Finding the Third Proportional to 16 and 40

The question asks us to find the third proportional to the numbers 16 and 40. Understanding the concept of proportion is key to solving this problem.

In mathematics, when three numbers are in continuous proportion, it means the ratio of the first number to the second number is equal to the ratio of the second number to the third number. If the three numbers are \(a\), \(b\), and \(c\), they are in continuous proportion if:

Here, \(c\) is called the third proportional to \(a\) and \(b\).

Calculating the Third Proportional

In this problem, the given numbers are 16 and 40. We can consider 16 as the first term (\(a\)) and 40 as the second term (\(b\)). We need to find the third proportional, which we will call \(c\).

Using the formula for continuous proportion, we set up the equation:

To find the value of \(c\), we can cross-multiply:

Now, we solve for \(c\) by dividing both sides of the equation by 16:

So, the third proportional to 16 and 40 is 100.

Verifying the Proportion

Let's check if 16, 40, and 100 are in continuous proportion:

Ratio of the first to the second term: \(\frac{16}{40} = \frac{2 \times 8}{5 \times 8} = \frac{2}{5}\)

Ratio of the second to the third term: \(\frac{40}{100} = \frac{2 \times 20}{5 \times 20} = \frac{2}{5}\)

Since both ratios are equal (\(\frac{2}{5}\)), the numbers 16, 40, and 100 are indeed in continuous proportion, and 100 is the third proportional.

Final Answer Selection

Based on our calculation, the third proportional to 16 and 40 is 100. We look at the given options to find 100.

• 40
• 10
• 640
• 100

The value 100 matches the fourth option.

Revision Table: Understanding Proportion Concepts

Additional Information on Proportions

Proportions are a fundamental concept in mathematics and are used in various fields, including scaling recipes, map making, financial calculations, and physics problems. Understanding ratios and proportions helps in comparing quantities and solving problems involving relative sizes.

The property used to solve for an unknown term in a proportion is that the product of the means equals the product of the extremes. In the proportion \(a:b :: c:d\), \(b\) and \(c\) are the means, and \(a\) and \(d\) are the extremes. So, \(ad = bc\).

In the case of continuous proportion \(a:b :: b:c\), \(b\) and \(b\) are the means, and \(a\) and \(c\) are the extremes. So, \(a \times c = b \times b\), or \(ac = b^2\). This confirms that \(c = \frac{b^2}{a}\) is the formula for the third proportional when \(a\) and \(b\) are given, and \(b = \sqrt{ac}\) is the formula for the mean proportional between \(a\) and \(c\).

Using the formula \(c = \frac{b^2}{a}\) for our problem with \(a=16\) and \(b=40\):

This provides another way to calculate the third proportional and confirms our previous result.