# (solved)Question 4 SSC-CGL 2018 June 4 Shift 1

Four years ago, the ratio of the ages of A and B was 4: 5. Eight years from now, the ratio of the ages of A and B will be 11: 13. What is the sum of the present age of both of them?
(Rev. 24-Aug-2023)

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### Question 4SSC-CGL 2018 June 4 Shift 1

Four years ago, the ratio of the ages of A and B was 4: 5. Eight years from now, the ratio of the ages of A and B will be 11: 13. What is the sum of the present age of both of them?

1. 76
2. 72
3. 80
4. 96

### Solution in Short

Ages of A and B 4 years back: 4k, 5k.

After 8 years from today, their ages will be 4k + 12 and 5k + 12. This ratio is given as 11 : 13. Form an equation and get k = 8.

Present ages are 4k + 4 and 5k + 4 = 36 and 44, giving the sum as 80 answer!

### Solution in Detail

Ages 4 years back: $\displaystyle 4k$ and $\displaystyle 5k$

Ages 8 years from today: 4k + 12, 5k + 12

Ratio given as $\displaystyle \frac{4k + 12 }{5k + 12} = \frac{11}{13}$

$\displaystyle \implies k = 8$

Present age of $\displaystyle \text{A}$: $\displaystyle 4k + 4$

$\displaystyle \text{i.e., } = 4 \times 8 + 4 = 36$ years

Similarly, of $\displaystyle \text{B}$ is $\displaystyle 5k + 4 = 44$

Their sum is 80 Ans.

### Solution by Clever Trick

ADVICE: my advice is that you should spend some time in understanding these tricks. Nothing like that if these can be learnt as a habit.

Four years ago, sum of their ages are $\displaystyle 4k + 5k = 9k$.

Hence, the sum of ages four years back has to be a multiple of $\displaystyle 9$.

Check first option: sum 4 years back is 76 - 8 = 68, not a multiple of 9. Rejected.

Check second option: sum 4 years back is 72 - 8 = 64, not a multiple of 9. Rejected.

Check third option: sum 4 years back is 80 - 8 = 72. Possible.

Similarly, fourth option also impossible.

Hence the answer is $\displaystyle 80$

### Solution by Fundamentals

Don't waste time on understanding this method if you can't. Given here for academic interest only.

Use this fact: the difference of the ages of two persons always remains same. For example, if you are two years elder to your sister, then you will always remain two years elder.

The difference in ages of A and B as per the ratio 11 : 13 is 2 [i.e., 13 - 11].

Now rewrite the ratio 4 : 5 as 8 : 10 so that the difference is 2 for this ratio also.

Unitary method: If increase in age of A is (11 - 8) = 3, then his age four years back was 8.

If the increase in age of A is 12 [4 years back + 8 years later], then the age four years back should be $\displaystyle \frac{8}{3} \times 12 = 32$

Similarly, age of B four years back should be $\displaystyle 40$

Sum of present ages = $\displaystyle (40 + 4) + (32 + 4) = 80$ years Ans.

### Solution by another Clever Trick

From the ratio of their ages four years ago, we can infer that their present ages are $\displaystyle 4k + 4$ and $\displaystyle 5k + 4$

Sum of ages is $\displaystyle 9k + 8$, where $\displaystyle k$ IS A NATURAL WHOLE NUMBER.

Check the first option [i.e., 76]. The sum of ages as $\displaystyle 9k + 8 = 76$ gives $\displaystyle k = 68/9$ which contradicts the fact that $\displaystyle k$ should be a whole number. This option is rejected.

Check second option, $\displaystyle 9k + 8 = 72$ is possible for $\displaystyle k = 7$.

But then their present ages will be $\displaystyle 4 k + 4 \equiv 32$ and $\displaystyle 5 k + 4 \equiv 39$

Ratio $\displaystyle 8$ years later will be $\displaystyle \frac{32 + 8}{39 + 8} \ne \frac{11}{13}$

Hence second option is not possible.

Check third option, $\displaystyle 9k + 8 = 80$ is possible for $\displaystyle k = 8$.

Present ages will be $\displaystyle 36$ and $\displaystyle 44$

Ratio $\displaystyle 8$ years later will be $\displaystyle \frac{36 + 8}{44 + 8} = \frac{11}{13}$

Hence third option is correct. On similar lines, we can verify that fourth is not possible.