Finding the Fourth Proportional

The question asks us to find the fourth proportional to the numbers 50, 35, and 20.

When four numbers, say a, b, c, and d, are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. This relationship is expressed as:

a : b = c : d

This can also be written in fraction form as:

\(\frac{a}{b} = \frac{c}{d}\)

In this problem, we are given the first three numbers: a = 50, b = 35, and c = 20. We need to find the fourth proportional, which we will call d (or x).

So, we set up the proportion:

50 : 35 = 20 : x

Writing this in the fraction form gives us the equation:

\(\frac{50}{35} = \frac{20}{x}\)

To solve for x, we can cross-multiply:

\(50 \times x = 35 \times 20\)

\(50x = 700\)

Now, to find the value of x, we need to divide both sides of the equation by 50:

\(x = \frac{700}{50}\)

We can simplify the fraction:

\(x = \frac{70}{5}\)

Dividing 70 by 5 gives us:

\(x = 14\)

Therefore, the fourth proportional to 50, 35, and 20 is 14.

Let's verify the proportion with the value x = 14:

\(\frac{50}{35} = \frac{10 \times 5}{7 \times 5} = \frac{10}{7}\)

\(\frac{20}{14} = \frac{10 \times 2}{7 \times 2} = \frac{10}{7}\)

Since both ratios are equal to \(\frac{10}{7}\), the numbers 50, 35, 20, and 14 are indeed in proportion.

Looking at the options provided, the value 14 matches one of the options.