Ratio & Proportion
Scenario
You are splitting a Rs1200 Swiggy bill among 3 roommates who ate different amounts.
How do you divide a single total into uneven, relative slices without losing track of the sum?
Mental model
Ratios are relative slices of a pie. Proportion means two pies are sliced exactly the same way.
A ratio compares two or more quantities of the same kind. A proportion equates two ratios, letting you scale one up if you know the other.
Formula
If a:b = c:d (proportional), then ad = bc (cross-product rule). Value of 1 part = Total / Sum of ratio parts. Combining ratios: if A:B = x:y and B:C = m:n, then A:B:C = (x x m) : (y x m) : (y x n). Fourth proportional to a, b, c is d = bc/a. Mean proportional of a and b is sqrt(a x b).
The product of extremes equals the product of means. To merge three-part ratios (A:B:C from A:B and B:C), match the B terms by multiplying. For fractions of a whole, total ratio parts always sum to the whole.
Pattern discovery
Divide different totals according to the exact same 2:3 ratio.
- Ratio 2:3, Total 50 -> Parts: 20, 30
- Ratio 2:3, Total 100 -> Parts: 40, 60
- Ratio 2:3, Total 500 -> Parts: 200, 300
To find the value of one 'part', what do you divide the total quantity by?
Rule: You divide the total by the sum of the ratio parts. A ratio of 2:3 means the pie is cut into 5 equal slices.
Worked example
Problem: Divide Rs1260 among A, B, C in the ratio 3:4:5. How much does each get?
- Sum of ratio parts = 3 + 4 + 5 = 12.
- Value of 1 part = 1260 / 12 = Rs105.
- A = 3 x 105 = Rs315. B = 4 x 105 = Rs420. C = 5 x 105 = Rs525.
- Sanity-check: 315 + 420 + 525 = 1260. Correct.
Answer: A gets Rs315, B gets Rs420, C gets Rs525.
Problem: A:B = 2:3 and B:C = 4:5. Find A:C.
- To merge, make B terms equal. A:B = 2:3. Multiply by 4: A:B = 8:12. B:C = 4:5. Multiply by 3: B:C = 12:15.
- Now B = 12 in both. A:B:C = 8:12:15.
- A:C = 8:15.
- Sanity-check: A/C = (A/B) x (B/C) = (2/3) x (4/5) = 8/15. Matches.
Answer: A:C = 8:15.
Shortcuts
- Merging Ratios: If A:B = x:y and B:C = m:n, write them stacked. A:B:C = (x x m) : (y x m) : (y x n). (A:B = 2:3, B:C = 4:5. A:B:C = (2x4) : (3x4) : (3x5) = 8:12:15.)
- Unitary Method: Divide the total value by the total number of ratio parts to find the value of 1 unit, then scale up. (Divide Rs600 in 1:2:3. Total parts = 6. 1 part = 100. Shares: 100, 200, 300.)
Common mistakes
- Adding Constants to Ratios: If a ratio is 2:3, you cannot assume the values are 2 and 3. They are 2x and 3x. Do not add actual values to ratio numbers.
- Area and Volume Scaling: If sides are in ratio 1:2, areas are in ratio 1:4 (squares), and volumes are 1:8 (cubes). Do not apply the linear ratio to all dimensions.
Glossary
- proportion
- A mathematical statement that two ratios are exactly equal.
- extremes
- The outside numbers in a proportion.
- ratios
- A way to compare two amounts, showing how much of one thing there is compared to another.
Recall questions
- If A:B = 3:4 and B:C = 8:9, what is A:C?
- Divide 70 in the ratio 3:4. What is the larger part?
- Find the fourth proportional to 4, 9, 12.
Questions & answers
Two numbers are in the ratio 3:5. If 9 is subtracted from each, the new numbers are in the ratio 12:23. Find the smaller number.
33
Approach: (3x - 9)/(5x - 9) = 12/23. Cross-multiply: 69x - 207 = 60x - 108. 9x = 99 -> x = 11. Smaller number = 3x = 33.
A bag contains 50p, 25p and 10p coins in the ratio 5:9:4, amounting to Rs206. Find the number of 50p coins.
200
Approach: Values in Rs: 5x/2 + 9x/4 + 4x/10 = 206. LCM 20: (50x + 45x + 8x)/20 = 206 -> 103x = 4120 -> x = 40. 50p coins = 5x = 200.
Continue learning
Next: Mixtures & Alligation
Next: Partnership
Next: Averages