 32, 56 → x = 5
Mean proportional read as an average	`always wrong`	15, not 17

Both formulas come from one equationProduct of extremes equals product of means. Leave the last term blank and repeat the second and you get b²/a; leave the middle blank and you get √(ac). Writing the chain with the blank in place beats memorising two formulas.

The make-it-proportional trap is linearCross-multiplying (a+x)(d+x) = (b+x)(c+x) puts x² on both sides, so it cancels. Every question of this shape reduces to one linear equation, however alarming it looks.

The geometric mean sits below the averageFor 9 and 25 it is 15 against 17; for 4 and 49 it is 14 against 26.5. The gap widens as the two numbers move apart, which makes it a useful sanity check on any mean-proportional answer.

06 Where & why

Where this shows up

These are short, high-yield questions. The value here is entirely in speed and in not confusing the two formulas under pressure.

SSC CGL · Bank Prelims

One-line proportional questions

“Find the third proportional to 18 and 36.” A mark each, ten seconds each with the multiplier jump. They come in clusters, so the time saved compounds.

SSC CGL Tier 2

The combination question

Two formulas in one question, asked as a ratio. Doing the raw arithmetic is genuinely faster here than eliminating options, which is worth knowing in advance.

Algebraic versions

“Third proportional to x² − y² and x + y”

Factor and cancel for (x+y)/(x−y), or put x = 3 and y = 2 and test the options. The substitution route is faster and just as safe when the options are distinct.

Mains-level traps

“What must be added to each of …”

Looks quadratic, is linear. The ratio-of-differences route handles the subtraction version in about twenty seconds, which is the difference between attempting it and skipping it.

Two habits: write the chain with the blank in the right place instead of recalling a formula, and check every answer with one cross product. Both are quick, and between them they remove the confusion this model is built to exploit.

07 Interview questions

What gets asked

Ten. The second and third are the pair an interviewer uses to see whether you actually distinguish the two formulas.

Find the third proportional to 12 and 30.

75. The chain is 12 : 30 :: 30 : x, so x = 30²/12 = 900/12 = 75. Faster: the step from 12 to 30 is a factor of 2.5, so one more step gives 30 × 2.5 = 75.

Find the mean proportional between 9 and 25.

15. The mean proportional is √(9 × 25) = √225 = 15. It is the geometric mean, so it is not 17 — the average of two unequal numbers is always larger than their geometric mean.

Third proportional and mean proportional — how do you keep them apart?

By where the blank goes. The third proportional leaves the last term blank and repeats the second: a : b :: b : x, so x = b²/a. The mean proportional leaves the middle blank: a : x :: x : c, so x = √(ac). Both come from the same cross-product rule.

Explain the multiplier-jump shortcut.

A continued proportion is a geometric chain, so the ratio between consecutive terms is constant. For 18 and 36 that factor is 2, so the third proportional is 36 × 2 = 72. It avoids squaring a large number and dividing, and it is much the faster route whenever the factor is a whole number.

Find the mean proportional between 0.08 and 0.18.

0.12. Ignore the decimals and split the product into perfect squares: 8 × 18 = 4 × 2 × 9 × 2, whose root is 2 × 3 × 2 = 12. There are four decimal places under the root, so two come out, giving 0.12.

Find the ratio between the third proportional to 12 and 30 and the mean proportional between 9 and 25.

5 : 1. The third proportional is 75 and the mean proportional is 15, and 75 : 15 simplifies to 5 : 1. On a paper this is one of the rare cases where raw calculation beats option elimination, because both numbers come out in a single step.

What number must be added to each of 10, 16, 22 and 32 so that they are in proportion?

8. Setting (10+x)/(16+x) = (22+x)/(32+x) and cross-multiplying puts x² on both sides, so it cancels and you are left with 4x = 32. Check: 18 : 24 and 30 : 40 both reduce to 3 : 4.

Why is that question not quadratic?

Because both sides of the cross-multiplication contain exactly one x² term, and they are identical, so they subtract away. That is true for every question of this shape, which is why they can be set at prelims level despite looking like they need the quadratic formula.

If x is subtracted from 23, 39, 32 and 56, the results are in proportion. Find x.

Hard

If x is subtracted from each of 23, 39, 32 and 56, the four results are in proportion. (a) Find x using the ratio-of-differences method. (b) Verify it by cross-multiplying the original expressions. (c) Hence find the mean proportional between (x + 4) and (3x + 1). (d) Explain why the answer options for a question like this cannot be tested by substitution.

Follow-up

Part (d) is what makes this a hard problem rather than a long one. The options are values of the final mean proportional, not values of x, so the usual plug-the-options tactic is unavailable and you have to actually solve for x first. Recognising when a shortcut is closed off is as valuable as knowing the shortcut.

Show the hint

For (a) the two gaps are 39 − 23 and 56 − 32; reduce their ratio and cross-multiply the two numerators with it.