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Aptitude & Reasoning · A — Quantitative Aptitude

Averages, Ages & Mixtures

Average is a balance point. See it that way and age problems and alligation stop being separate topics.

Test weight: MediumTime / question: 50–70 secDifficulty: Medium

An average is the value each member would have if the total were shared equally. The master equation — total equals average times count — solves almost everything.

Foundations

Average = Total / Count.   Total = Average × Count.

Alligation (mixing two prices/values)

Cheaper qty : Dearer qty = (Dearer − Mean) : (Mean − Cheaper).

How to Approach It

Picture an average as a balance point, and treat 'total = average × count' as your master equation. Nearly every question in this chapter is that one identity in disguise.

  • Turn averages into totals. Multiply the average by the count to get the total, which is far easier to manipulate. Convert back to an average only at the very end.
  • Track only what changes. When a value is added, removed or replaced, the change in the total equals the count times the change in the average. Use this instead of recomputing the whole set.
  • In age problems, age everyone equally. Add the same number of years to every person, since ratios shift over time. Write the 'after' condition as its own equation and solve.
  • For mixtures, set up alligation. Place the two values on either side of the mean; quantities come in the ratio (dearer − mean) : (mean − cheaper). This gives a ratio of amounts, not the amounts themselves.

Techniques & Methods

  • Total = average × count. Turn averages into totals, do the arithmetic, convert back. e.g. Average 27 of 5 numbers → total 135.
  • New-value shortcut. Score needed for a new average = n×(new avg) − (n−1)×(old avg). e.g. 11×44 − 10×40 = 84.
  • Uniform shift. Add k to every value and the average simply rises by k. e.g. +4 to each, average 30 → 34.
  • Alligation. cheaper : dearer = (dearer − mean) : (mean − cheaper). e.g. 15 and 20 to make 18 → 2 : 3.
The Edge
Don't recompute the whole average when one value changes. The shift in the total equals count × change-in-average. If 10 players average 40 and a new score lifts the average of 11 to 44, the new score = 11×44 − 10×40 = 84. One line, no list.
Worked example
The average of 5 numbers is 27. When one number is removed, the average of the remaining 4 becomes 25. Find the removed number.
  1. Convert averages to totals: total of all 5 = 5 × 27 = 135.
  2. Total of the remaining 4 = 4 × 25 = 100.
  3. Removed number = 135 − 100 = 35.
  4. Check: 135 − 35 = 100, and 100 ÷ 4 = 25 — the stated new average.
Worked example
In what ratio must rice at Rs 15/kg be mixed with rice at Rs 20/kg to get a mixture worth Rs 18/kg?
  1. This is alligation — blending two prices to hit a target average.
  2. Place the mean (18) between the prices and take cross-differences.
  3. cheaper : dearer = (20 − 18) : (18 − 15) = 2 : 3.
  4. Read as: for every 2 parts of Rs 15 rice, use 3 parts of Rs 20 rice — a ratio of quantities.

Worked Drills

Worked example
The average age of 30 students is 12 years. Including the teacher, it becomes 13. The teacher's age is:
  1. Total of 31 people = 31 × 13 = 403; total of 30 = 30 × 12 = 360.
  2. Teacher = 403 − 360 = 43.
Worked example
Five years ago the average age of a family of 5 was 28. A baby is born and the average is still 28. The baby's age is:
  1. 5 years ago total = 5 × 28 = 140; now the 5 members total = 140 + 25 = 165.
  2. With baby, 6 members average 28 → total 168; baby = 168 − 165 = 3.
Worked example
Three numbers average 4000. A fourth person joins and the average drops to 3500. The fourth value is:
  1. Fourth = 4×3500 − 3×4000 = 14000 − 12000 = 2000.
Worked example
The average of 11 results is 50. The average of the first 6 is 49 and of the last 6 is 52. The 6th result is:
  1. First 6 + last 6 counts the 6th twice: 6×49 + 6×52 = 294 + 312 = 606.
  2. Subtract the total of all 11: 606 − 11×50 = 606 − 550 = 56.
Worked example
A man's age is three times his son's. Five years ago it was four times. The son's present age is:
  1. Let son = s, man = 3s. Then 3s − 5 = 4(s − 5).
  2. Solve: 3s − 5 = 4s − 20 → s = 15.
Worked example
The average of 50 numbers is 30. Two numbers are removed and the average of the remaining 48 becomes 28.5. The average of the two removed numbers is:
  1. Sum removed = 50×30 − 48×28.5 = 1500 − 1368 = 132.
  2. Average of the two = 132 / 2 = 66.
Worked example
60 litres of a mixture has milk and water in 2:1. The water to be added to make it 1:1 is:
  1. Milk = 40 L, water = 20 L. For 1:1 we need water = milk = 40 L.
  2. Add 40 − 20 = 20 L.
Worked example
The average weight of A, B, C is 45 kg; of A, B is 40 kg; of B, C is 43 kg. B's weight is:
  1. A+B+C = 135. A+B = 80 → C = 55. B+C = 86 → A = 49.
  2. B = 135 − 49 − 55 = 31 kg.
Worked example
The average of the first five multiples of 7 is:
  1. Multiples: 7, 14, 21, 28, 35; sum = 105.
  2. Average = 105 / 5 = 21.
⚠ Watch out
  • The average of a set need not be one of its members.
  • In age problems, every person ages by the same number of years — add it to each, not once.
  • Alligation gives the ratio of quantities, not the quantities themselves.
Takeaways
  • Total = average × count. Work in totals; convert back only at the end.
  • Track the change, not the set. Δtotal = count × Δaverage handles add/remove/replace.
  • Age everyone equally. Add the same years to each person and write the 'after' equation.
  • Alligation gives a ratio. Cross-differences around the mean — amounts come after.
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