Find two consecutive natural numbers such that the difference of their reciprocals is 1/5 the reciprocal of the smaller number. (Enter your answers as a comma-separated list.)

Answer 1

(1/x) - (1/(x+1)) = (1/(5x))

Note: you know that x is the smaller number because

x < x + 1 for all values of x.

NOTE that x cannot equal 0 because you cannot divide by 0. Since x is a natural number it cannot be 0. Note also that x+1 cannot be 0, since it is also natural number.

Find a common denominator and combine the rational fractions on the left-hand side if the equation.

( ( x + 1) - x ) / ( x ( x + 1)) = 1/ (5x)

NOTE that this must be the order of subtraction, since otherwise we would get a negative number as a result, and that could not equal one-fifth of the reciprocal of a positive number.

1 / (x² + x ) = 1 / (5x)

Cross multiply:

5x = x² + x

Subtract 5x from both sides.

0 = x² -4x

0 = x(x-4)

So x=0 or x=4 are possible solutions. x cannot be 0 because it is a natural number and because we cannot use it in the equation above (division by 0).

If x=4, then (1/4) - (1/5) = (5/20) - (4/20) = 1/20

(1/5) × (1/4) = 1/20

AND thus 4 and 5 are the solution pair for this problem.

REFRESHER:

NATURAL NUMBERS are whole numbers (no fractions) greater than 1. In other words, 1, 2, 3, ...

COUNTING NUMBERS means the same thing as NATURAL NUMBERS. Let the 2 N's in CouNting Numbers remind you that they are they same as Natural Numbers.

WHOLE NUMBERS are natural numbers AND ZERO. In other words, 0, 1, 2, 3 ...

Let the O in WHOLE remind you that they include zero.

INTEGERS are positive and negative natural numbers plus zero. In other words, 0, ±1, ±2, ±3...

Answer 2

Let x = smaller number

1 / x - 1 / (x+1) = (1/5)(1/x)

1 / x - 1/(x+1) = 1/(5x)

5(x + 1) - 5x= x + 1

x = 4

The numbers are 4 and 5.