Understanding Continued Proportion

The question deals with the concept of continued proportion. Three quantities, say a, b, and c, are said to be in continued proportion if the ratio of the first to the second is equal to the ratio of the second to the third. Mathematically, this is expressed as a : b :: b : c.

This proportion can be written as fractions:

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

By cross-multiplication, we get the relationship:

\(ac = b^2\)

In this relationship, 'b' is called the mean proportional between 'a' and 'c'.

Applying the Continued Proportion Formula

We are given that a, b, and c are in continued proportion, and the value of the mean proportional, b, is 96.

So, we have the relation:

\(ac = b^2\)

Substitute the given value of b = 96:

\(ac = 96^2\)

Let's calculate \(96^2\):

\(96^2 = 96 \times 96 = 9216\)

So, the condition for the pair (a, c) is that their product must be equal to 9216. We need to check the given options to find which pair of (a, c) satisfies the condition \(a \times c = 9216\).

Evaluating the Options

Let's test each option by multiplying the given values of 'a' and 'c'.

Conclusion

From the evaluation, only Option 2, where \(a = 32\) and \(c = 288\), satisfies the condition \(a \times c = 9216\). Therefore, the possible pair of values for a and c is 32 and 288.

In summary, for three quantities to be in continued proportion \(a : b :: b : c\), the square of the middle term (the mean proportional) must be equal to the product of the first and third terms (\(b^2 = ac\)). Given \(b = 96\), we required \(ac = 96^2 = 9216\), and the pair \(a = 32, c = 288\) is the only option where the product \(32 \times 288\) equals 9216.

Revision Table: Key Concepts

Additional Information: Properties of Proportion

Understanding the properties of proportion can be helpful when solving such problems.

• Product of Extremes and Means: In any proportion \(a : b :: c : d\), the product of the extremes (a and d) is equal to the product of the means (b and c). So, \(ad = bc\). In continued proportion \(a : b :: b : c\), the extremes are a and c, and the means are both b. Thus, \(ac = b \times b = b^2\).
• Invertendo: If \(a : b = c : d\), then \(b : a = d : c\).
• Alternendo: If \(a : b = c : d\), then \(a : c = b : d\).
• Componendo: If \(a : b = c : d\), then \((a+b) : b = (c+d) : d\).
• Dividendo: If \(a : b = c : d\), then \((a-b) : b = (c-d) : d\).
• Componendo and Dividendo: If \(a : b = c : d\), then \((a+b) : (a-b) = (c+d) : (c-d)\).

These properties are derived from basic algebraic manipulation of the fractional form of proportion and are useful tools in solving more complex problems involving ratios and proportions, including continued proportion scenarios.