# Average Age

During the 1^{st} Math2Shine International Vedic Mathematics Competition, only 41 out of 210 participants correctly answered this question. This is probably because the calculations involved in the traditional method will result in big products.

The average is usually computed as

A = (75 x 28 + 125 x 24)/ (75 + 125)

We can use doubling and halving together to quickly get

75 x 28 = 150 x 14 = 300 x 7 = 2100

125 x 24 = 250 x 12 = 500 x 6 = 1000 x 3 = 3000

(2100 + 3000)/ (75 + 125) = 5100/ 200 = 51/2 = 25.5

We can also use 24 as a base to simplify computations:

24 → 0 and 28 → 28 – 24 = 4

A = 24 + (75 x 4 + 125 x 0)/ (75 + 125) = 24 + 300/200 = 24 + 1.5 = 25.5

The 3^{rd} technique is what I call the “see-saw” method. To balance the weights in a see-saw, the heavier weight must be placed nearer the center or fulcrum than a lighter weight on the opposite side. Weights W1 and W2 should be placed at distances D1 and D2 respectively from the center such that W1 x D1 = W2 x D2 so that they will be balanced. This is equivalent to W1/W2 = D2/D1.

The difference between the male and female averages is 28 – 24 or 4 years. The total number of males and females are in the ratio of 75:125 or 3: 5. If we partition 4 years in the ratio of 3: 5 we get 1.5: 2.5.

Since there are more female agents in the call center the average must be nearer 24, so the group average is 24 + 1.5 = 25.5 years