Understanding Proportions and Solving for A

This problem involves understanding the concepts of fourth proportion and third proportion and then using the given information to find the value of A.

What is a Proportion?

A proportion is a statement that two ratios are equal. If four quantities a, b, c, and d are in proportion, we write it as $a:b :: c:d$ or $\frac{a}{b} = \frac{c}{d}$. Here, d is called the fourth proportion to a, b, and c.

What is a Third Proportion?

If three quantities a, b, and c are in continued proportion, we write it as $a:b :: b:c$ or $\frac{a}{b} = \frac{b}{c}$. Here, c is called the third proportion to a and b, and b is called the mean proportion between a and c.

Step-by-Step Solution

Step 1: Calculate the Fourth Proportion

We are given the numbers 12, 24, and 27, and we need to find their fourth proportion. Let the fourth proportion be $x$.

According to the definition of fourth proportion:

$\frac{12}{24} = \frac{27}{x}$

Now, we solve for $x$:

$12 \times x = 24 \times 27$

$x = \frac{24 \times 27}{12}$

$x = 2 \times 27$

$x = 54$

So, the fourth proportion to 12, 24, and 27 is 54.

Step 2: Calculate the Third Proportion

We are told that the fourth proportion calculated in Step 1 (which is 54) is the same as the third proportion to A and 36. Let the third proportion be $y$. In this case, $y = 54$.

According to the definition of third proportion, for quantities A, 36, and $y$ in continued proportion:

$\frac{A}{36} = \frac{36}{y}$

We know $y = 54$, so we substitute this value:

$\frac{A}{36} = \frac{36}{54}$

Step 3: Solve for A

Now, we solve the equation from Step 2 for A:

$A = \frac{36 \times 36}{54}$

We can simplify this expression:

$A = \frac{36 \times 36}{9 \times 6}$

$A = \frac{4 \times 9 \times 6 \times 6}{9 \times 6}$

Cancel out the common terms (9 and 6):

$A = 4 \times 6$

$A = 24$

The value of A is 24.

Conclusion

The value of A is 24.

Revision Table: Key Proportion Concepts

Additional Information on Proportions

Proportions are a fundamental concept in mathematics used to compare quantities. They are widely applied in various fields like geometry (similar figures), physics, chemistry, and in everyday life (like scaling recipes or maps).

• Direct Proportion: Two quantities are directly proportional if an increase in one quantity leads to a proportional increase in the other, and vice-versa. Their ratio is constant ($y = kx$).
• Inverse Proportion: Two quantities are inversely proportional if an increase in one quantity leads to a proportional decrease in the other, and vice-versa. Their product is constant ($xy = k$).
• Continued Proportion: When three or more quantities are such that the ratio of the first to the second is equal to the ratio of the second to the third, and so on, they are said to be in continued proportion. For a, b, c in continued proportion, $\frac{a}{b} = \frac{b}{c}$.