 QUESTION

# Two numbers are in the ratio 3:5. If each number is increased by 10, the ratio becomes 5:7. The numbers are __________.a)3, 5b)7, 9c)13, 22d)15, 25

Hint: Assume the two numbers are multiple of ‘x’ and write their values in terms of ‘x’. then add 10 in both the numbers and take the ratio of two numbers. Then equate the ratio with 5:7 to get the value of ‘x’. Use the value of ‘x’ to get numbers.

To solve the given question we will write the given data, therefore,

Two numbers are in the ratio = 3:5 ………………………………………………………….. (1)

The ratio of two numbers after increasing them by 10 = 5:7 = $\dfrac{5}{7}$ ………………………………….. (2)

Now, we will assume the two numbers are in the multiple of ‘x’, therefore by using equation (1) we can write the numbers as,

First number = 3x And Second number = 5x ……………………………. (3)

If we increase the numbers by 10 we will get the numbers as,

First number + 10 = 3x + 10

Second number+ 10 = 5x + 10

If we divide first number by second number we will get,

The ratio of two numbers after increasing them by 10 = $\dfrac{3x+10}{5x+10}$

If we compare above equation with equation (2) we will get,

$\therefore \dfrac{5}{7}=\dfrac{3x+10}{5x+10}$

If we do cross multiplication in the above equation we will get,

$\therefore 5\left( 5x+10 \right)=7\left( 3x+10 \right)$

By simplifying the above equation we will get,

$\therefore 25x+50=21x+70$

If we shift 50 on the right hand side and 21x on the left hand side of the equation we will get,

Therefore, 25x – 21x = 70 – 50

Therefore, 4x = 20

If we shift 4 on the right hand side of the equation we will get,

$\therefore x\text{ }=\text{ }\dfrac{20}{4}$

Therefore, x = 5

If we put the above value of ‘x’ in equation (3) we will get,

First number = 3 (5) and Second number = 5 (5)

Therefore, First number = 15 and Second number = 25

Therefore the two numbers are 15 and 25.

Therefore the correct answer is option (d).

Note: Many students commit the mistake of assuming 3x + 10 = 5 and will get a wrong answer as they have given the ratio after adding 10 in given numbers. Still if you want to solve it like that then you should assume another variable says ‘y’ and therefore get the simultaneous equations in two variables.