Question

One number is 4 times another number. The sum of their reciprocals is 5. What are the two numbers?

Answer 1

Answer:

The two number are:

$y=1,\:x=\frac{1}{4}$

Step-by-step explanation:

• Let 'x' be the first number

• let 'y' be the second number

Given that one number is 4 times another number, the first equation is

$y = 4x$

Next, the sum of their reciprocals is 5, so

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

Now, we have the system of equations

$\begin{bmatrix}y=4x\\ \frac{1}{x}+\frac{1}{y}=5\end{bmatrix}$

Let us solve the system of equations

$\begin{bmatrix}y=4x\\ \frac{1}{x}+\frac{1}{y}=5\end{bmatrix}$

substitute y = 4x in the equation $\frac{1}{x}+\frac{1}{y}=5$

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

Least Common Multiple of x, 4x:  4x

Adjust fractions based on L.C.M

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

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

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

Multiply both sides by 4x

$\frac{5}{4x}\cdot \:4x=5\cdot \:4x$

Simplify

$5=20x$

switch sides

$20x=5$

Divide both sides by 20

$\frac{20x}{20}=\frac{5}{20}$

Simplify

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

For y = 4x, substitute $x=\frac{1}{4}$

$y = 4x$

$y=4\cdot \frac{1}{4}$

$y=1$

Therefore, the two number are:

$y=1,\:x=\frac{1}{4}$

Verification:

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

substituting $y=1,\:x=\frac{1}{4}$

$\:\frac{1}{\frac{1}{4}}+\frac{1}{1}=\:5\:\:$

$4+1=5$

$5 = 5$

L.H.S = R.H.S

and

$y = 4x$

substituting $y=1,\:x=\frac{1}{4}$

$y\:=\:4\left(\frac{1}{4}\right)$

$y=1$