A ratio compares quantities of the same kind; a proportion states that two ratios are equal. The recurring skill is converting a chain of ratios to a single common scale.
Chaining ratios
If a:b = 2:3 and b:c = 4:5, scale b to a common value → a:b:c = 8:12:15.
Partnership
Profit share ratio = (Capital × Time) for each partner.
How to Approach It
A ratio is a recipe and your job is to scale it. Almost every question is one of two things: chaining ratios together, or splitting a total into parts.
- Express the total in parts. Add the ratio terms to find the total number of parts, then work out the value of one part. Every share is that part value times its ratio term.
- Chain through the shared term. To combine a:b with b:c, scale b to the same number in both ratios (its LCM). Only once b matches can you read off a:c reliably.
- Weight partnership by time. Profit follows capital multiplied by duration, not capital alone. Compute each partner's money×months product, then reduce to a ratio.
- Reduce, then match the question. Simplify the final ratio to lowest terms before comparing options, and confirm whether the question wants a share, a difference, or the ratio itself.
Techniques & Methods
- Scale the common term. To chain a:b and b:c, raise b to the LCM of its two values. e.g. 2:3 and 4:5 → 8:12:15.
- Parts method. Treat the total as a sum of parts; one part = total ÷ (sum of ratio terms). e.g. 880 in 3:5 → part 110 → 330 and 550.
- Capital × time. Partnership profit divides as (money × months) for each partner. e.g. 12000×6 : 9000×8 = 72000:72000 = 1:1.
- Difference of parts. If two shares differ by k, that equals (difference in ratio terms) × one part. e.g. 7:9 differing by 8 → 1 part = 4 → 28 and 36.
The Edge
In partnership, never compare money alone — compare capital × months. A partner who invests less but stays in longer can still earn the bigger share. If everyone invests for the same duration, the time term cancels and profit simply follows the capital ratio.Worked example
Divide Rs 880 between two people in the ratio 3 : 5.
- Treat the ratio as parts: 3 + 5 = 8 parts in all.
- Value of one part: 880 ÷ 8 = 110.
- Each share = parts × part value: 3 × 110 = 330 and 5 × 110 = 550.
- Check: 330 + 550 = 880 and 330 : 550 reduces to 3 : 5.
Answer: Rs 330 and Rs 550
Worked example
A invests Rs 12000 for 6 months, B invests Rs 9000 for 8 months. Find their profit-sharing ratio.
- Profit follows capital × time, so weight each investment by its months.
- A's weight = 12,000 × 6 = 72,000.
- B's weight = 9,000 × 8 = 72,000.
- Ratio = 72,000 : 72,000 = 1 : 1 — B's longer duration evens out A's larger capital.
Answer: Profit ratio = 1 : 1
Worked Drills
Worked example
If a:b = 2:3 and b:c = 4:5, then a:c is:
- Scale b to a common value: a:b:c = 8:12:15.
- So a:c = 8:15.
Answer: 8:15 (option a)
Worked example
Partners A, B, C invest in the ratio 4 : 6 : 9 and the profit is Rs 5700. C's share is:
- Total parts = 4 + 6 + 9 = 19.
- C's share = 9/19 × 5700 = 2700.
Answer: Rs 2700 (option c)
Worked example
If x : y = 3 : 4, then (2x + 3y) : (3x + 2y) equals:
- Put x = 3, y = 4: 2x + 3y = 6 + 12 = 18; 3x + 2y = 9 + 8 = 17.
- Ratio = 18:17.
Answer: 18:17 (option a)
Worked example
If A:B = 2:3, B:C = 4:5 and C:D = 6:7, then A:D is:
- Chain to A:B:C = 8:12:15; scale C:D so C matches → A:B:C:D = 16:24:30:35.
- A:D = 16:35.
Answer: 16:35 (option a)
Worked example
Rs 5600 is divided among A, B, C with A:B = 2:3 and B:C = 4:5. C's share is:
- Chain: A:B:C = 8:12:15, total 35 parts.
- C = 15/35 × 5600 = 2400.
Answer: Rs 2400 (option c)
Worked example
Equal capitals are invested: A for 12 months, B for 9 months, C for 6 months. The profit ratio is:
- Equal capital → profit follows time: 12:9:6.
- Reduce: 4:3:2.
Answer: 4:3:2 (option a)
Worked example
Milk and water are in the ratio 5:1. Adding 5 litres of water makes it 5:2. The original milk is:
- Let parts be milk = 5k, water = k. New ratio: 5k/(k+5) = 5/2 → k = 5.
- Milk = 5k = 25 L.
Answer: 25 L (option b)
Worked example
Two numbers are in ratio 3:5. Increasing each by 10 makes the ratio 5:7. The smaller number is:
- Let the numbers be 3x and 5x. Then (3x+10)/(5x+10) = 5/7 → x = 5.
- Smaller = 3x = 15.
Answer: 15 (option a)
Worked example
If A:B = 3:4, then 20% of A is what percent of B?
- Let A = 3, B = 4. 20% of A = 0.2 × 3 = 0.6.
- 0.6 / 4 = 0.15 = 15%.
Answer: 15% (option b)
⚠ Watch out
- Adding the same number to both terms of a ratio changes the ratio (2:3 + 4 each = 6:7).
- In partnership, equal capital does not mean equal profit if the durations differ.
- Always reduce the final ratio to lowest terms before matching options.
Takeaways
- Parts method splits totals. One part = total ÷ sum of terms; multiply back for each share.
- Chain through the shared term. Scale b to its LCM to combine a:b with b:c.
- Partnership = capital × time. Longer duration can beat larger capital.
- Reduce before matching. Always simplify the final ratio against the options.