Finding the Fourth Proportional

The question asks us to find the fourth proportional to the numbers 10, 12, and 15. Let's first understand what a proportion is and what the fourth proportional means.

Understanding Proportion and Fourth Proportional

A proportion is a statement that two ratios are equal. If four quantities, say a, b, c, and d, are in proportion, it is written as \(a : b :: c : d\). This means the ratio of the first two quantities is equal to the ratio of the last two quantities. Mathematically, this is expressed as:

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

In this proportion, 'a' is the first proportional, 'b' is the second proportional, 'c' is the third proportional, and 'd' is the fourth proportional.

When we are given three numbers and asked to find the fourth proportional, it means we need to find the value 'd' such that the given three numbers (in order) and 'd' are in proportion.

Calculating the Fourth Proportional

Given the three numbers 10, 12, and 15, let the fourth proportional be \(x\). According to the definition of proportion, these four numbers will be in proportion as follows:

\(10 : 12 :: 15 : x\)

This can be written in terms of ratios as:

\(\frac{10}{12} = \frac{15}{x}\)

To find the value of \(x\), we can use cross-multiplication. The product of the means (12 and 15) is equal to the product of the extremes (10 and \(x\)).

\(10 \times x = 12 \times 15\)

Now, we solve for \(x\):

\(10x = 180\)

Divide both sides of the equation by 10:

\(x = \frac{180}{10}\)

\(x = 18\)

So, the fourth proportional to 10, 12, and 15 is 18.

Step-by-Step Solution

1. Identify the given numbers as the first three terms of the proportion: a = 10, b = 12, c = 15.
2. Let the fourth proportional be \(x\).
3. Set up the proportion: \(a : b :: c : x\) or \(\frac{a}{b} = \frac{c}{x}\).
4. Substitute the given values: \(\frac{10}{12} = \frac{15}{x}\).
5. Cross-multiply: \(10 \times x = 12 \times 15\).
6. Simplify: \(10x = 180\).
7. Solve for \(x\): \(x = \frac{180}{10} = 18\).

Thus, the fourth proportional is 18.

Revision Table: Ratios and Proportion

Additional Information: Types of Proportion

Understanding the different types of proportion can be helpful:

• Direct Proportion: Two quantities are in direct proportion if an increase in one quantity leads to a proportional increase in the other quantity, and vice versa. For example, the cost of purchasing more items (at a fixed price per item). If quantity A is directly proportional to quantity B, then \(\frac{A}{B} = k\) (where k is a constant).
• Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity leads to a proportional decrease in the other quantity, and vice versa. For example, the speed of a vehicle and the time taken to cover a fixed distance. If quantity A is inversely proportional to quantity B, then \(A \times B = k\) (where k is a constant).
• Continued Proportion: Three quantities a, b, and c are in continued proportion if \(a : b :: b : c\). In this case, 'b' is called the mean proportional between 'a' and 'c'. Mathematically, \(\frac{a}{b} = \frac{b}{c}\) or \(b^2 = ac\).

The problem of finding the fourth proportional involves setting up a simple proportion between four terms.