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# [Smart Math] Algebra Problem 19

Here’s and example of a SMART MATH problem for ALGEBRA. ### Problem

Three years ago A’s age was double that of B’s. Seven years later, the sum of their united ages will be 83 years. How old is A?

1. 24 years
2. 35 years
3. 45 years
4. 48 years
5. 54 years

### The Usual Method

Let the present age of A be ‘1’ years.

And the present age of B be ‘2’ years. $\therefore \left( a-3 \right)=2\left( b-3 \right)$

and $a+7+b+7=83$

Hence $a-3=2b-6$ $\therefore a-2b=-3$                                         …… Eq. 1

and $a+b=69$                                        …… Eq. 2

Solving both the equations simultaneously, we get: $a-2b=-3$

+ $2a+2b=138$ (Multiplying Eq. 2 by 2 on both sides)

3a = 135 $\therefore a=45$

(Ans: 3)

Estimated Time to arrive at the answer = 60 seconds.

### Using Technique

Note that the age of A three years earlier should have been as even number $\because \left( a-3 \right)=2\left( b-3 \right)$. From the options, the various values of (a – 3) that we get are:

Option # Value of (a – 3)

a                      24 – 3 = 21 (Odd)

b                      35 – 3 = 32 (Even)

c                      45 – 3 = 42 (Even)

d                      48 – 3 = 45 (Odd)

e                      54 – 3 = 51 (Odd)

Options ‘1’, ‘4’ and ‘5’ are eliminated as there yield odd values.

Note that half of 83 = 41.5. From the given conditions, we can also deduce that a > b. Hence, a > 41.5. This leaves us only with option ‘3’.

(Ans: 3)

Estimated Time to arrive at the answer = 15 seconds.
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