A natural number, $N$ , is $4$ less than another natural number, $M$ . The sum of the reciprocals of $M$ and $N$ is $5$ times the reciprocal of twice the value of $M$ . What are the two numbers?

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Q. A natural number,

N

, is

4

less than another natural number,

M

. The sum of the reciprocals of

M

and

N

5

times the reciprocal of twice the value of

M

. What are the two numbers?

Denote and Express: Let's denote the natural number $M$ and express $N$ in terms of $M$ . $N = M - 4$
Write Sum Equation: Now, let's write the equation for the sum of the reciprocals of $M$ and $N$ being $5$ times the reciprocal of twice the value of $M$ . $\newline$ $\frac{1}{M} + \frac{1}{N} = \frac{5}{2M}$
Substitute and Simplify: Substitute $N$ with $M - 4$ in the equation. $\newline$ $\frac{1}{M} + \frac{1}{M - 4} = \frac{5}{2M}$
Find Common Denominator: Find a common denominator for the left side of the equation. $\newline$ $\frac{(M - 4) + M}{M(M - 4)} = \frac{5}{2M}$
Clear Denominators: Simplify the numerator on the left side of the equation. $\newline$ $(2M - 4) / [M(M - 4)] = 5/(2M)$
Expand Equation: Multiply both sides of the equation by $2M(M - 4)$ to clear the denominators. $\newline$ $2M(2M - 4) = 5M(M - 4)$
Set Equation to Zero: Expand both sides of the equation. $\newline$ $4M^2 - 8M = 5M^2 - 20M$
Combine Like Terms: Move all terms to one side to set the equation to zero. $\newline$ $5M^2 - 4M^2 - 20M + 8M = 0$
Factor Out M: Combine like terms. $M^2 - 12M = 0$
Solve for $M$ : Factor out $M$ from the equation. $M(M - 12) = 0$
Find N: Set each factor equal to zero and solve for M. $\newline$ $M = 0$ or $M - 12 = 0$
Find N: Set each factor equal to zero and solve for M. $\newline$ $M = 0$ or $M - 12 = 0$ Since $M$ is a natural number, $M$ cannot be $0$ . So we solve for the second factor. $\newline$ $M = 12$
Find N: Set each factor equal to zero and solve for M. $\newline$ $M = 0$ or $M - 12 = 0$ Since $M$ is a natural number, $M$ cannot be $0$ . So we solve for the second factor. $\newline$ $M = 12$ Now that we have the value of $M$ , we can find $N$ . $\newline$ $N = M - 4$ $\newline$ $N = 12 - 4$ $\newline$ $M - 12 = 0$ $0$