Finding the Second Term in a Proportion

A proportion is a statement that two ratios are equal. It can be written as \(a:b :: c:d\) or \(\frac{a}{b} = \frac{c}{d}\). In a proportion, the terms are arranged in order: the first term, the second term, the third term, and the fourth term.

The given information provides the first, third, and fourth terms of a proportion:

• First term = 8
• Third term = 16
• Fourth term = 6

We need to find the second term. Let's represent the second term by the variable \(x\).

Setting up the Proportion Equation

Using the definition of a proportion, we can set up the relationship between the terms:

First term : Second term :: Third term : Fourth term

Substituting the given values and our variable \(x\), we get:

\(8 : x :: 16 : 6\)

This proportion can be written as an equation involving two equal ratios:

\(\frac{8}{x} = \frac{16}{6}\)

Solving for the Unknown Second Term

To find the value of \(x\) in the equation \(\frac{8}{x} = \frac{16}{6}\), we can use the property of proportions called cross-multiplication. This property states that the product of the means (the two inner terms, \(x\) and 16) is equal to the product of the extremes (the two outer terms, 8 and 6).

Applying cross-multiplication:

\(8 \times 6 = x \times 16\)

\(48 = 16x\)

Now, we need to isolate \(x\). We can do this by dividing both sides of the equation by 16:

\(x = \frac{48}{16}\)

Performing the division:

\(x = 3\)

So, the second term of the proportion is 3.

Verification

Let's check if the proportion holds true with the second term being 3:

\(8 : 3 :: 16 : 6\)

As ratios:

\(\frac{8}{3}\) and \(\frac{16}{6}\)

Simplify the second ratio \(\frac{16}{6}\) by dividing the numerator and denominator by their greatest common divisor, which is 2:

\(\frac{16 \div 2}{6 \div 2} = \frac{8}{3}\)

Since \(\frac{8}{3} = \frac{8}{3}\), the ratios are equal, and the proportion is correct. The second term is indeed 3.

The equation derived is \(\frac{8}{x} = \frac{16}{6}\), which simplifies to \(48 = 16x\), leading to \(x=3\).

Revision Table: Understanding Proportions

Additional Information: Types of Proportions and Solving Techniques

Besides finding missing terms, proportions are used in various real-world applications. Understanding the relationship between the quantities is key.

• Direct Proportion: Two quantities are directly proportional if an increase in one quantity causes a proportional increase in the other, and vice versa. The ratio between them is constant. Example: Distance traveled and time taken at a constant speed.
• Inverse Proportion: Two quantities are inversely proportional if an increase in one quantity causes a proportional decrease in the other, and vice versa. Their product is constant. Example: Speed of a vehicle and the time taken to cover a fixed distance.

Solving for an unknown term in a proportion \(\frac{a}{b} = \frac{c}{d}\) can always be done using cross-multiplication to get \(ad = bc\) and then solving for the unknown variable.

For example, if you need to find the fourth term \(d\), given \(a, b, c\):

\(ad = bc\)

\(d = \frac{bc}{a}\)

Similarly, for the second term \(b\), given \(a, c, d\):

\(bc = ad\)

\(b = \frac{ad}{c}\)

In this problem, we found the second term \(x\) using the formula \(x = \frac{ad}{c}\) where \(a=8\), \(c=16\), and \(d=6\), resulting in \(x = \frac{8 \times 6}{16} = \frac{48}{16} = 3\).