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Ratios and averages for LIC AAO

Dividing in a ratio, combining ratios, ratios that change when something is added, partnership, averages from totals, replacement, weighted averages and alligation. The methods and why they work, eight worked examples and a practice set.

2 Oct 2026 6 min read

In this guide
  1. Ratios: think in parts
  2. Averages: think in totals
  3. Worked examples
  4. Common mistakes
  5. Practice set
  6. What to do next

Ratios and averages turn up everywhere in LIC AAO Quant. A caselet says "in the ratio 7 : 5", a DI question asks for the average premium per branch, and a word problem replaces one person in a group and asks how the average moved. Both topics come down to one habit: work with parts for ratios and with totals for averages. Candidates who try to juggle averages directly, without converting to totals, are the ones who get stuck.

Ratios: think in parts

A ratio of 4 : 5 means the whole is split into 4 + 5 = 9 equal parts. Once you know the value of one part, everything else follows.

  • Dividing an amount. To split ₹2,700 in 4 : 5, one part is 2,700 ÷ 9 = 300. The shares are 1,200 and 1,500.
  • Combining ratios. If A : B = 5 : 6 and B : C = 4 : 7, the B values differ. Scale both so B is the same: B's LCM is 12. So A : B = 10 : 12 and B : C = 12 : 21, giving A : B : C = 10 : 12 : 21. This works because multiplying both terms of a ratio by the same number does not change it.
  • Ratios that change. If the same amount is added to both terms, write the new terms as (3k + 10) and (5k + 10) and set their ratio equal to the new one. Then solve for k.

Partnership

Profit is shared in the ratio of capital × time. A partner who puts in twice the money for half the time has the same claim as one who puts in the original sum for the full time. This is ratio work with one extra multiplication.

Averages: think in totals

Average = sum ÷ count, so sum = average × count. Almost every average problem is solved by converting averages to totals, doing the arithmetic on the totals, and converting back.

  • Removing an item. Old total minus new total gives the removed value.
  • Replacing an item. If one member is replaced and the average of n rises by d, the new member is heavier (or older, or higher) by n × d. The whole group's total rose by n × d, and only one member changed.
  • Weighted average. Combine totals, not averages. The average of two groups is (n₁a₁ + n₂a₂) ÷ (n₁ + n₂), which is closer to the larger group's average.
  • Consecutive numbers. The average of equally spaced numbers is the middle one, or the mean of the first and last.

Alligation: a weighted average in reverse

If groups averaging 60 and 70 combine to give 64, the groups are in the ratio (70 − 64) : (64 − 60) = 6 : 4 = 3 : 2. The group nearer the combined average is the bigger one. This is the weighted-average formula solved for the ratio of counts.

SituationConvert toKey step
Divide in a ratioPartsValue of one part = total ÷ sum of ratio terms
Combine two ratiosCommon middle termScale both to the LCM
Item removed from a groupTotalsOld total − new total
Item replacedTotalsNew value = old value + n × change in average
Two groups combinedTotalsAdd totals, add counts, divide
Ratio of two groups from their averagesAlligationCross the differences

Worked examples

Example 1. Divide ₹2,700 in the ratio 4 : 5.
One part = 2,700 ÷ 9 = 300. Shares: ₹1,200 and ₹1,500.

Example 2. A : B = 5 : 6 and B : C = 4 : 7. Find A : B : C.
Make B equal to 12: 10 : 12 and 12 : 21. 10 : 12 : 21.

Example 3. The average of 8 numbers is 25. One is removed and the average of the rest is 24. Find the removed number.
Old total = 200. New total = 7 × 24 = 168. Removed = 32.

Example 4. The average weight of 10 people rises by 1.5 kg when a person weighing 60 kg is replaced by a new person. Find the new person's weight.
The total rises by 10 × 1.5 = 15 kg. New person = 60 + 15 = 75 kg.

Example 5. 30 students average 60 marks and 20 students average 70. Find the overall average.
(30 × 60 + 20 × 70) ÷ 50 = (1,800 + 1,400) ÷ 50 = 64. By alligation, the ratio 3 : 2 matches 30 : 20.

Example 6. A invests ₹60,000 for 12 months and B invests ₹80,000 for 6 months. The profit is ₹44,000. Find each share.
Capital × time: 7,20,000 and 4,80,000, which is 3 : 2.
A = 3/5 × 44,000 = ₹26,400; B = ₹17,600.

Example 7. Two numbers are in the ratio 3 : 5. If 10 is added to each, the ratio becomes 5 : 7. Find the numbers.
(3k + 10) ÷ (5k + 10) = 5/7, so 21k + 70 = 25k + 50 and 4k = 20. k = 5.
The numbers are 15 and 25. Check: 25 : 35 = 5 : 7.

Example 8. After 16 innings, a batter scores 87 in the 17th and raises the average by 3. Find the new average.
Let the old average be x. Then 16x + 87 = 17(x + 3), so x = 87 − 51 = 36.
New average = 39. Check: 16 × 36 + 87 = 663, and 663 ÷ 17 = 39.

Common mistakes

  • Combining ratios without first making the common term equal.
  • Adding a number to a ratio's terms directly (3 : 5 plus 10 is not 13 : 15 in parts; it is 3k + 10 : 5k + 10).
  • Ignoring time in partnership questions.
  • Forgetting that a replacement changes the total by n × change in average, not by the change alone.

Practice set

  1. Divide ₹1,800 in the ratio 3 : 4 : 5. ₹450, ₹600, ₹750. One part = 1,800 ÷ 12 = 150.
  2. Average of the first 15 natural numbers? 8. The middle term of 1 to 15.
  3. A batch of 40 averages 55 and another of 60 averages 65. Combined average? 61. (2,200 + 3,900) ÷ 100.
  4. A : B = 2 : 3 and B : C = 4 : 5. Find A : B : C. 8 : 12 : 15. Make B equal to 12.
  5. The average of five consecutive even numbers is 36. Find the largest. 40. The numbers are 32, 34, 36, 38, 40.
  6. Two salaries are in the ratio 3 : 5. Each rises by ₹4,000 and the ratio becomes 5 : 7. Find the original salaries. ₹6,000 and ₹10,000. 21k + 28,000 = 25k + 20,000, so k = 2,000.
  7. A invests ₹40,000 for the full year. B joins after 4 months with ₹60,000. The year's profit is ₹34,000. Find each share. ₹17,000 each. 40,000 × 12 = 60,000 × 8 = 4,80,000.
  8. Groups averaging 50 and 80 combine to average 60. Find the ratio of their sizes. 2 : 1. (80 − 60) : (60 − 50) = 20 : 10.

What to do next

  • Rewrite every average question you meet this week as a totals question before solving it.
  • Do five combining-ratio and five partnership questions.
  • Use these skills in caselet DI, where ratios hide inside sentences.
  • Next, work through time, work and speed.

A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Life Insurance Corporation of India website .

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