Understanding the Ratio Problem

This problem involves finding the ratio of three quantities, A, C, and B, given a relationship between them expressed as an equality of fractions, percentages, and decimals. The key is to convert all parts of the equation into a consistent format, like fractions, and then determine the relationship between A, B, and C.

Step-by-Step Solution

The given relationship is:

$\frac{2}{3} \text{ of } A = 75\% \text{ of } B = 0.6 \text{ of } C$

Converting to Fractions

First, let's convert the percentage and decimal to fractions:

• $75\% = \frac{75}{100} = \frac{3 \times 25}{4 \times 25} = \frac{3}{4}$
• $0.6 = \frac{6}{10} = \frac{3 \times 2}{5 \times 2} = \frac{3}{5}$

Now, substitute these back into the original equation:

$\frac{2}{3} A = \frac{3}{4} B = \frac{3}{5} C$

Finding the Relationship Between A, B, and C

Let's set each part of the equation equal to a constant value, say $k$. This makes it easier to express A, B, and C in terms of $k$:

• $\frac{2}{3} A = k$
• $\frac{3}{4} B = k$
• $\frac{3}{5} C = k$

Now, solve for A, B, and C in terms of $k$:

• From $\frac{2}{3} A = k$, multiply both sides by $\frac{3}{2}$: $A = k \times \frac{3}{2} = \frac{3}{2}k$
• From $\frac{3}{4} B = k$, multiply both sides by $\frac{4}{3}$: $B = k \times \frac{4}{3} = \frac{4}{3}k$
• From $\frac{3}{5} C = k$, multiply both sides by $\frac{5}{3}$: $C = k \times \frac{5}{3} = \frac{5}{3}k$

Determining the Ratio A:B:C

The ratio $A:B:C$ is the ratio of their expressions in terms of $k$:

$A:B:C = \frac{3}{2}k : \frac{4}{3}k : \frac{5}{3}k$

We can divide each term by $k$ (assuming $k \neq 0$):

$A:B:C = \frac{3}{2} : \frac{4}{3} : \frac{5}{3}$

To get rid of the fractions, we multiply each term by the Least Common Multiple (LCM) of the denominators (2, 3, 3). The LCM of 2, 3, and 3 is 6.

Multiply each part of the ratio by 6:

$A:B:C = \left(\frac{3}{2} \times 6\right) : \left(\frac{4}{3} \times 6\right) : \left(\frac{5}{3} \times 6\right)$

$A:B:C = 9 : 8 : 10$

Finding the Ratio A:C:B

The question asks for the ratio $A:C:B$. We found that $A:B:C = 9:8:10$. This means:

• A is proportional to 9
• B is proportional to 8
• C is proportional to 10

Therefore, the ratio $A:C:B$ is obtained by arranging the corresponding values in the requested order:

$A:C:B = 9:10:8$

Summary of Calculations

Ratio Calculation Revision Table

Additional Information on Ratio and Proportion

Ratio and proportion are fundamental concepts in mathematics used to compare quantities. A ratio $a:b$ can be written as a fraction $\frac{a}{b}$. A proportion is an equality between two ratios, e.g., $\frac{a}{b} = \frac{c}{d}$.

In this problem, we used the idea of proportionality. If $\frac{2}{3} A = \frac{3}{4} B = \frac{3}{5} C$, it means that A, B, and C are in certain proportions relative to each other. By setting each part equal to a constant $k$, we expressed A, B, and C as multiples of $k$. This method is useful for finding the ratio when quantities are related by such complex equations.

Understanding how to convert between fractions, decimals, and percentages is crucial for solving many quantitative problems. Each format represents a part of a whole and can be interchanged as needed for calculation.