Ratio, Proportion, Percentage & Averages for GATE Aptitude 2027: Formulas & Solved PYQs

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Quick Summary: Ratio, Proportion, Percentage and Averages together give 2 to 3 marks in GATE General Aptitude 2027 every year, across DA, RA, CS, ME, EE, CE and EC. The questions are short, the arithmetic is light, and the mark is lost only to careless setup. This guide gives you the exact formulas, the successive-percentage and alligation shortcuts, 8 solved previous-year and GATE-style questions, a 5-question practice set, a one-page formula sheet and the traps that turn a sure mark into a negative.

🎯 Score 15/15 in General Aptitude. Practise this topic with 100+ exam-level questions, detailed solutions and All-India ranking in the Best GATE Aptitude Test Series 2027 by PiyushAI. Topic-wise tests + full-length GA mocks, valid till GATE 2027.

📋 Table of Contents

Why this topic matters in GATE GA
What the official GA syllabus says
Ratio and proportion
Percentage and successive change
Averages and weighted averages
Mixtures and alligation
Shortcuts and traps
Solved PYQs and GATE-style questions
Practice set
One-page formula sheet
Common mistakes
FAQs

Why this topic matters in GATE GA

General Aptitude is 15 marks in every GATE paper, and Quantitative Aptitude is the biggest slice of it. Within Quant, ratio, percentage and averages are the “connective tissue”: they appear on their own as 1-mark questions, and they hide inside 2-mark word problems on profit, mixtures, data interpretation and work-time. If your ratio and percentage reflexes are weak, you lose marks in four other chapters too.

The good news is that GATE never asks anything beyond Class 8 arithmetic here. The question is a short paragraph, the numbers are friendly, and the answer is usually an integer or a clean fraction. The mark is lost to one of three things: reading “of” as “by”, mixing up the base of a percentage, or adding ratios that have different totals. This guide fixes all three.

Here is roughly how much this cluster has contributed in recent years. Serious aspirants aiming at 15/15 in GA treat these as guaranteed marks.

Year	Marks from this topic (approximate, across papers)	Typical question type
2019	2 to 3	Cost sharing, average of integers with a product constraint
2020	1 to 2	Percentage change, profit and loss
2021	2 to 3	Ratio of boys to girls, percentage overlap
2022	1 to 2	Percentage in a word problem, ratio of areas
2023	1 to 3	Averages, successive percentage change
2024	2 to 3	Ratio inside a data table, mixture
2025	1 to 2	Percentage of a percentage, weighted average
2026	2 to 3	Proportion in a geometry setting, average speed

What the official GA syllabus says

Official GATE GA syllabus, Quantitative Aptitude: “Data interpretation: data graphs (bar graphs, pie charts, and other graphs representing data), 2- and 3-dimensional plots, maps, and tables. Numerical computation and estimation: ratios, percentages, powers, exponents and logarithms, permutations and combinations, and series. Mensuration and geometry. Elementary statistics and probability.”

“Ratios, percentages” is the line this post covers, together with averages, which GATE files under “elementary statistics”. The full syllabus, weightage and the paper-wise strategy are in the GATE General Aptitude 2027 hub. This post is the fifth in the series and the first Quant chapter; the previous post covered Narrative Sequencing and Sentence Completion.

Ratio and proportion

A ratio a : b is just the fraction a/b written without a denominator. Three rules carry every GATE ratio question.

Scaling: a : b = ka : kb for any k > 0. If a : b = 7 : 3, the actual numbers are 7k and 3k, and the total is 10k. The total must be a multiple of 10.
Combining ratios: If A : B = 2 : 3 and B : C = 4 : 5, make B equal in both. LCM of 3 and 4 is 12, so A : B = 8 : 12 and B : C = 12 : 15, giving A : B : C = 8 : 12 : 15.
Proportion: a : b = c : d means ad = bc. The mean proportional between a and b is √(ab). The third proportional to a and b is b²/a.

Direct proportion: y = kx (double x, double y). Inverse proportion: xy = k (double x, halve y). Work-time, speed-time and pipe problems are all inverse proportion in disguise, which is why this chapter feeds the Speed-Time-Distance and Work-Time chapter directly.

✅ Solved Example (GATE 2023 style): The incomes of P and Q are in the ratio 5 : 4 and their expenditures are in the ratio 3 : 2. If each saves Rs 6,000, what is the income of P?
Solution: Let incomes be 5x and 4x, expenditures 3y and 2y. Savings: 5x − 3y = 6000 and 4x − 2y = 6000. From the second, 2x − y = 3000, so y = 2x − 3000. Substitute: 5x − 6x + 9000 = 6000, so x = 3000. Income of P = 5 × 3000 = Rs 15,000. (Check: Q earns 12,000, y = 3000, expenditures 9,000 and 6,000, both save 6,000.)

Percentage and successive change

“Per cent” means “per hundred”. x% of N = xN/100. The whole chapter is about one question: percentage of what? The base is always the quantity that comes after “of” or the original value in “increased by”.

Percentage change = (New − Old)/Old × 100. The denominator is the old value, never the new one.

Successive changes: a change of a% followed by b% gives a net change of a + b + ab/100 (use signs). Increase 20% then decrease 20%: 20 − 20 − 400/100 = −4%, a net decrease of 4%. This formula replaces half of all profit-loss and discount questions.

Multiplier method: write +25% as ×1.25 and −20% as ×0.80. A price that rises 25% and then falls 20% becomes 1.25 × 0.80 = 1.00 times the original: no change.

Reverse percentage: if price rises by x%, consumption must fall by 100x/(100 + x)% to keep expenditure constant. For x = 25, the fall is 2500/125 = 20%.

✅ Solved Example (GATE 2023 style): The population of a town increased by 10% in the first year and decreased by 10% in the second year. If the population at the end of two years is 59,400, what was it at the start?
Solution: Net multiplier = 1.10 × 0.90 = 0.99. So 0.99 × P = 59,400, giving P = 59,400/0.99 = 60,000. Note the net change is a 1% decrease, matching a + b + ab/100 = 10 − 10 − 1 = −1.

💡 GATE Tip: Memorise the fraction-percentage pairs: 1/8 = 12.5%, 1/6 = 16.67%, 1/3 = 33.33%, 3/8 = 37.5%, 5/8 = 62.5%, 2/3 = 66.67%, 7/8 = 87.5%. GATE options are usually built from these, so recognising 37.5% as 3/8 saves a minute of division.

Averages and weighted averages

Average = Sum / Count. In GATE the trick is to work with the sum, not the average. If the average of 11 numbers is 30, the only thing you really know is that they add to 330.

If a number x joins a group of n numbers with average A and the new average is B, then x = (n + 1)B − nA.
If one member is replaced and the average of n members rises by d, the newcomer is heavier by nd.
Average of consecutive or evenly spaced numbers = (first + last)/2. Average of 1 to 100 is 50.5.
Weighted average = (n₁A₁ + n₂A₂)/(n₁ + n₂). Never take the plain average of two averages unless the groups are equal in size.
Average speed for equal distances at u and v = 2uv/(u + v), not (u + v)/2.

✅ Solved Example (GATE 2023 style): The average weight of 8 persons increases by 2.5 kg when one of them weighing 65 kg is replaced by a new person. What is the weight of the new person?
Solution: Total weight rises by 8 × 2.5 = 20 kg. The newcomer therefore weighs 65 + 20 = 85 kg.

✅ Solved Example (GATE 2023 style): A class of 40 students has an average mark of 60. The 25 boys average 56. What is the average mark of the girls?
Solution: Total = 40 × 60 = 2400. Boys contribute 25 × 56 = 1400. Girls (15 of them) contribute 1000, so their average is 1000/15 = 66.67.

Mixtures and alligation

Alligation is the weighted-average formula drawn as a cross. If two ingredients priced c₁ and c₂ (c₁ < c₂) are mixed to give a mean price m, then

Quantity of cheaper : Quantity of dearer = (c₂ − m) : (m − c₁)

For a container of volume V from which x units are removed and replaced by water, repeated n times, the quantity of the original liquid left is V(1 − x/V)ⁿ.

✅ Solved Example (GATE 2023 style): In what ratio must rice at Rs 60 per kg be mixed with rice at Rs 90 per kg so that the mixture costs Rs 72 per kg?
Solution: Cheaper : dearer = (90 − 72) : (72 − 60) = 18 : 12 = 3 : 2. Check: (3 × 60 + 2 × 90)/5 = 360/5 = 72.

Want 40 such mixture and percentage questions with timed solutions? They are in the GATE Aptitude Test Series topic test on Ratio and Percentage.

Shortcuts and traps

💡 GATE Tip: When a ratio is given, write the quantities as 7k and 3k immediately. Any question about “an acceptable total” is then a question about multiples of 10k.

💡 GATE Tip: Convert every percentage change into a multiplier before touching the numbers. Multipliers chain by multiplication; percentages do not chain by addition.

💡 GATE Tip: If A is x% more than B, then B is 100x/(100 + x)% less than A. If A is 25% more than B, B is 20% less than A. The two percentages are never equal.

💡 GATE Tip: For averages, assume a convenient base. If 10 numbers average 42.3, treat the base as 40 and work with deviations (+2.3 each, total +23). Mental arithmetic becomes trivial.

❌ Adding ratios from different groups (2 : 3 in one class and 1 : 1 in another is not 3 : 4 overall unless the classes are equal).
❌ Taking the percentage change on the new value instead of the old value.
❌ Assuming +x% followed by −x% returns to the original. It leaves you x²/100 % lower.
❌ Averaging two averages without weights.
❌ Reading “x% of y” as “x% more than y”.
❌ Forgetting that the mean proportional is √(ab), not (a + b)/2.
❌ Using 22/7 for π when the options are written in terms of π.

Solved previous-year and GATE-style questions

✅ Solved Example (GATE 2021, CS): The ratio of boys to girls in a class is 7 to 3. Among the options below, an acceptable value for the total number of students in the class is: (A) 21 (B) 37 (C) 50 (D) 73
Solution: Boys = 7k, girls = 3k, total = 10k. The total must be a multiple of 10. Only 50 qualifies. Answer: (C) 50.

✅ Solved Example (GATE 2018, CS): In a party, 60% of the invited guests are male and 40% are female. If 80% of the invited guests attended the party and all the female guests attended, what is the ratio of males to females among the attendees?
Solution: Take 100 invitees: 60 males, 40 females. Attendees = 80. All 40 females came, so 40 males came. Ratio = 40 : 40 = 1 : 1.

✅ Solved Example (GATE 2019, CS): Ten friends planned to share equally the cost of buying a gift for their teacher. When two of them decided not to contribute, each of the other friends had to pay Rs 150 more. The cost of the gift was Rs ____.
Solution: Let the cost be C. Then C/8 − C/10 = 150. LCM 40: (5C − 4C)/40 = 150, so C = Rs 6,000. (Check: 750 each for 8, 600 each for 10, difference 150.)

✅ Solved Example (GATE 2013): In the summer of 2012, in New Delhi, the mean temperature of Monday to Wednesday was 41°C and of Tuesday to Thursday was 43°C. If the temperature on Thursday was 15% higher than that of Monday, then the temperature on Thursday was:
Solution: M + T + W = 123 and T + W + Th = 129. Subtracting, Th − M = 6. Also Th = 1.15M, so 0.15M = 6, M = 40 and Th = 46°C.

✅ Solved Example (GATE 2016): A person moving through a tuberculosis-prone zone has a 50% probability of becoming infected. However, only 30% of infected people develop the disease. What percentage of people moving through the zone remain infected but do not show symptoms?
Solution: Infected = 50%. Of these, 70% show no symptoms. 50% × 70% = 35%. (Percentage of a percentage: multiply the fractions.)

✅ Solved Example (GATE-style): The price of a laptop is increased by 20% and then reduced by 20%. The net change in price is:
Solution: Net = 20 − 20 − (20 × 20)/100 = −4%. Multiplier check: 1.2 × 0.8 = 0.96. Answer: a 4% decrease.

✅ Solved Example (GATE-style): The average of 11 numbers is 30. The average of the first six numbers is 28 and that of the last six numbers is 33. What is the sixth number?
Solution: Total = 330. First six sum to 168, last six sum to 198. The sixth number is counted in both: 168 + 198 − 330 = 36.

✅ Solved Example (GATE-style): A 40-litre mixture contains milk and water in the ratio 3 : 1. How much water must be added so that the ratio becomes 3 : 2?
Solution: Milk = 30 L, water = 10 L. Milk stays 30, so water must become 20 L. Add 10 litres.

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Practice set

Attempt these in 6 minutes before opening the key.

If A : B = 2 : 3 and B : C = 4 : 5, then A : B : C is (A) 2 : 3 : 5 (B) 8 : 12 : 15 (C) 6 : 8 : 10 (D) 4 : 6 : 5
The price of petrol rises by 25%. By what percentage must a motorist reduce consumption to keep expenditure unchanged? (A) 25% (B) 20% (C) 30% (D) 12.5%
The average of 5 numbers is 20. When one number is removed, the average of the remaining four is 18. The removed number is (A) 22 (B) 26 (C) 28 (D) 30
If 30% of x equals 45% of 60, then x is (A) 60 (B) 75 (C) 90 (D) 120
Two successive discounts of 10% and 20% are equivalent to a single discount of (A) 30% (B) 28% (C) 29% (D) 27%

Show answer key

(B) Make B equal: LCM of 3 and 4 is 12, so A : B : C = 8 : 12 : 15.
(B) Reduction = 100 × 25/(100 + 25) = 20%.
(C) Total 100, remaining total 72, removed = 28.
(C) 0.3x = 27, so x = 90.
(B) Net = −10 − 20 + (10 × 20)/100 = −28%. Multiplier check: 0.9 × 0.8 = 0.72.

One-page formula sheet

1. a : b = ka : kb; quantities are ak and bk, total (a + b)k.
2. a : b = c : d means ad = bc. Mean proportional = √(ab). Third proportional to a, b = b²/a.
3. x% of N = xN/100. Percentage change = (New − Old)/Old × 100.
4. Successive changes a%, b%: net = a + b + ab/100. Three changes: multiply the multipliers.
5. +x% then −x%: net = −x²/100 %.
6. Price up x%: cut consumption by 100x/(100 + x)%. Price down x%: raise consumption by 100x/(100 − x)%.
7. A is x% more than B means B is 100x/(100 + x)% less than A.
8. Average = Sum/Count. Newcomer x = (n + 1)B − nA. Replacement changes total by n × (change in average).
9. Weighted average = Σn_iA_i / Σn_i. Average speed over equal distances = 2uv/(u + v).
10. Alligation: cheaper : dearer = (c₂ − m) : (m − c₁).
11. Repeated replacement: liquid left = V(1 − x/V)ⁿ.
12. Profit % = (SP − CP)/CP × 100; discount % is on the marked price.

Common mistakes

❌ Computing profit % on selling price instead of cost price.
❌ Treating a 3 : 2 ratio as “3 of A and 2 of B” when the total is given as 40 (it is 24 and 16).
❌ Averaging percentages across groups of different sizes.
❌ Confusing “percentage points” (43 − 41 = 2 points) with “percentage change” (2/41 = 4.9%).
❌ Answering a NAT with the ratio when the question asks for the actual quantity.
❌ Rounding too early in a multi-step percentage chain.

FAQs

How many questions on ratio, percentage and averages appear in GATE GA?

Typically 1 to 2 direct questions worth 2 to 3 marks, plus the arithmetic hidden in data interpretation and work-time questions. Across every GATE paper the pattern is the same because the GA section is common.

Are these questions MCQ or NAT?

Both. Ratio and percentage questions are often MCQ with four numeric options; averages and cost-sharing questions are frequently NAT where you type an integer. NAT has no negative marking, so always attempt it.

What is the fastest way to handle successive percentage changes?

Convert every change into a multiplier (1.2 for +20%, 0.85 for −15%) and multiply. The a + b + ab/100 formula is the same thing for two steps. Never add the percentages.

Is the PiyushAI Aptitude Test Series enough for GA?

Yes. The Best GATE Aptitude Test Series 2027 has a dedicated topic test on ratio, percentage and averages, tests on every other GA chapter, full-length GA mocks in the GATE interface, solved PYQs from 2010 to 2026 and All-India rank analysis. Combined with the free guides on this site, it covers the GA section completely.

Do I need to learn alligation for GATE?

It is optional but useful. Every alligation question can be solved by writing the weighted-average equation. Alligation just makes it a 15-second question instead of a 60-second one.

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