The sum of two numbers is 10 and the difference in their squares is 60. What are the two numbers?

Correct option is B

Given:

Sum of two numbers = 10

Difference of their squares = 60

Concept Used:

Algebraic equations and solving them.

a² - b² = (a + b)(a - b)

Solution:

Let the two numbers be x and y.

x + y = 10 (Equation 1)

x² - y² = 60 (Equation 2)

x² - y² = (x + y)(x - y)

Substitute the given value: (x + y)(x - y) = 60

Substitute the value of (x + y) from Equation 1:

10(x - y) = 60

x - y = 6 (Equation 3)

Solve Equations 1 and 3 simultaneously:

x + y = 10

x - y = 6

Add the two equations: 2x = 16

x = 8

Substitute the value of x in Equation 1:

8 + y = 10

y = 2

Therefore, the two numbers are 8 and 2.

Alternate Solution:

From x+y=10, we can say x=10-y.

Substitute that into x²-y²=60.

(10-y)²-y²=60

100-20y+y²-y²=60

100-20y=60

40=20y

y=2

Substitute y=2 into x+y=10.

x+2=10

x=8.

The numbers are 8 and 2.