Question

Seven years ago the ratio of the ages of P and Q
(in years) was 7: 6. Which of the following cannot be the ratio of their ages 6 years from now?
(a) 13:11
(b) 15 : 14
(c) 13:12
(d) 16:15
​

Answer 1

Answer:

Option (a) 13 : 11

Step-by-step explanation:

Given :

• Seven years ago the ratio of the ages of P and Q  (in years) was 7 : 6

To find :

which of the given options cannot be the ratio of their ages 6 years from now

Solution :

 Let the present age of P be x years and the present age of Q be y years.

Seven years ago,

P's age = (x - 7) years

Q's age = (y - 7) years

P's age/Q's age = 7/6

  $\sf \dfrac{x-7}{y-7}=\dfrac{7}{6} \\\\ \sf 6(x-7)=7(y-7) \\\\ \sf 6x-42=7y-49 \\\\ \sf 7y-6x=49-42 \\\\ \sf 7y-6x=7 \rightarrow [1]$

After 6 years,

P's age = (x + 6) years

Q's age = (y + 6) years

Now, we have to check each option.

 (a) 13 : 11

  $\sf \dfrac{x+6}{y+6}=\dfrac{13}{11} \\\\ \sf 11(x+6)=13(y+6) \\\\ \sf 11x+66=13y+78 \\\\ \sf 11x-13y=78-66 \\\\ \sf 11x-13y=12 \rightarrow [2]$

Multiplying equation [1] by 11 and equation [2] by 6, we get

 77y - 66x = 77

 66x - 78y = 72

Adding both the equations,

 77y - 78y = 149

   -y = 149

   y = -149

Age can not be negative.

Hence, this option can not be the ratio of their ages after 6 years

 

(b) 15 : 14

   $\sf \dfrac{x+6}{y+6}=\dfrac{15}{14} \\\\ \sf 14(x+6)=15(y+6) \\\\ \sf 14x+84=15y+90 \\\\ \sf 14x-15y=90-84 \\\\ \sf 14x-15y=6 \rightarrow [3]$

Multiplying the equation [1] by 7 and equation [3] by 3, we get

49y - 42x = 49

42x - 45y = 18

Adding both the equations,

49y - 45y = 67

  4y = 67

   y = 67/4

   y = 16.75 years

This can be possible since we got the value positive.

(c) 13 : 12

 $\sf \dfrac{x+6}{y+6}=\dfrac{13}{12} \\\\ \sf 12(x+6)=13(y+6) \\\\ \sf 12x+72=13y+78 \\\\ \sf 12x-13y=78-72 \\\\ \sf 12x-13y=6 \rightarrow [4]$

Multiplying the equation [1] by 2, we get

14y - 12x = 14

12x - 13y = 6

Adding both the equations,

14y - 13y = 20

   y = 20 years

This can also be possible since we got the positive value.

(d) 16 : 15

  $\sf \dfrac{x+6}{y+6}=\dfrac{16}{15} \\\\ \sf 15(x+6)=16(y+6) \\\\ \sf 15x+90=16y+96 \\\\ \sf 15x-16y=96-90 \\\\ \sf 15x-16y=6 \rightarrow [5]$

Multiplying the equation [1] by 5 and equation [5] by 2, we get

35y - 30x = 35

30x - 32y = 12

Adding both the equations,

 3y = 47

  y = 47/3

  y = 15.67 years

This can also be possible since we got the positive value.

So, the answer is option (a)