Question

Problem solving involving rational equation

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 $\frac{12}{35}$

Solution:

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

From given question,

$\text{ reciprocal of 5} = \frac{1}{5}$

$\text{ reciprocal of 7} = \frac{1}{7}$

Given that,

reciprocal of 5 + reciprocal of 7 = ?

$\frac{1}{5} + \frac{1}{7} = x$

On cross-multiplying we get,

$\frac{1}{5} + \frac{1}{7} = \frac{7+5}{5 \times 7} = \frac{12}{35}$

Thus reciprocal is $\frac{35}{12}$

So the reciprocal of 5 plus the reciprocal of 7 is the reciprocal of $\frac{12}{35}$

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 $\frac{1}{72}$

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

reciprocal of the product of two consecutive integers = $\frac{1}{72}$

$\frac{1}{x^2 + x} = \frac{1}{72}\\\\x^2 + x = 72\\\\x^2 + x - 72 = 0$

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