Question

One number is four times another number. The sum of their reciprocals is 1/4 . What is the answer

Answer 1

Answer:

The required numbers are 5 and 20.

Step-by-step explanation:

Given : One number is four times another number.

Let the first number is 'x'

The reciprocal of the first number is $\frac{1}{x}$

and another number is '4x'

The reciprocal of the another number is $\frac{1}{4x}$

The sum of their reciprocals is $\frac{1}{4}$

i.e. $\frac{1}{x}+\frac{1}{4x}=\frac{1}{4}$

Solving the equation,

$\frac{4+1}{4x}=\frac{1}{4}$

$\frac{5}{4x}=\frac{1}{4}$

Cross multiply,

$5\times 4=1\times 4x$

$20=4x$

$x=\frac{20}{4}$

$x=5$

Another number is $4x=5\times 4=20$

Therefore, the required numbers are 5 and 20.

Answer 2

Using the information from the question, one can construct two equations:

$4 z = x$  ...  (1)

$\frac{1}{z} + \frac{1}{x}$   ... (2)

by substituting (1) into (2) to find x

$\frac{1}{z} + \frac{1}{4z} = \frac{1}{4}$

$\frac{5}{4z} = \frac{1}{4}$

\frac{5}{4z}  =   \frac{1}{4}  

$\frac{4z}{5} = \frac{4}{1}$

$\frac{4z}{1} = \frac{20}{1}$

$z = \frac{20}{4}$

⇒ z = 5

By substituting value of z into (1)

⇒ 4 (5) = x

⇒ x = 20

Thus the two numbers are 5 & 20