Ratio & Proportion, formulas + CAT PYQs
An A-to-Z formula sheet, 12 basics for practice, and every CAT ratio / proportion / variation question I could source from 2002-2025. All answers independently solved and code-verified.
Formula & Concept Sheet (A → Z)
Ratio, proportion, variation, partnership, and the classic identities.
- a:b = a/b; scale both terms freely.
- A ratio gives proportions, not values, you need a total or difference.
- Chain a:b and b:c via a common middle term.
- a:b=2:3, b:c=4:5 ⇒ 8:12:15.
- Share = (term/Σterms) × total.
- Difference of parts: 1 part = D/(p−q).
- a:b=c:d ⇒ ad=bc.
- Mean prop = √(ab); third (b²/a); fourth (bc/a).
- a/b=c/d ⇒ (a+b)/(a−b) = (c+d)/(c−d).
- Write each as (term)×k; apply transfer/add; set new ratio.
- Total stays constant in pure transfers.
- Direct y=kx; inverse xy=k; joint z=kxy.
- Power: price ∝ weight² ⇒ k·w².
- Profit share ∝ investment × time.
- a/(b+c)=b/(c+a)=c/(a+b)=r ⇒ 1/2 (sum≠0) or −1 (sum=0).
- Ages (constant gap); 3-way distribution; group composition; queue position (p+q parts + 1).
- Mixing to a target = alligation: (d−m):(m−c).
- Ratio alone ⇒ no values; combine before comparing; "x% more" ≠ ratio.
Basics for practice, 12 questions
Warm-ups before the CAT papers.
Simplify the ratio 18 : 24.
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Divide ₹600 in the ratio 2 : 3.
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If a : b = 2 : 3 and b : c = 4 : 5, find a : b : c.
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Find the mean proportional of 4 and 9.
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Find the fourth proportional to 3, 4, 9.
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Find the third proportional to 4 and 8.
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Two numbers are in the ratio 5 : 7 and their sum is 96. Find them.
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Two numbers are in the ratio 3 : 5 and their difference is 10. Find them.
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If A : B = 2 : 3 and B : C = 4 : 5, find A : C.
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x varies directly as y; x = 12 when y = 4. Find x when y = 7.
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x varies inversely as y; x = 8 when y = 5. Find x when y = 10.
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Divide ₹1000 among A, B, C in the ratio 2 : 3 : 5.
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Practice questions generated · up to 100
Original easy-hard warm-up drills (not CAT PYQs). Pick the levels, generate a set, reveal answers.
All CAT Ratio Questions, 25 PYQs (2002-2025)
Every clean CAT ratio / proportion / variation question, by year. Solved & code-verified. = hard.
Classic CAT (2002-2006)1. If a/(b+c) = b/(c+a) = c/(a+b) = r then r cannot take any value except __________.
- (1) 1/2
- (2) −1
- (3) 1/2 or −1
- (4) −1/2 or −1
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2. An airline charges for excess luggage at a fixed rate per kg. Raja and Praja together carry 60 kg and are charged ₹1200 and ₹2400. Had all 60 kg belonged to one person, the charge would be ₹5400. Praja's luggage (kg) is.
- (A) 20
- (B) 25
- (C) 30
- (D) 35
- (E) 40
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3. (Same setup as #2.) The free luggage allowance (kg) is.
- (A) 10
- (B) 15
- (C) 20
- (D) 25
- (E) 30
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4. Let a, b, c, d and e be integers such that a = 6b = 12c, and 2b = 9d = 12e. Then which of the following pairs contains a number that is not an integer.
- (1) [a/27, a/e]
- (2) [a/36, c/e]
- (3) [a/12, bd/18]
- (4) [a/6, c/d]
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5. Suppose, C1, C2, C3, C4, and C5 are five companies. The profits made by C1, C2, and C3 are in the ratio 9 : 10 : 8 while the profits made by C2, C4, and C5 are in the ratio of 18 : 19 : 20. If C5 has made a profit of ₹19 crore more than C1, then the total profit (in ₹) made by all five companies is
- (1) 438 crore
- (2) 435 crore
- (3) 348 crore
- (4) 345 crore
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6. A stall sells popcorn and chips in packets of three sizes: large, super, and jumbo. The numbers of large, super, and jumbo packets in its stock are in the ratio 7 : 17 : 16 for popcorn and 6 : 15 : 14 for chips. If the total number of popcorn packets in its stock is the same as that of chips packets, then the numbers of jumbo popcorn packets and jumbo chips packets are in the ratio
- (1) 1 : 1
- (2) 8 : 7
- (3) 4 : 3
- (4) 6 : 5
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7. If a, b, c are three positive integers such that a and b are in the ratio 3 : 4 while b and c are in the ratio 2 : 1, then which one of the following is a possible value of (a + b + c)?
- (1) 201
- (2) 205
- (3) 207
- (4) 210
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8. Raju and Lalitha originally had marbles in the ratio 4 : 9. Then Lalitha gave some of her marbles to Raju. As a result, the ratio of the number of marbles with Raju to that with Lalitha became 5 : 6. What fraction of her original number of marbles was given by Lalitha to Raju?
- (1) 1/5
- (2) 6/19
- (3) 1/4
- (4) 7/33
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9. The scores of Amal and Bimal in an examination are in the ratio 11 : 14. After an appeal, their scores increase by the same amount and their new scores are in the ratio 47 : 56. The ratio of Bimal's new score to that of his original score is
- (1) 4 : 3
- (2) 8 : 5
- (3) 5 : 4
- (4) 3 : 2
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10. In an examination, Rama's score was one-twelfth of the sum of the scores of Mohan and Anjali. After a review, the score of each of them increased by 6. The revised scores of Anjali, Mohan, and Rama were in the ratio 11 : 10 : 3. Then Anjali's score exceeded Rama's score by
- (1) 26
- (2) 32
- (3) 35
- (4) 24
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11. A sum of money is split among Amal, Sunil and Mita so that the ratio of the shares of Amal and Sunil is 3 : 2, while the ratio of the shares of Sunil and Mita is 4 : 5. If the difference between the largest and the smallest of these three shares is ₹400, then Sunil's share, in rupees, is :
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12. The amount Neeta and Geeta together earn in a day equals what Sita alone earns in 6 days. The amount Sita and Neeta together earn in a day equals what Geeta alone earns in 2 days. The ratio of the daily earnings of the one who earns the most to that of the one who earns the least is
- (1) 11 : 3
- (2) 3 : 2
- (3) 7 : 3
- (4) 11 : 7
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13. Anil, Bobby and Chintu jointly invest in a business and agree to share the overall profit in proportion to their investments. Anil's share of investment is 70%. His share of profit decreases by ₹420 if the overall profit goes down from 18% to 15%. Chintu's share of profit increases by ₹80 if the overall profit goes up from 15% to 17%. The amount, in ₹, invested by Bobby is
- (1) 2200
- (2) 2400
- (3) 1800
- (4) 2000
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14. One part of a hostel's monthly expenses is fixed, and the other part is proportional to the number of its boarders. The hostel collects ₹1600 per month from each boarder. When the number of boarders is 50, the profit of the hostel is ₹200 per boarder, and when the number of boarders is 75, the profit of the hostel is ₹250 per boarder. When the number of boarders is 80, the total profit of the hostel, in ₹, will be
- (1) 20200
- (2) 20500
- (3) 20000
- (4) 20800
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15. In a village, the ratio of number of males to females is 5 : 4. The ratio of number of literate males to literate females is 2 : 3. The ratio of the number of illiterate males to illiterate females is 4 : 3. If 3600 males in the village are literate, then the total number of females in the village is:
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16. Pinky is standing in a queue at a ticket counter. Suppose the ratio of the number of persons standing ahead of Pinky to the number of persons standing behind her in the queue is 3 : 5. If the total number of persons in the queue is less than 300, then the maximum possible number of persons standing ahead of Pinky is:
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17. The price of a precious stone is directly proportional to the square of its weight. Sita has a precious stone weighing 18 units. If she breaks it into four pieces with each piece having distinct integer weight, then the difference between the highest and lowest possible values of the total price of the four pieces will be 2,88,000. Then, the price of the original precious stone is
- (1) 9,72,000
- (2) 12,96,000
- (3) 19,44,000
- (4) 16,20,000
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18. In a company, 20% of the employees work in the manufacturing department. If the total salary obtained by all the manufacturing employees is one-sixth of the total salary obtained by all the employees in the company, then the ratio of the average salary obtained by the manufacturing employees to the average salary obtained by the non-manufacturing employees is
- (1) 4 : 5
- (2) 6 : 5
- (3) 5 : 6
- (4) 5 : 4
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19. The number of coins collected per week by two coin-collectors A and B are in the ratio 3 : 4. If the total number of coins collected by A in 5 weeks is a multiple of 7, and the total number of coins collected by B in 3 weeks is a multiple of 24, then the minimum possible number of coins collected by A in one week is _______.
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20. A fruit seller has a total of 187 fruits consisting of apples, mangoes and oranges. The number of apples and mangoes are in the ratio 5 : 2. After she sells 75 apples, 26 mangoes and half of the oranges, the ratio of number of unsold apples to number of unsold oranges becomes 3 : 2. The total number of unsold fruits is
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21. When Rajesh's age was same as the present age of Garima, the ratio of their ages was 3 : 2. When Garima's age becomes the same as the present age of Rajesh, the ratio of the ages of Rajesh and Garima will become
- (A) 4 : 3
- (B) 2 : 1
- (C) 3 : 2
- (D) 5 : 4
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22. Rajesh and Vimal own 20 hectares and 30 hectares of agricultural land, respectively, which are entirely covered by wheat and mustard crops. The cultivation area of wheat and mustard in the land owned by Vimal are in the ratio of 5 : 3. If the total cultivation area of wheat and mustard are in the ratio 11 : 9, then the ratio of cultivation area of wheat and mustard in the land owned by Rajesh is
- (A) 1 : 1
- (B) 7 : 9
- (C) 4 : 3
- (D) 3 : 7
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23. The ratio of the number of students in the morning shift and afternoon shift of a school was 13 : 9. After 21 students moved from the morning shift to the afternoon shift, this ratio became 19 : 14. Next, some new students joined the morning and afternoon shifts in the ratio 3 : 8 and then the ratio of the number of students in the morning shift and the afternoon shift became 5 : 4. The number of new students who joined is
- (A) 110
- (B) 88
- (C) 121
- (D) 99
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24. The ratio of expenditures of Lakshmi and Meenakshi is 2 : 3, and the ratio of income of Lakshmi to expenditure of Meenakshi is 6 : 7. If excess of income over expenditure is saved by Lakshmi and Meenakshi, and the ratio of their savings is 4 : 9, then the ratio of their incomes is
- (A) 5 : 6
- (B) 3 : 5
- (C) 7 : 8
- (D) 2 : 1
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25. The ratio of the number of coins in boxes A and B was 17:7. After 108 coins were shifted from box A to box B, this ratio became 37:20. The number of coins that needs to be shifted further from A to B, to make this ratio 1:1, is
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CAT 2024 & 2025, recent
Fresh questions distributed from the real CAT 2024 & CAT 2025 papers into this chapter.
- (A) 2 : 1
- (B) 1 : 2
- (C) 5 : 8
- (D) 8 : 5