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Ratio, Proportion & Partnership

Quantitative Aptitude / DI · Speed Calculation

Useful for: Railway RRB, IBPS Banking, CTET, OTET, OSSTET, REET, UPTET, BPSC TRE

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Overview

A ratio compares quantities; an average is the total divided by the count — most items reduce to managing totals.

Key points

To combine a:b and b:c, make b equal: a:b = 2:3, b:c = 4:5 → 8:12:15.
Mean proportional of a and b = √(ab); third proportional to a and b = b²/a.
Average = sum ÷ number of items; sum = average × number.
Average of first n natural numbers = (n + 1)/2; of first n odd numbers = n.
Weighted average = Σ(value × weight)/Σ weights.

Explanation

When one item is replaced, the change in sum = new − old; change in average = that change ÷ n.
Adding a person changes the average: new sum − old sum gives the new person’s value.
Ages: present ratio with past/future ages gives two equations — use the ratio parts as unknowns.
Dividing an amount in a ratio: share = amount × part ÷ total parts.

Step by step

Write totals.
Apply the change.
Divide back to the average or share.

Examples

A:b = 2:3 and b:c = 4:5 → a:b:c = 8:12:15.
Average of 5 numbers is 20. One number 25 is replaced by 40. New average = (100 − 25 + 40)/5 = 23.
₹480 divided in the ratio 6:2:4 → second share = 480 × 2/12 = ₹80.

Traps and advanced points

Trap: Adding ratios of different bases directly.
Trap: Averaging averages without weights (unequal group sizes).
Trap: Forgetting that the number of items changes when someone joins or leaves.

Quick revision

Work with totals, not averages.
Equalise the common term to merge ratios.
Weighted average for unequal groups.

Practice MCQs

₹600 is divided in the ratio 4:4:7. Find the second share.
1. 240
2. 200
3. 160
4. 120
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Answer: (c) 160. One ratio unit = 600/(15) = 40.
₹660 is divided in the ratio 5:4:2. Find the second share.
1. 360
2. 180
3. 300
4. 240
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Answer: (d) 240. One ratio unit = 660/(11) = 60.
₹720 is divided in the ratio 7:3:2. Find the second share.
1. 120
2. 300
3. 180
4. 240
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Answer: (c) 180. One ratio unit = 720/(12) = 60.
₹800 is divided in the ratio 5:5:6. Find the second share.
1. 250
2. 350
3. 200
4. 300
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Answer: (a) 250. One ratio unit = 800/(16) = 50.
₹1020 is divided in the ratio 7:7:3. Find the second share.
1. 420
2. 480
3. 360
4. 540
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Answer: (a) 420. One ratio unit = 1020/(17) = 60.
The average of 5 numbers is 67. If 4 of them are 35, 32, 85, 75, find the remaining number.
1. 118.8
2. 129.6
3. 97.2
4. 108
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Answer: (d) 108. Total = average × count.
The average of 6 numbers is 49. If 5 of them are 58, 68, 80, 24, 43, find the remaining number.
1. 27
2. 21
3. 24
4. 18
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Answer: (b) 21. Total = average × count.
The average of 6 numbers is 64. If 5 of them are 90, 81, 53, 44, 23, find the remaining number.
1. 93
2. 83.7
3. 102.3
4. 111.6
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Answer: (a) 93. Total = average × count.
The average of 7 numbers is 30. If 6 of them are 35, 33, 18, 27, 28, 20, find the remaining number.
1. 58.8
2. 53.9
3. 49
4. 44.1
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Answer: (c) 49. Total = average × count.
The average of 7 numbers is 61. If 6 of them are 68, 21, 44, 88, 54, 44, find the remaining number.
1. 97.2
2. 118.8
3. 129.6
4. 108
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Answer: (d) 108. Total = average × count.
₹440 is divided in the ratio 2:5:4. Find the second share.
1. 240
2. 280
3. 160
4. 200
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Answer: (d) 200. One ratio unit = 440/(11) = 40.
₹540 is divided in the ratio 4:2:3. Find the second share.
1. 120
2. 60
3. 180
4. 240
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Answer: (a) 120. One ratio unit = 540/(9) = 60.
₹640 is divided in the ratio 7:5:4. Find the second share.
1. 200
2. 160
3. 240
4. 280
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Answer: (a) 200. One ratio unit = 640/(16) = 40.
₹700 is divided in the ratio 2:7:5. Find the second share.
1. 300
2. 450
3. 400
4. 350
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Answer: (d) 350. One ratio unit = 700/(14) = 50.
₹780 is divided in the ratio 3:4:6. Find the second share.
1. 180
2. 300
3. 360
4. 240
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Answer: (d) 240. One ratio unit = 780/(13) = 60.