Question

Problem solving involving rational equation

1. The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what number?

2. The reciprocal of the product of two consecutive intergers is 1/72

Answer 1

Question 1

1. The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what number?

Answer:

The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of
`(12)/(35)`

Solution:

1. The reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what number?

From given question,

`\text{ reciprocal of 5} = (1)/(5)`

`\text{ reciprocal of 7} = (1)/(7)`

Given that,

reciprocal of 5 + reciprocal of 7 = ?

`(1)/(5) + (1)/(7) = x`

On cross-multiplying we get,

`(1)/(5) + (1)/(7) = (7+5)/(5 * 7) = (12)/(35)`

Thus reciprocal is
`(35)/(12)`

So the reciprocal of 5 plus the reciprocal of 7 is the reciprocal of
`(12)/(35)`

Question 2

2. The reciprocal of the product of two consecutive integers is 1/72

Answer:

The value of two consecutive numbers are 8 and 9

Solution:

Let the two consecutive integers be x and x + 1

Given that reciprocal of product of two consecutive integers is
`(1)/(72)`

product of two consecutive integers = x(x + 1) =
`x^2 + x`

reciprocal of the product of two consecutive integers =
`(1)/(72)`

`(1)/(x^2 + x) = (1)/(72)\\\\x^2 + x = 72\\\\x^2 + x - 72 = 0`

Solve the above quadratic equation by grouping method

`x^2 + x - 72 = 0\\\\x^2 -8x + 9x - 72 = 0\\\\x^2 + 9x + (-8x - 72) = 0\\\\x(x + 9) -8(x + 9) = 0\\\\(x + 9)(x - 8) = 0`

Thus x = -9 or 8

Ignoring negative value,

x = 8

Thus two consecutive integers are x = 8 and x + 1 = 8 + 1 = 9

8 and 9 are two consecutive integers