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The average age of P and Q is 19 years. If R replaces P, the average will be 20 years and if R replaces Q, the average would be 21. What are the ages of P, Q, and R?

The average age of P and Q is 19 years. If R replaces P, the average will be 20 years and if R replaces Q, the average would be 21. What are the ages of P, Q, and R? Correct Answer 20, 18, 22

Trick:

Let's decode two statements

1) The average age of P and Q is 19 years. If R replaces P, the average will be 20 years

Since R is replacing P and average is increasing by 1

It means, R is 2 years greater than P

2) The average age of P and Q is 19 years and if R replaces Q, the average would be 21

Since R is replacing Q and average is increasing by 2

It means R is 4 years greater than Q

Only option 2 reflects the same, hence it is the answer

Detailed solution:

Let’s assume that the ages of P, Q, and R are p, q, and r years respectively.

We know that,

Average = (Sum of all quantities)/(Number of quantities)

Average of age of people = (Sum of age of people)/(Number of people)

∵ Average of P and Q is 19 years,

19 = (Sum of ages of P and Q)/2 = (p + q)/2

⇒ p + q = 19 × 2 = 38      ----(i)

∵ Average of R and Q is 20 years,

20 = (Sum of ages of R and Q)/2 = (q + r)/2

⇒ q + r = 20 × 2 = 40

⇒ q = (40 - r)       ----(ii)

∵ Average of P and R is 21 years,

21 = (Sum of ages of P and R)/2

⇒ p + r = 21 × 2 = 42

⇒ p = (42 - r)      ----(iii)

Now, substituting the values of p and q from Equations (ii) and (iii) in Equation (i),

∴ (42 - r) + (40 - r) = 38

⇒ 82 - 2r = 38

⇒ 2r = 44

⇒ r = 22

∴ p = 42 - r

⇒ p = 42 - 22 = 20

Also, q = 40 - r

⇒ q = 40 - 22 = 18

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