CSAT· Prelims
Ratio and Proportion for CSAT
Ratio and proportion for CSAT: simplifying ratios, fourth and mean proportionals, direct and inverse variation, and ratio methods inside ages and partnership questions, with MCQs.
Ratio is a comparison of two quantities of the same kind, written a:b and read as 'a to b'. It tells how many times one quantity contains the other: the ratio 3:4 means that for every 3 units of the first there are 4 units of the second. Proportion is the statement that two ratios are equal, written a:b = c:d, which is true exactly when a x d = b x c (the product of the extremes equals the product of the means). Ratio and proportion matter for CSAT because Paper II is a qualifying paper needing a 33 percent score, and ratio questions sit inside ages, partnership, mixture and time-work problems: master the ratio, and four chapters get easier at once.
Reading and simplifying ratios
A ratio a:b is the fraction a/b in disguise. To simplify a ratio, divide both terms by their HCF, just like reducing a fraction: 18:24 simplifies to 3:4. To compare two ratios, cross-multiply: 3:4 versus 5:7 gives 3 x 7 = 21 against 4 x 5 = 20, so 3:4 is the larger. To split a quantity in the ratio a:b, give the first part a/(a+b) of the total and the second part b/(a+b).
Example 1. Divide Rs. 1,260 between A and B in the ratio 5:4. The total parts are 5 + 4 = 9. A's share = (5/9) x 1,260 = 700; B's share = (4/9) x 1,260 = 560. Check: 700 + 560 = 1,260.
Proportion and the rule of three
Four numbers a, b, c, d are in proportion when a:b = c:d, and the test is a x d = b x c. Here a and d are the extremes and b and c are the means. If three of the four are known, the fourth follows from the rule of three: d = (b x c) / a. A special case is the mean proportional between two numbers: the mean proportional between a and b is the number x with a:x = x:b, which gives x = the square root of (a x b).
Example 2. Find the fourth proportional to 4, 9 and 12. Set 4:9 = 12:x. Cross-multiplying: 4x = 108, so x = 27. The fourth proportional is 27.
Example 3. Find the mean proportional between 8 and 18. The mean proportional is the square root of (8 x 18) = square root of 144 = 12. Check: 8:12 = 12:18 both reduce to 2:3.
Direct and inverse proportion
Two quantities are in direct proportion when one increasing makes the other increase in the same ratio: more workers, more work done in a day. They are in inverse proportion when one increasing makes the other decrease: more workers, fewer days to finish a fixed job. In symbols, direct means y = kx (y/x is constant) and inverse means y = k/x (x x y is constant). The classic CSAT test is: if 12 men do a job in 15 days, how long do 20 men take? Men and days are inversely proportional, so 12 x 15 = 20 x days, giving 9 days.
Example 4. A car travels 240 km on 16 litres of fuel. How far will it go on 25 litres? Distance and fuel are directly proportional, so 240:16 = x:25. Cross-multiplying: 16x = 6,000, so x = 375. The car will go 375 km.
Ratios in ages and partnership
Ratio is the engine inside ages questions and partnership questions. In ages, a ratio like 4:5 for present ages means the ages are 4k and 5k for some number k; any condition (six years hence, difference of 8 years) gives an equation for k. In partnership, profit is shared in the ratio of (capital x time invested): if A invests Rs. 4,000 for 12 months and B invests Rs. 6,000 for 8 months, the profit ratio is (4,000 x 12):(6,000 x 8) = 48,000:48,000 = 1:1.
Example 5. The present ages of two persons are in the ratio 4:5. Six years hence the ratio will be 6:7. Find their present ages. Let the ages be 4k and 5k. Then (4k + 6)/(5k + 6) = 6/7. Cross-multiplying: 7(4k + 6) = 6(5k + 6), so 28k + 42 = 30k + 36, giving 2k = 6 and k = 3. Their present ages are 12 and 15 years.
Common traps and speed tips
- Trap 1: ratio has no units. 3:4 is a pure number. Rs. 30:Rs. 40 is fine, but Rs. 30:40 cm is meaningless. Both terms must be the same kind.
- Trap 2: order matters. The ratio 3:4 is not the same as 4:3. In mixture and age questions, match each term to the right person or item before solving.
- Trap 3: adding ratios directly. If A:B = 2:3 and B:C = 4:5, you cannot write A:C = 2:5. First make the B terms equal: A:B = 8:12 and B:C = 12:15, so A:C = 8:15.
- Tip 1: use k for the common multiplier. Writing 4:5 as 4k and 5k turns every ratio equation into a one-variable linear equation.
- Tip 2: cross-multiply before simplifying. In a:b = c:d, write ad = bc immediately; it avoids fraction mistakes under time pressure.
Situation | Terms | Ratio of shares |
|---|---|---|
Split Rs. 900 in ratio 2:3 | parts = 2 + 3 = 5 | 600 and 300 |
A:B = 2:3, B:C = 4:5 | make B = 12 | A:B:C = 8:12:15 |
Profit sharing | share in ratio capital x time | weight = C x T |
Direct proportion | y/x constant | double x, double y |
Inverse proportion | x x y constant | double x, halve y |
If A:B = 3:4 and B:C = 2:5, then A:C equals:
Show answer
Answer: (B) Make the B terms equal: A:B = 3:4 = 6:8 and B:C = 2:5 = 8:20, so A:B:C = 6:8:20 and A:C = 6:20 = 3:10.
The fourth proportional to 6, 8 and 15 is:
Show answer
Answer: (C) Set 6:8 = 15:x; cross-multiplying gives 6x = 120, so x = 20.
Two numbers are in the ratio 5:7. If each is increased by 6, the ratio becomes 3:4. The larger number is:
Show answer
Answer: (A) Let the numbers be 5k and 7k. Then (5k+6)/(7k+6) = 3/4, so 4(5k+6) = 3(7k+6), giving 20k + 24 = 21k + 18, so k = 6. The larger number is 7 x 6 = 42.
A sum of Rs. 2,400 is divided among A, B and C in the ratio 2:3:5. B's share is:
Show answer
Answer: (B) Total parts = 2 + 3 + 5 = 10. B's share = (3/10) x 2,400 = Rs. 720.
If 15 workers can build a wall in 48 days, how many days will 20 workers take (same rate)?
Show answer
Answer: (A) Workers and days are inversely proportional: 15 x 48 = 20 x days, so days = 720/20 = 36.
The mean proportional between 12 and 27 is:
Show answer
Answer: (A) Mean proportional = square root of (12 x 27) = square root of 324 = 18.
Answer key
- Q1 - (b). Make the B terms equal: A:B = 3:4 = 6:8 and B:C = 2:5 = 8:20, so A:B:C = 6:8:20 and A:C = 6:20 = 3:10.
- Q2 - (c). Set 6:8 = 15:x; cross-multiplying gives 6x = 120, so x = 20.
- Q3 - (a). Let the numbers be 5k and 7k. Then (5k+6)/(7k+6) = 3/4, so 4(5k+6) = 3(7k+6), giving 20k + 24 = 21k + 18, so k = 6. The larger number is 7 x 6 = 42.
- Q4 - (b). Total parts = 2 + 3 + 5 = 10. B's share = (3/10) x 2,400 = Rs. 720.
- Q5 - (a). Workers and days are inversely proportional: 15 x 48 = 20 x days, so days = 720/20 = 36.
- Q6 - (a). Mean proportional = square root of (12 x 27) = square root of 324 = 18.
Can two ratios with different units be equal?
No. A ratio compares two quantities of the same kind, so it is a pure number with no units. Writing 3 kg : 4 m is meaningless; only 3 kg : 4 kg (or both in metres) forms a valid ratio.
Why do ages in ratio 4:5 become 4k and 5k?
The ratio 4:5 means the first age divided by the second equals 4/5. Calling the actual ages 4k and 5k keeps that ratio for every value of k, so any extra condition (difference, future ratio) becomes a simple equation in one unknown k.
How is ratio used in partnership profit sharing?
Profit is shared in the ratio of each partner's (capital multiplied by time invested). If A puts in Rs. 5,000 for 12 months and B puts in Rs. 4,000 for 9 months, the profit ratio is 60,000:36,000 = 5:3.
What is the difference between ratio and fraction?
A ratio a:b is a comparison of two quantities, while a fraction a/b is a single number. Every ratio can be read as a fraction (a:b = a/b), but a ratio always describes two related quantities, such as two shares of a total.