#### Question

The sum of two number *a* and *b* is 15, and the sum of their reciprocals `1/a` and `1/b` is 3/10. Find the numbers *a* and *b*.

#### Solution

Given that a and b be two numbers in such a way that b = (15 - a).

Then according to question

`1/a+1/b=3/10`

`(b + a)/(ab)=3/10`

`(a + b)/(ab)=3/10`

By cross multiplication

10a + 10b = 3ab ........ (1)

Now putting the value of *b* in equation (1)

10a + 10(15 - a) = 3a(15 - a)

10a + 150 - 10a = 45a - 3a^{2}

150 = 45a - 3a^{2}

3a^{2} - 45a + 150 = 0

3(a^{2} - 15a + 50) = 0

(a^{2} - 15a + 50) = 0

a^{2} - 10a - 5a + 50 = 0

a(a - 10) - 5(a - 10) = 0

(a - 10)(a - 5) = 0

a - 10 = 0

a = 10

Or

a - 5 = 0

a = 5

Therefore,

When* a = 10 *then

b = 15 - a = 15 - 10 = 5

And when* a = 5 *then

b = 15 - a = 15 - 5 = 10

Thus, two consecutive number be either a = 5, b = 10 or a = 10, b = 5.