Aptitude · Ratio and Proportion · Translation toolkit

Fifteen sentences, and the ratio each one hides

Almost no exam question hands you a ratio. It hands you an English sentence that contains one. This lesson is the lookup table: what each phrasing means, which quantity becomes the base, and where the numbers swap places.

Pick a sentence pattern and watch it become a ratio →

Find the base first. “More than B” or “less than B” or “% of B” all make B the base, so set B = 100 and read the other quantity off the sentence.

01 The idea

The sentence is the question

“P earns 25% more than Q.” That sentence contains the ratio 5 : 4, and extracting it takes about four seconds once you know the move. Set the base to 100, build the other quantity, write them in the order the sentence named them, simplify. Q = 100, P = 125, so P : Q = 125 : 100 = 5 : 4.

The base is the whole game. “More than Q” makes Q the base. “40% of B” makes B the base. “Less than Y’s speed” makes Y the base. Pick the wrong one and every number afterwards is wrong, while looking entirely reasonable — which is why this is a lesson rather than a footnote.

A second family of sentences has no base at all, because it gives you an equation. “20% of X equals 30% of Y” is 20X = 30Y, so X : Y = 30 : 20 = 3 : 2 — each letter takes the other letter’s number. But “one third of A equals one quarter of B” is A/3 = B/4, so A : B = 3 : 4, and here nothing swaps. Whether the number multiplies the letter or divides it decides the direction, and that one distinction is the most expensive detail in the whole toolkit.

The third family is deliberately treacherous. “A is 3 times as efficient as B” is 3 : 1. “A is 3 times more efficient than B” ought to mean B plus three more of B, which is 4 : 1 — and that is what this course’s source key and many exam keys mark. Plenty of papers use the same phrase loosely to mean 3 : 1. The phrase is broken English, not a rule to memorise, so the honest strategy is to check the options.

Set the base to 100, build the other quantity from the words, write the pair in the order the sentence named them, then simplify. Four steps, fifteen sentence types, one method.

The baseThe quantity the sentence compares everything against — the one after “than” or “of”. Set it to 100 for percentages, 1 for “times”, and the denominator for fractions.

The swapWhen the sentence multiplies both letters, as in 20%X = 30%Y, the coefficients trade places: X : Y = 30 : 20. When it divides them, as in A/3 = B/4, they do not: A : B = 3 : 4.

Naming orderThe ratio is written in the order the sentence names the quantities, not the order the numbers appear. “Girls are 20% fewer than boys” asked as “girls : boys” is 4 : 5, and as “boys : girls” is 5 : 4.

02 Worked example

“P earns 25% more than Q”, worked in full

One sentence runs this lesson, because the method never changes. P earns 25% more than Q. Find P : Q, and check your answer by rewriting the sentence with Q as the subject.

Find the baseThe sentence says “more than Q”, so Q is what P is being measured against. Q is the base, and a percentage base is always set to 100.Q = 100

Build the other quantity from the wordsP starts level with Q and gains 25% of Q on top. Note that the 25% is taken on Q, never on P — that is what “more than Q” means.P = 100 + 25 = 125

Write the pair in the order asked, then simplifyThe question asks for P : Q, and P was named first in the sentence, so the 125 goes first. Twenty-five divides both terms.P : Q = 125 : 100 = 5 : 4

Rewrite the sentence the other way roundQ falls short of P by 25 out of P’s 125. That is 1/5, or 20% — not 25%, because the base has changed from Q to P.25/125 = 1/5 = 20% ⇒ “Q earns 20% less than P”

Read the ratio back as a percentageA ratio a : b always says the first quantity exceeds the second by (a − b)/b. Doing this in reverse is the check that costs five seconds and catches a wrong base.(5 − 4)/4 = 1/4 = 25% ✓ matches the sentence

“25% more” and “20% less” describe the same pair of numbers, because they use different bases. That is not a curiosity — it is the reason the last step exists. Any answer you produce here can be read back out as a sentence, and if the sentence you get is not the one you were given, you used the wrong base. This is also the bridge to the percentages module, which runs the same conversion in the other direction.

03 The method

All fifteen patterns

The three levels below are the source’s own grouping: placement basics, competitive level, then the mains-level compounds. Read the middle column, not the answer — the middle column is what you have to be able to produce under time pressure.

Percentage sentences: base = 100. “x% of B” gives x : 100. “x% more than B” gives (100+x) : 100. “x% less than B” gives (100−x) : 100. Equation sentences: swap or do not swap. pA = qB gives A : B = q : p, but A/p = B/q gives A : B = p : q.

The “times more than” trap, stated honestly. “A is 3 times as efficient as B” is unambiguous: 3 : 1. “A is 3 times more efficient than B” is not. Read strictly it is B + 3B = 4B, giving 4 : 1, which is what this course’s source material and many SSC and Bank answer keys mark as correct. But the phrase is also used loosely, by plenty of setters and in ordinary speech, to mean exactly the same as “3 times as much” — 3 : 1. Practical rule: look at the options. If only one of 3 : 1 and 4 : 1 is listed, that is the intended reading. If both are listed, the word “more” is there on purpose and 4 : 1 is wanted. Never assume; the phrase itself does not decide it.

#	Sentence	Mental translation	Ratio
1	Selected students are 3/8 of the unselected	first name takes the top, second takes the bottom	`3 : 8`
2	A’s investment is 40% of B’s	B = 100, A = 40	`2 : 5`
3	3/7 of the total employees are male	total = 7, male = 3, so female = 4	`3 : 4`
4	P earns 25% more than Q	Q = 100, P = 125	`5 : 4`
5	X’s speed is 20% less than Y’s	Y = 100, X = 80	`4 : 5`
6	A is 3 times as efficient as B	B = 1, A = 3	`3 : 1`
7	A is 3 times MORE efficient than B	B = 1, A = 1 + 3 strictly — but see the warning above	`4 : 1` or `3 : 1`
8	1/3 of A’s marks equals 1/4 of B’s	A/3 = B/4, denominators stay put	`3 : 4`
9	20% of X equals 30% of Y	2X = 3Y, so the numbers swap	`3 : 2`
10	Girls are 16.66% fewer than boys	16.66% = 1/6, so boys = 6 and girls = 5	`5 : 6`
11	A receives 2/5 of what B and C receive together	A = 2, B+C = 5, total 7 parts	`2 : 5`
12	Half of A, a third of B and a quarter of C are equal	A/2 = B/3 = C/4, denominators are the answer	`2 : 3 : 4`
13	A spends 20%, and what is left equals B’s income	80% of A = 100% of B, so 4A = 5B	`5 : 4`
14	The difference is 20% of the sum	5(A−B) = A+B, so 4A = 6B	`3 : 2`
15	A is half of B, and B is twice C	start from C = 1, work upwards	`1 : 2 : 1`

05 Cheat sheet

The translation table you want the morning of the exam

Seven rows that between them cover most of what a paper will throw at you, plus the three facts that decide the close calls.

Sentence shape	Becomes	Example
A is x% of B	`x : 100`	40% of B → 2 : 5
A is x% more than B	`(100+x) : 100`	25% more → 5 : 4
A is x% less than B	`(100−x) : 100`	20% less → 4 : 5
A is x/y of B	`x : y`	3/8 of B → 3 : 8
pA = qB (the numbers multiply)	`q : p — they swap`	20%A = 30%B → 3 : 2
A/p = B/q (the numbers divide)	`p : q — no swap`	A/3 = B/4 → 3 : 4
A is n times MORE than B	`ambiguous: (n+1) : 1 or n : 1`	3 times more → 4 : 1 or 3 : 1

Recurring percentages are fractions16.66% is 1/6, 33.33% is 1/3, 8.33% is 1/12, 14.28% is 1/7. Converting first turns “16.66% fewer girls than boys” into boys = 6, girls = 5 with no arithmetic at all.

“Fraction of the total” needs a subtraction“3/7 of employees are male” gives male : total = 3 : 7, so male : female is 3 : 4, not 3 : 7. Read whether the question wants a part-to-part or a part-to-whole ratio.

A ratio is a percentage in disguise9 : 8 is a 12.5% gap because 1/8 = 12.5%; 5 : 4 is 25%; 6 : 5 is 20%. The percentages module runs this the other way, turning a stated percentage change into a multiplier.

06 Where & why

Where this shows up

This is the lesson with the widest reach in the module, because the translation step sits at the front of questions that are not otherwise about ratios at all.

TCS NQT · Wipro · Infosys

Patterns 1 to 5, on their own

One-mark questions that are pure translation. Under twenty seconds each once the base habit is automatic, and they appear in almost every paper.

Bank Prelims · SSC CGL

Patterns 6 to 10, inside longer questions

The translation is line one of a five-line question about profit or savings. Getting it wrong costs the whole question, not one mark.

Mains-level compounds

Patterns 11 to 15

“A receives 2/5 of what B and C receive together” sets up a three-way split whose total is 7 parts, not 5. That single reading is worth several marks.

Data interpretation

Caption sentences under charts

“Sales in 2023 were 20% below 2022” is pattern 5. DI sets are full of these and they are usually the only place the actual numbers come from.

Learn the base habit and the swap rule and you have most of this lesson. Learn that “times more than” is a defective phrase rather than a rule, and you will stop losing the one mark that everybody argues about afterwards.

07 Interview questions

What gets asked

Twelve, because this is the lesson interviewers use to check whether you read carefully or pattern-match.

“A’s investment is 40% of B’s.” Find A : B.

2 : 5. Set B = 100, so A = 40, and 40 : 100 simplifies to 2 : 5. The phrase “of B” is what tells you B is the base.

“P earns 25% more than Q.” Find P : Q.

5 : 4. Q is the base at 100, so P is 125, and 125 : 100 is 5 : 4. Read it back as a check: 5 : 4 means P is 1/4 above Q, which is 25%.

Is “25% more” the same as “20% less”?

They describe the same pair of numbers from opposite ends. If P is 25% more than Q, then Q is 20% less than P — 25 out of 125 is 1/5. The percentage changes because the base changes, and that is the whole reason you must identify the base first.

“20% of X equals 30% of Y.” Find X : Y.

3 : 2. The percentage signs cancel to give 2X = 3Y, so X : Y = 3 : 2 — each letter takes the other letter’s number. It is worth sanity-checking: a smaller percentage of X matches a larger percentage of Y, so X must be the bigger quantity.

“One third of A equals one quarter of B.” Find A : B.

3 : 4. Here A/3 = B/4, so A : B = 3 : 4 with the denominators staying under their own letters. Nothing swaps, because the numbers are dividing rather than multiplying. Compare with the previous question — the contrast is the point.

When do the numbers swap and when do they not?

They swap when the numbers multiply the letters: pA = qB gives A : B = q : p. They do not swap when the numbers divide the letters: A/p = B/q gives A : B = p : q. Percentages multiply, so they swap; fractions of the form “one third of” divide, so they do not.

“A is 3 times more efficient than B.” What is the ratio?

This phrase is ambiguous and you should say so. Strictly it means B plus three more of B, giving 4 : 1, and that is what most Indian exam keys mark. Many setters use it loosely to mean 3 : 1, the same as “3 times as efficient”. Check which of the two appears in the options; if both do, the word “more” was deliberate and 4 : 1 is wanted.

“3/7 of the employees are male.” Find male : female.

3 : 4. The 3/7 is a part of the whole, so the total is 7 parts and male is 3 of them, leaving 4 for female. Answering 3 : 7 answers a different question — male to total, not male to female.

“A receives 2/5 of what B and C receive together.” How many parts is the whole?

Seven. A is 2 parts and B + C together are 5, so the whole amount is 2 + 5 = 7 parts. Treating the total as 5 parts is the standard error and it makes every subsequent number wrong by a factor of 7/5.

“Half of A, one third of B and one quarter of C are equal.” Find A : B : C.

2 : 3 : 4. From A/2 = B/3 = C/4 the denominators are the ratio directly. To see why, set each equal to 1: A = 2, B = 3, C = 4. The three-term version behaves exactly like the two-term one.

“The difference between two numbers is 20% of their sum.” Find the ratio.

Hard

In a company, the number of engineers is 3/5 of the number of technicians, and the number of technicians is 25% more than the number of clerks. (a) Find engineers : technicians : clerks. (b) If there are 96 more technicians than engineers, find the total staff. (c) State what percentage of the staff are clerks, correct to two decimal places.

Follow-up

Two translations that have to be joined on the shared quantity, which is the connecting model in this module arriving one lesson early. Part (c) then converts a ratio part back into a share of the whole, which is where a correct ratio still gets the wrong final answer.

Show the hint

Translate each sentence separately first, then scale so that the technicians count matches in both.