Finding the Third Proportional to 24 and 60

The question asks us to find the third proportional to the numbers 24 and 60. Understanding the concept of proportionality is key here.

When three numbers, say \(a\), \(b\), and \(c\), are in continued proportion, it means that the ratio of the first to the second is equal to the ratio of the second to the third. This can be written as:

\[\frac{a}{b} = \frac{b}{c}\]

In this continued proportion, \(a\) is the first proportional, \(b\) is the mean proportional, and \(c\) is the third proportional.

In our problem, the two given numbers are 24 and 60. If 24, 60, and the unknown third proportional (\(c\)) are in continued proportion, then:

• The first number is \(a = 24\).
• The second number (which is also the mean proportional in this context) is \(b = 60\).
• The third proportional is the number we need to find, let's call it \(c\).

So, we can set up the proportion as follows:

\[\frac{24}{60} = \frac{60}{c}\]

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

Applying cross-multiplication to our equation:

\[24 \times c = 60 \times 60\]

Now, we simplify the equation:

\[24c = 3600\]

To isolate \(c\), we divide both sides of the equation by 24:

\[c = \frac{3600}{24}\]

Let's perform the division:

\[c = \frac{3600}{24} = \frac{1800}{12} = \frac{900}{6} = \frac{450}{3} = 150\]

So, the third proportional to 24 and 60 is 150.

We can verify this by checking if 24, 60, and 150 are in continued proportion:

\[\frac{24}{60} = \frac{24 \div 12}{60 \div 12} = \frac{2}{5}\]

\[\frac{60}{150} = \frac{60 \div 30}{150 \div 30} = \frac{2}{5}\]

Since both ratios are equal to \(\frac{2}{5}\), the numbers 24, 60, and 150 are indeed in continued proportion, and 150 is the correct third proportional.

Revision Table: Key Concepts in Proportion

Additional Information: Exploring Proportionality

Proportionality is a fundamental concept in mathematics that describes how two quantities are related. Beyond continued proportion, there are other important types of proportionality, such as direct proportion and inverse proportion.

• Direct Proportion: Two quantities are in direct proportion if they increase or decrease at the same rate. If \(y\) is directly proportional to \(x\), we write \(y \propto x\), which means \(y = kx\) for some constant \(k\). For example, the cost of purchasing apples is directly proportional to the number of apples bought.
• Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity leads to a decrease in the other, and vice versa, such that their product is constant. If \(y\) is inversely proportional to \(x\), we write \(y \propto \frac{1}{x}\), which means \(y = \frac{k}{x}\) or \(xy = k\) for some constant \(k\). For example, the time taken to complete a job is inversely proportional to the number of workers, assuming they work at the same rate.

Understanding these different types of proportionality helps in solving a wide range of problems in various fields, including physics, chemistry, economics, and everyday life.