Question

10. Two numbers are in the ratio 5:6. If 9 is added to each of the numbers, then the ratio becomes 6:7. Find
the numbers.​

Answer 1

Let the numbers be x and y.

According to the question,

1. The ratio of the numbers is 5:6.

⇒ x / y = 5 / 6

⇒ 6x = 5y

⇒ 6x - 5y = 0 ...(1)

Also,

2. If 9 is added to both the numbers, the ratio becomes 6:7.

⇒ (x + 9) / (y + 9) = 6 / 7

⇒ 7x + 63 = 6y + 54

⇒ 7x - 6y = -9 ...(2)

Multiply eq.(1) by 6 and eq.(2) by 5, In order to make the coefficient of y same.

The new equations we get would be,

▶ 36x - 30y = 0 ...(3)

▶ 35x - 30y = -45 ...(4)

Subtract (4) from (3), we get

⇒ 36x - 30y - 35x + 30y = 0 - (-9)

⇒ 36x - 35x = 45

⇒ x = 45

On substituting the value of x in any of the above equations we will get the value of y.

Put x = 45 in (1)

⇒ 6(45) - 5y = 0

⇒ -5y = -6(45)

⇒ 5y = 6×45

⇒ y = 6×9

⇒ y = 54

We assumed the numbers to be x and y. Hence The numbers are 45 and 54.

Answer 2

Answer:

Given :-

• Two numbers are in the ratio of 5 : 6.
• If9 is added to each of the numbers, then the ratio becomes 6 : 7.

To Find :-

• What are the numbers.

Solution :-

Let, the first number be 5x

And, the second number will be 6x

According to the question,

↦ $\sf \dfrac{5x + 9}{6x + 9} =\: \dfrac{6}{7}$

By doing cross multiplication we get,

↦ $\sf 6(6x + 9) =\: 7(5x + 9)$

↦ $\sf 36x + 54 =\: 35x + 63$

↦ $\sf 36x - 35x =\: 63 - 54$

➠ $\sf\bold{\green{x =\: 9}}$

Hence, the required numbers are :

✪ First number :

↦ $\sf 5x$

↦ $\sf 5(9)$

↦ $\sf 5 \times 9$

➦ $\sf\bold{\red{45}}$

And,

✪ Second number :

↦ $\sf 6x$

↦ $\sf 6(9)$

↦ $\sf 6 \times 9$

➦ $\sf\bold{\red{54}}$

$\therefore$ The numbers are 45 and 54.