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Ratio, Proportion & Average

Quantitative Aptitude · Ratio, Average, SI & CI

Useful for: SSC CGL

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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

The average of 11 numbers is 50. If the average of first 6 is 49 and last 6 is 52, the 6th number is:
1. 56
2. 54
3. 52
4. 58
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Answer: (a) 56. Sum of 11 = 550.
Two numbers are in ratio 3:5. If 8 is added to each, the ratio becomes 2:3. The smaller number is:
1. 24
2. 30
3. 18
4. 40
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Answer: (a) 24. Numbers = 3k, 5k.
The average age of a class of 30 students is 14 years. If the teacher's age is included, the average becomes 15. The teacher's age is:
1. 46 years
2. 40 years
3. 45 years
4. 44 years
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Answer: (c) 45 years. Students sum = 30×14 = 420.
If a:b = 2:3 and b:c = 4:5, then a:b:c is:
1. 8:3:5
2. 8:12:15
3. 2:3:5
4. 2:12:15
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Answer: (b) 8:12:15. B ko common banao: b=3 aur b=4 → LCM 12.
The average of first 50 natural numbers is:
1. 25.5
2. 25
3. 26
4. 24.5
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Answer: (a) 25.5. Average of first n natural numbers = (n+1)/2 = 51/2 = 25.5.
Rs 6300 is divided among A, B, C in the ratio 1/2 : 1/3 : 1/6. A's share is:
1. Rs 2800
2. Rs 3150
3. Rs 2100
4. Rs 1050
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Answer: (b) Rs 3150. Ratio 1/2:1/3:1/6 → LCM 6 se multiply → 3:2:1.
The average of 5 consecutive even numbers is 16. The largest number is:
1. 20
2. 18
3. 22
4. 16
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Answer: (a) 20. Consecutive numbers ka average = middle number.
The ratio of two numbers is 3:4 and their HCF is 5. Their LCM is:
1. 20
2. 60
3. 30
4. 45
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Answer: (b) 60. Numbers = 3×5=15, 4×5=20 (HCF=5=k).
The mean of 20 observations is 17. If each observation is multiplied by 3 and then 2 is added, the new mean is:
1. 53
2. 51
3. 49
4. 55
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Answer: (a) 53. New mean = 3×(old mean) + 2 = 3×17 + 2 = 51+2 = 53.
In a ratio which is equal to 5:8, if the antecedent is 40, the consequent is:
1. 56
2. 72
3. 64
4. 60
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Answer: (c) 64. 5:8 = 40:x → 5/8 = 40/x → x = 40×8/5 = 64.