Question

The sum of two numbers is 60. The ratio of their product to the sum of their square is 2:5. Find the

smallest number.​

Answer 1

Step-by-step explanation:

Let the smaller no. = x and greater no. =y

A/q

x+y=60

x=60-y. ...........(i)

Product of the no. = xy

sum of square of the numbers = x^2 + y^2

A/q

$\frac{xy}{ {x}^{2} + {y}^{2} } = \frac{2}{5} \\ 5xy = 2 {x}^{2} + 2 {y}^{2} \: \: \: \: \: \: \: \: ......(ii)$

put the value of x from equation (i) in equation (ii)

$5(60 - y)y = 2 ({60 - y})^{2} + 2 {y}^{2} \\ 300y - 5 {y}^{2} = 7200 + 2 {y}^{2} - 240y + 2 {y}^{2} \\ 9 {y}^{2} - 540y + 7200 = 0 \\ or \: take \: 9 \: as \: common \\ {y }^{2} - 60y + 800 = 0 \\ factorise \: this \: equation \\ {y}^{2} - 40y - 20y + 800 = 0\\ y(y - 40) - 20(y - 40) = 0 \\ (y - 20)(y - 40)$

if you take

y-20=0

y=20

or

y-40=0

y=40

we suppose y as greater number so here we take

y=40

put the value of y in equation (i)

x= 60-y

= 60-40

=20

So smallest number = x=20

I hope it will help you.

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Thank you.