Understanding Third and Fourth Proportional Concepts

This problem requires us to first find two values, 'p' and 'q', based on the concepts of third proportional and fourth proportional, respectively. Then, we need to calculate the value of the expression (2p + q).

What is a Third Proportional?

The third proportional to two numbers, say 'a' and 'b', is a number 'c' such that the ratio of 'a' to 'b' is equal to the ratio of 'b' to 'c'. This is also known as continued proportion. Mathematically, this is written as:

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

Here, 'c' is the third proportional.

What is a Fourth Proportional?

The fourth proportional to three numbers, say 'a', 'b', and 'c', is a number 'd' such that the ratio of 'a' to 'b' is equal to the ratio of 'c' to 'd'. Mathematically, this is written as:

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

Here, 'd' is the fourth proportional.

Calculating the Third Proportional (p)

The question states that 'p' is the third proportional to 8 and 20. Using the formula for the third proportional \( \frac{a}{b} = \frac{b}{c} \), where a = 8, b = 20, and c = p, we get:

\( \frac{8}{20} = \frac{20}{p} \)

To solve for p, we can cross-multiply:

\( 8 \times p = 20 \times 20 \)

\( 8p = 400 \)

Now, divide both sides by 8:

\( p = \frac{400}{8} \)

\( p = 50 \)

So, the value of the third proportional, p, is 50.

Calculating the Fourth Proportional (q)

The question states that 'q' is the fourth proportional to 3, 5, and 24. Using the formula for the fourth proportional \( \frac{a}{b} = \frac{c}{d} \), where a = 3, b = 5, c = 24, and d = q, we get:

\( \frac{3}{5} = \frac{24}{q} \)

To solve for q, we cross-multiply:

\( 3 \times q = 5 \times 24 \)

\( 3q = 120 \)

Now, divide both sides by 3:

\( q = \frac{120}{3} \)

\( q = 40 \)

So, the value of the fourth proportional, q, is 40.

Final Calculation of (2p + q)

Now that we have the values of p and q, we can find the value of the expression (2p + q).

Substitute p = 50 and q = 40 into the expression:

\( 2p + q = 2(50) + 40 \)

\( 2p + q = 100 + 40 \)

\( 2p + q = 140 \)

Therefore, the value of (2p + q) is 140.

Revision Table: Proportionality Formulas

Additional Information on Ratio and Proportion

Ratio is a comparison of two quantities of the same kind by division. For example, the ratio of 8 to 20 is \( \frac{8}{20} \), which simplifies to \( \frac{2}{5} \) or 2:5.

Proportion is an equality of two ratios. When two ratios are equal, they are said to be in proportion. If \( \frac{a}{b} = \frac{c}{d} \), then a, b, c, and d are in proportion. Here:

• 'a' and 'd' are called the extreme terms.
• 'b' and 'c' are called the mean terms.

In a proportion, the product of the extremes is equal to the product of the means. That is, \( a \times d = b \times c \).

This property is what we used for cross-multiplication when solving for the unknown proportional terms.

Continued proportion occurs when the mean terms are the same, like in the case of the third proportional (a : b :: b : c).