Proportion and Its Types

Ratio and Proportion · 25 min

Aptitude · Ratio and Proportion · Proportion

One multiplication decides whether two ratios match

A proportion is two ratios that are equal. You never have to simplify either of them to check — multiply the outside pair and the inside pair, and see whether the two products agree. Mean, third and fourth proportional are then three ways of asking for a missing term.

Test a proportion and fill any missing term →
Product of the extremes = product of the means. In 4 : 6 :: 6 : 9 both products are 36, so it is a proportion — and because the middle term repeats, it is a continued one.

01 The idea

Equal ratios, and the test that avoids the arithmetic

Two tonnes of timber sells for ₹10 crore; four tonnes sells for ₹20 crore. The first deal is 2 : 10, the second is 4 : 20, and both cut down to 1 : 5. Because they are equal, the four numbers are said to be in proportion, written 2 : 10 :: 4 : 20 and read “2 is to 10 as 4 is to 20”.

Simplifying both sides to check is slow, and with numbers like 143 and 187 it is worse than slow. There is a test that never needs it. In a : b :: c : d the outer terms a and d are the extremes and the inner terms b and c are the means, and the proportion holds exactly when ad = bc. For 2 : 10 :: 4 : 20 that is 2 × 20 = 40 and 10 × 4 = 40. One multiplication each side, done.

The interesting case is when the two middle terms are the same number. 4 : 6 :: 6 : 9 passes the test, since 4 × 9 = 36 and 6 × 6 = 36. Three numbers linked like this are in continued proportion, and the repeated middle term 6 is the mean proportional between 4 and 9. It comes straight out of the test: b² = ac, so b = √(ac) = √36 = 6.

Exams then ask for the same chain from three directions. Give the two outer terms and ask for the middle one: mean proportional. Give the first two and ask for the last: third proportional, which is b²/a. Give any three of four terms in a non-repeating proportion and ask for the fourth: fourth proportional, which is bc/a. Three names, one cross-multiplication.

In a : b :: c : d, multiply the outside pair and the inside pair. If the two products match, it is a proportion — and every missing-term question is that same equation solved for the gap.
Extremes and meansIn a : b :: c : d, the extremes are a and d, the means are b and c. The proportion holds exactly when ad = bc.
Continued proportiona : b :: b : c — the middle term repeats, so b² = ac. The three numbers form a geometric chain: 4, 6, 9 with each term 3/2 of the last.
Mean, third, fourth proportional√(ac), b²/a and bc/a. Which one is wanted depends only on which term of the chain the question left blank.

02 Worked example

4 : 6 :: 6 : 9, taken apart

One chain runs the whole lesson. Show that 4 : 6 :: 6 : 9 is a proportion. Name the mean proportional and the third proportional in it, and find the fourth proportional to 4, 6 and 9.

1
Test it with one multiplication each sideExtremes are 4 and 9, means are 6 and 6. No simplifying and no division — compare the two products directly.4 × 9 = 36    6 × 6 = 36  ⇒  it is a proportion
2
The repeated middle term is the mean proportionalBecause b² = ac, the middle term is the square root of the product of the outer two. This is the geometric mean, not the average.√(4 × 9) = √36 = 6  (the average of 4 and 9 is 6.5, a different number)
3
The same chain read forwards gives the third proportionalGiven 4 and 6, the third proportional is what completes 4 : 6 :: 6 : x. Write the 6 twice, then solve.4x = 6 × 6  ⇒  x = 36/4 = 9 ✓
4
The fourth proportional extends it one more termNow nothing repeats: 4 : 6 :: 9 : x. Cross-multiply and divide. It is allowed to be a fraction.4x = 6 × 9 = 54  ⇒  x = 13.5
5
It was one geometric chain all along4, 6, 9, 13.5 — each term is 3/2 of the one before it. Mean, third and fourth proportional are three different blanks in that single chain.4 × 3/2 = 6,  6 × 3/2 = 9,  9 × 3/2 = 13.5

Once you see the chain, none of the three formulas needs memorising. The mean proportional is the term between two you were given; the third proportional is the term after two you were given; the fourth proportional is the term that keeps two separate ratios equal. Anyone who memorises b²/a without seeing the chain will eventually apply it to the wrong pair of numbers, and that is by far the most common error in this model.

03 The method

The formulas, and the direct-versus-inverse trap

Four formulas for the missing-term questions, and one distinction that decides whether you cross-multiply at all.

a : b :: c : d ⇔ ad = bc. Mean proportional between a and c is √(ac). Third proportional to a and b is b²/a. Fourth proportional to a, b and c is bc/a.
Decide direct or inverse before you write anything. If the two quantities rise together — more lorries, more timber carried — it is direct, so you cross-multiply: 5/50 = 8/x gives x = 80. If one rises as the other falls — more workers, fewer days — it is inverse, so you multiply straight across: 10 × 20 = 40 × x gives x = 5. Students who cross-multiply an inverse relationship get 80 days for 40 workers, and the answer is absurd on sight. Sanity-check the direction before the arithmetic.
TypeGivenFormulaOn 4, 6, 9
Mean proportionalthe two extremes√(ac)√36 = 6
Third proportionalthe first two termsb²/a36/4 = 9
Fourth proportionalany three termsbc/a54/4 = 13.5
Continued proportionthe chain itselfb² = ac6² = 4×9 ✓
Direct proportionboth rise togethercross-multiply5 : 50 = 8 : 80
Inverse proportionone rises, one fallsM₁D₁ = M₂D₂10×20 = 40×5
Cross-multiplying an inverse pairthe classic wrong turnnevergives 40 workers 80 days

05 Cheat sheet

Proportion on one page

The test, the three missing-term formulas, and the two traps that cost marks in every sitting.

CaseRuleOn 4, 6, 9
Is it a proportion?ad = bc4×9 = 6×6 = 36
Mean proportional√(ac)√36 = 6
Third proportionalb²/a36/4 = 9
Fourth proportionalbc/a54/4 = 13.5
Continued proportiona, b, c is a GP4, 6, 9 — step 3/2
Mean proportional as an averagewrong√36 = 6, not (4+9)/2 = 6.5
Third proportional to two numbersrepeat the second term4 : 6 :: 6 : x, not 4 : 6 :: x
The test needs no simplifyingTwo multiplications settle a proportion however ugly the numbers. 143 : 187 :: 13 : 17 holds because 143 × 17 = 2431 = 187 × 13, and you never have to notice that 143 = 11 × 13.
Mean proportional is a geometric meanIt is √(ac), which sits below the plain average for any two unequal numbers. For 4 and 9 it is 6 against an average of 6.5, and the gap widens as the two numbers move apart.
Inverse proportions multiply straight acrossMore workers means fewer days, so M₁D₁ = M₂D₂. Cross-multiplying instead gives an answer that moves the wrong way, which makes it easy to catch if you glance at the direction first.

06 Where & why

Where this shows up

Proportion questions are short, direct and reliably set. The value here is speed — each one should take under thirty seconds.

SSC CGL · Bank Prelims
“Find the mean proportional between 4 and 49”

14. One multiplication and one root. These come in pairs with third-proportional questions, and the only real risk is applying the wrong formula to the right numbers.

TCS NQT · Infosys
Unitary-method questions in disguise

“If 5 lorries carry 50 tonnes, how much do 8 carry?” is a fourth-proportional question. Setting it up as a proportion is faster than finding the per-lorry rate.

Time and work · speed
Inverse proportion

Every men-and-days and speed-and-time question is an inverse proportion. Getting the direction right here removes a whole class of error later in the syllabus.

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

The same b²/a with factorisation instead of arithmetic: (x+y)²/[(x+y)(x−y)] = (x+y)/(x−y). Substituting small numbers for x and y checks it in seconds.

If a question hands you three numbers and asks for a fourth, you are in this lesson. Decide which term of the chain is missing, write the proportion with the blank in place, and cross-multiply. There is no faster route and no shortcut worth learning instead.

07 Interview questions

What gets asked

Ten. The definitions come first because interviewers open with them, but the direct-versus-inverse question in the middle is the one that separates candidates.

What is a proportion?
Two equal ratios, written a : b :: c : d. It holds exactly when the product of the extremes equals the product of the means, that is ad = bc. So 2 : 10 :: 4 : 20 because both products are 40.
State the rule of proportion.
Product of extremes equals product of means: a × d = b × c. Its value is that it needs no simplifying, so it settles a proportion with ugly numbers just as fast as with tidy ones.
Find the mean proportional between 4 and 9.
6. The mean proportional is √(ac), so √36 = 6. Note it is the geometric mean, not the arithmetic one — the average of 4 and 9 is 6.5, which is a different number and a common wrong answer.
Find the third proportional to 4 and 6.
9. The third proportional completes 4 : 6 :: 6 : x, so x = 6²/4 = 36/4 = 9. The middle term has to be written twice; forgetting that is what turns this into a fourth-proportional calculation with a missing number.
Third proportional versus fourth proportional — what is the difference?
The third proportional is given two numbers and repeats the second: a : b :: b : x, so x = b²/a. The fourth proportional is given three numbers with nothing repeated: a : b :: c : x, so x = bc/a. Count the numbers in the question and the choice makes itself.
What is a continued proportion?
Three numbers where the middle term serves both sides: a : b :: b : c, so b² = ac. 4, 6, 9 is one. It is the same thing as saying the three numbers form a geometric progression, here with common ratio 3/2.
If 5 lorries carry 50 tonnes, how much do 8 lorries carry?
80 tonnes. More lorries carry more timber, so this is direct: 5/50 = 8/x gives 5x = 400 and x = 80. Cross-multiplying is correct here precisely because the two quantities rise together.
If 10 workers take 20 days, how long do 40 workers take?
5 days. This is inverse — more workers means fewer days — so you multiply straight across: 10 × 20 = 40 × x gives x = 5. Four times the workers, a quarter of the time. Cross-multiplying here would give 80 days, which is wrong in a way you can spot without any arithmetic.
How do you tell a direct proportion from an inverse one?
Ask what happens to the second quantity when the first goes up. If it goes up too, it is direct and you cross-multiply. If it comes down, it is inverse and you multiply straight across. Doing that check first costs two seconds and prevents the single most common error in the chapter.
When would you use a proportion rather than just computing a unit rate?
When the unit rate is ugly or irrelevant. For 5 lorries and 50 tonnes the rate is a clean 10 tonnes each, so either route works. For 7 machines producing 913 items, setting up 7 : 913 :: 11 : x avoids ever computing 130.43 items per machine, and avoids the rounding error that comes with it.

08 Practice problems

Six on proportion

The first two are formula drills. The middle three are where the direction and the decimals bite, and the last one runs a chain in both directions.

Two formulas, one line each

Easy
Find the mean proportional between 4 and 49, and the third proportional to 16 and 24.
Follow-up
Two different formulas on two different pairs, side by side, so you have to pick the right one twice rather than settling into one habit.
Show the hint
One needs a square root of a product; the other needs a square divided by the first term.

Units inside a proportion

Easy
Find the fourth proportional to 3 hours, 1 hour 30 minutes and 12 hours.
Follow-up
The middle term is written in mixed units, so it has to be converted before it enters the formula. The answer is a time, and it should be stated as one.
Show the hint
Put all three quantities in the same unit first, then apply bc/a.

Which direction?

Medium
(a) If 6 pumps fill a tank in 14 hours, how long do 7 pumps take? (b) If 6 pumps deliver 1,260 litres, how much do 7 pumps deliver? Give both answers and say which one you cross-multiplied.
Follow-up
The same two numbers in both halves, and only the relationship changes. One is inverse and one is direct, so getting both right proves you checked the direction rather than reaching for a formula.
Show the hint
For each half, ask whether raising the number of pumps raises or lowers the quantity asked about.

Decimals under a root

Medium
Find the mean proportional between 0.08 and 0.18, and between 1.21 and 0.09.
Follow-up
Both products are perfect squares once you clear the decimals, but the number of decimal places in the answer is the part that goes wrong. Counting places under the root is the skill being tested.
Show the hint
Turn each decimal into a fraction over a power of ten, multiply, then root the numerator and the denominator separately.

Make them proportional

Medium
What number must be added to each of 10, 16, 22 and 32 so that the four results are in proportion?
Follow-up
The unknown appears in all four terms, so the setup is (10+x) : (16+x) :: (22+x) : (32+x). The x² terms cancel when you cross-multiply, which is the only reason this is a linear problem rather than a quadratic one.
Show the hint
Cross-multiply, then cancel the x² terms on both sides before collecting.

Both ends of one chain

Hard
(a) 36 is the third proportional to 4 and x — find x. (b) Find the mean proportional between that x and 27. (c) Show that 4, 12, 36 and 12, 18, 27 are each in continued proportion, and state the common ratio of each chain.
Follow-up
Part (a) runs the third-proportional formula backwards, which is where a memorised formula stops helping. Part (c) is the payoff: the same number 12 sits in two different geometric chains, one stepping by 3 and one by 3/2, which is what “continued proportion” actually means.
Show the hint
For (a) write 4 : x :: x : 36 and solve for x². For (c) divide each term by the one before it.