Understanding Proportions: Solving for the Unknown

The question asks us to find the value of \(x\) in the given proportion: 14 ∶ 30 ∶∶ 7 ∶ x.

A proportion is a statement that two ratios are equal. The notation \(a : b :: c : d\) means that the ratio \(a : b\) is equal to the ratio \(c : d\). This can be written as a fraction:

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

In our given proportion, 14 ∶ 30 ∶∶ 7 ∶ x, we have \(a = 14\), \(b = 30\), \(c = 7\), and \(d = x\). So, we can write the proportion as:

\[\frac{14}{30} = \frac{7}{x}\]

Solving the Proportion to Find the Value of x

To find the value of \(x\), we can use the property of proportions that the product of the means is equal to 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. Thus, \(a \times d = b \times c\).

Applying this to our equation \(\frac{14}{30} = \frac{7}{x}\), we cross-multiply:

\[14 \times x = 30 \times 7\]

Now, we simplify the right side of the equation:

\[14x = 210\]

To isolate \(x\), we divide both sides of the equation by 14:

\[x = \frac{210}{14}\]

Performing the division:

\[x = 15\]

So, the value of \(x\) that satisfies the proportion 14 ∶ 30 ∶∶ 7 ∶ x is 15.

Verification

We can check our answer by substituting \(x = 15\) back into the original proportion:

\[\frac{14}{30} = \frac{7}{15}\]

We can simplify the left side fraction by dividing the numerator and denominator by their greatest common divisor, which is 2:

\[\frac{14 \div 2}{30 \div 2} = \frac{7}{15}\]

\[\frac{7}{15} = \frac{7}{15}\]

Since both sides of the equation are equal, our value of \(x = 15\) is correct.

Revision Table: Key Concepts of Ratio and Proportion

Additional Information: Types of Proportions

While the problem deals with a direct proportion, it's useful to know about different types:

• Direct Proportion: Two quantities are in direct proportion if an increase in one quantity leads to a proportional increase in the other, and a decrease in one leads to a proportional decrease in the other. Their ratio is constant. For example, if you buy more apples, the total cost increases proportionally. \(y \propto x\) or \(y = kx\).
• Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity leads to a proportional decrease in the other, and vice versa. Their product is constant. For example, as the speed of a car increases, the time taken to cover a fixed distance decreases proportionally. \(y \propto \frac{1}{x}\) or \(xy = k\).

The problem 14 ∶ 30 ∶∶ 7 ∶ x represents a direct proportion between the first pair of numbers (14, 30) and the second pair (7, x). We solved it by setting the ratios equal and using cross-multiplication.