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The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. What are the numbers?

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± 12 and 18

± 12 and 20

± 2 and 18

± 13 and 18

answer is A.

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Detailed Solution

Given that the difference of squares of two numbers is 180.
Also, the square of the smaller number is 8 times the larger number.
Let the smaller number be n.
Let the larger number be m.
Square of smaller number

= n 2

.
Also, the square of smaller number is 8 times the larger number,

⇒ n 2 = 8 m

Square of larger number

.
The difference of squares of numbers is 180.
So,

⇒ m 2 − n 2 = 180 ⇒ m 2 − n 2 − 180 = 0

We know that,

n 2 = 8 m

.
Now, using the above relation,

m 2 − 8 m − 180 = 0

.
Solve the above quadratic equation using factorisation method.

m 2 − 8 m − 180 = 0 ⇒ m 2 − 18 m + 10 m − 180 = 0

Take out common terms.

⇒ m (m − 18) + 10 (m − 18) = 0 ⇒ (m − 18) (m + 10) = 0 ⇒ m − 18 o r m + 10 = 0 ⇒ m = 18 o r m = − 10

As the larger number cannot be negative, m = 18.
Smaller number,

n 2 = 8 m

⇒ n 2 = 8 × 18 ⇒ n 2 = 144

Take square root on both sides,

⇒ n = 144 ⇒ n = ± 12

Smaller number, n=

± 12

.
Larger number, m=18.
The numbers can be 18 and 12 or 18 and -12.
Therefore, the correct option is 1.

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