Question

The difference of two numbers is 4.If the difference of their reciprocal is 4/21.Find the numbers.

Answer 1

Let x be one no. and y be the other.
Then according to the question, we get

y-x=4 ...(1) and,
1/x-1/y=4/21 ...(2)

Solving these equations, we get
x = 3 and y = 7

Answer 2

The numbers are -3, -7 or 7, 3

Step-by-step explanation:

Let the two numbers be x and y such that x>y (say)

Given that (x-y) = 4 (as x > y)

[If we consider y>x then y - x = 4]

Also, if x > y, $\frac{1}{x}$ will be less than $\frac{1}{y}$

Given that reciprocal difference is $\frac{4}{21}$

$\frac{1}{y}-\frac{1}{x}=\frac{4}{21}$

$\frac{x-y}{x y}=\frac{4}{21}(\text { Take } L C M)$

$\frac{4}{x y}=\frac{4}{21}$

$x y=21$

We know that,

$(a+b)^{2}=a^{2}+b^{2}+2 a b$

$(a-b)^{2}=a^{2}+b^{2}-2 a b$

$(a+b)^{2}-(a-b)^{2}=a^{2}+b^{2}+2 a b-a^{2}-b^{2}+2 a b=4 a b$

Therefore,

$(x+y)^{2}-(x+y)^{2}=4 x y=4(21)$

$(x+y)^{2}-16=84$

$(x+y)^{2}=100$

$(x+y)=10 \text { or }-10$

Thus, x+y=10, x-y=4

Adding both gives 2x=14 $\Rightarrow$ x=7; y=3

Consider x+y=-10; x-y=4

Adding both gives 2x = -6 $\Rightarrow$ x=-3; y=-7