Lesson 23 of 100 | Quantitative Aptitude / Ratio and Proportion / अनुपात और समानुपात
Dividing quantities in a ratio
Learning outcome
Divide totals among recipients and recover quantities from differences, preserving the stated order and units.
Understanding ratio shares
Ratio terms count equal-sized parts, not actual amounts. To divide a positive total T in the positive ratio a:b, assign a parts to the first share and b to the second: a + b parts altogether.
Each part has value T/(a + b). Multiplying by a and b gives the respective shares. The common part size guarantees a:b. Their sum is T because every part has been allocated.
For a:b:c, use a + b + c total parts. The reasoning extends to more recipients. Simplifying the ratio first shortens arithmetic without changing the shares.
Distinguish part-to-part from part-to-whole comparisons. In a:b, the first share is a/b times the second, but a/(a + b) of the total. Choosing the wrong denominator answers a different question.
For a known positive difference D, with a > b, one part equals D/(a − b). The larger share contains a − b extra parts, explaining division by the difference rather than the sum.
Convert measurements to common units before combining them. Check both the stated total or difference and the ratio. Shares of indivisible objects must also be whole numbers.
Worked example 1
Divide ₹1560 between A and B in the ratio 5:7.
Total parts = 5 + 7 = 12. One part = 1560/12 = ₹130. Therefore A receives 5 × 130 = ₹650 and B receives 7 × 130 = ₹910. Their sum is ₹1560, and dividing both amounts by 130 recovers 5:7.
Worked example 2
Split 4.2 kg of mixture into three portions in the ratio 2:3:5.
Convert 4.2 kg to 4200 g. There are 2 + 3 + 5 = 10 parts, so each part weighs 4200/10 = 420 g. The portions weigh 840 g, 1260 g and 2100 g respectively. They sum to 4200 g and contain the required numbers of equal parts.
Worked example 3
Two positive lengths are in the ratio 7:4. The longer exceeds the shorter by 18 cm. Find both.
The difference represents 7 − 4 = 3 parts. One part is 18/3 = 6 cm. Hence the lengths are 7 × 6 = 42 cm and 4 × 6 = 24 cm. Their difference is 18 cm, as required.
Common mistakes
Do not divide the total by just one ratio term. Keep the recipients in order, and distinguish totals from differences. An allocation can add up correctly yet still have the wrong ratio.
Practice questions
- Divide ₹1980 in the ratio 4:5.
- Divide 3.6 L in the ratio 1:2:3. Give each portion in millilitres.
- Share 84 identical notebooks among A, B and C in the ratio 3:4:5.
- Two positive money amounts are in the ratio 9:5. The larger exceeds the smaller by ₹320. Find both.
Worked answers
- There are 9 parts. Each is 1980/9 = ₹220, so the shares are ₹880 and ₹1100, totalling ₹1980.
- Convert to 3600 mL. There are 6 parts, each 600 mL. The portions are 600 mL, 1200 mL and 1800 mL.
- There are 12 parts, each containing 84/12 = 7 notebooks. A, B and C receive 21, 28 and 35 respectively.
- The difference contains 9 − 5 = 4 parts. Each is 320/4 = ₹80. The amounts are ₹720 and ₹400; their difference is ₹320.