Question

The numerator of a rational number is less than its denominator by 3. If the

The numerator becomes three times greater, and the denominator is increased by 20. the new
number becomes Find the original number.

Answer 1

The original number is a fraction with a numerator 3 less than its denominator. After applying the given changes and solving the equations, the original number is found to be ⅔, which simplifies to ⅓.

Step-by-step explanation:

Let's define the original number to be a fraction where its numerator (which we'll call 'x') is 3 less than its denominator (which we'll call 'y'). Therefore, we can write the equation as x = y - 3. According to the given problem, if the numerator becomes three times greater, we get 3x, and the denominator increases by 20, we get y + 20.

The new fraction is said to be greater than 1, so we have the inequality:

\(\frac{3x}{y + 20} > 1\)

Substituting x with y - 3:

\(\frac{3(y - 3)}{y + 20} > 1\)

By cross-multiplying, we have:

3(y - 3) > y + 20

Simplifying the inequality we get:

3y - 9 > y + 20

2y > 29

y > 29/2

y > 14.5

Since y is the denominator of a fraction and typically, in the context of these problems, y should be a whole number, the smallest whole number greater than 14.5 is 15.

x would then be:

x = y - 3

x = 15 - 3

x = 12

The original number or fraction is \(\frac{12}{15}\), which simplifies to \(\frac{4}{5}\) when expressed in lowest terms.