Understanding Fourth Proportional Calculations

The question asks for the fourth proportional to the numbers 7, 15, and 21. When four numbers are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. If the four numbers are \(a, b, c, \text{ and } d\), they are in proportion if:

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

In this case, we are given the first three numbers: \(a = 7\), \(b = 15\), and \(c = 21\). We need to find the fourth proportional, which we will call \(d\) (or \(x\)). So, we can set up the proportion as:

\(\frac{7}{15} = \frac{21}{x}\)

Calculating the Fourth Proportional Step-by-Step

To find the value of \(x\), we can use cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction and setting it equal to the product of the denominator of the first fraction and the numerator of the second fraction.

From the proportion \(\frac{7}{15} = \frac{21}{x}\), cross-multiplication gives us:

\(7 \times x = 15 \times 21\)

Now, we need to solve for \(x\):

• Calculate the product on the right side: \(15 \times 21\).
• \(15 \times 21 = 15 \times (20 + 1) = 15 \times 20 + 15 \times 1 = 300 + 15 = 315\).
• So, the equation becomes: \(7x = 315\).
• To find \(x\), divide both sides of the equation by 7: \(x = \frac{315}{7}\).
• Perform the division: \(315 \div 7\).
• \(315 \div 7 = 45\).

Therefore, the fourth proportional to 7, 15, and 21 is 45.

Revision Table: Understanding Proportions

Additional Information: Other Types of Proportionals

Besides the fourth proportional, there are other related concepts:

• Third Proportional: This applies when you have only two numbers, say \(a\) and \(b\). The third proportional, \(c\), satisfies the proportion \(a:b = b:c\). The middle term \(b\) is repeated. So, \(\frac{a}{b} = \frac{b}{c}\), which means \(ac = b^2\), and \(c = \frac{b^2}{a}\). For example, the third proportional to 3 and 6 is \(c = \frac{6^2}{3} = \frac{36}{3} = 12\). The proportion is \(3:6 = 6:12\), which is \(\frac{3}{6} = \frac{1}{2}\) and \(\frac{6}{12} = \frac{1}{2}\).
• Mean Proportional (or Geometric Mean): This also applies to two numbers, say \(a\) and \(c\). The mean proportional, \(b\), is the middle term that satisfies the proportion \(a:b = b:c\). So, \(\frac{a}{b} = \frac{b}{c}\), which means \(b^2 = ac\), and \(b = \sqrt{ac}\) (taking the positive root for positive numbers). For example, the mean proportional between 4 and 9 is \(b = \sqrt{4 \times 9} = \sqrt{36} = 6\). The proportion is \(4:6 = 6:9\), which is \(\frac{4}{6} = \frac{2}{3}\) and \(\frac{6}{9} = \frac{2}{3}\).

Understanding these different types helps in solving various problems involving ratios and proportions.