of A = 75% of B = 0.6 of C, then A : B : C is

Question:

If $\frac{2}{3}$ of A = 75% of B = 0.6 of C, then A : B : C is

A. 2 : 3 : 3
B. 3 : 4 : 5
C. 4 : 5 : 6
D. 9 : 8 : 10

Core Concept

This question tests the concept of Proportions and Ratios, specifically when there are equal relationships between parts of the quantities involved. The task is to express the three variables $A$, $B$, and $C$ in the form of a single ratio.

Key Concepts:

1. Proportions: Proportions are equations that express the equality between two ratios. Here, the equality between parts of $A$, $B$, and $C$ is given.
2. Ratios: A ratio is a way to compare two or more quantities. The goal is to combine the relationships of $A$, $B$, and $C$ into a single ratio.

Formula:

The given relationships are:

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

To solve this, express each quantity in terms of a common variable.

Step-by-Step Solution

Step 1: Express the relationships in terms of a common variable

The equation provided is:

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

Let this common value be denoted by $k$. Therefore, we have the following equations:

$$\frac{2}{3}A = k, \quad 75\%B = k, \quad 0.6C = k$$

Step 2: Solve for $A$, $B$, and $C$

From these equations, we can express $A$, $B$, and $C$ in terms of $k$.

1. For $A$:

$$\frac{2}{3}A = k \quad \Rightarrow \quad A = \frac{3}{2}k$$

2. For $B$:

$$75\%B = k \quad \Rightarrow \quad B = \frac{k}{0.75} = \frac{4}{3}k$$

3. For $C$:

$$0.6C = k \quad \Rightarrow \quad C = \frac{k}{0.6} = \frac{5}{3}k$$

Step 3: Express the ratio of $A : B : C$

Now, we can express the ratio $A : B : C$ as follows:

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

Since $k$ is common in all terms, it can be cancelled out:

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

Step 4: Eliminate the fractions

To eliminate the fractions, find the least common multiple (LCM) of the denominators 2 and 3, which is 6. Multiply each term by 6:

$$A : B : C = 6 \times \frac{3}{2} : 6 \times \frac{4}{3} : 6 \times \frac{5}{3}$$ $$A : B : C = 9 : 8 : 10$$

Correct Answer and Option

The correct answer is:
D. 9 : 8 : 10

Answer Reasoning

By expressing each variable in terms of a common factor $k$, we were able to combine the given proportional relationships into a single ratio. After simplifying, we found that the ratio of $A : B : C$ is $9 : 8 : 10$.

Explanation of Incorrect Options

• A. 2 : 3 : 3: This option is incorrect because it does not correctly simplify the relationships between $A$, $B$, and $C$.
• B. 3 : 4 : 5: This is incorrect because it incorrectly assumes a proportional relationship between the variables that doesn’t align with the given conditions.
• C. 4 : 5 : 6: This is incorrect because, after correctly applying the relationships and simplifying, the correct ratio is $9 : 8 : 10$, not $4 : 5 : 6$.

Common Mistakes to Avoid

1. Incorrectly simplifying the fractions: Ensure the ratios are simplified by multiplying all terms by the LCM to eliminate fractions.
2. Misinterpreting the percentages: Convert the percentages (like 75%) properly into fractional form (i.e., 75% = 0.75) before applying the formula.

Simplest Way to Solve (Shortcut or Insight)

The most efficient approach is to express each of $A$, $B$, and $C$ in terms of a common variable $k$, solve for each variable, and then combine the ratios. This method avoids complex fraction manipulation and directly leads to the correct ratio.

Visual Aid Suggestion

Here’s a simple table to visualize the process of combining the ratios:

| Step                           | A              | B                | C                | Ratio of A : B : C |
|--------------------------------|----------------|------------------|------------------|--------------------|
| Given Relation                 | 2/3A = k       | 75% of B = k     | 0.6 of C = k     |                    |
| Express in terms of k          | 3/2k           | 4/3k             | 5/3k             |                    |
| Simplified Ratio               | 3/2 : 4/3 : 5/3|                  |                  | 9 : 8 : 10         |

Explanation:

• Step 1: The table shows how the relations are given.
• Step 2: Each variable is expressed in terms of $k$.
• Step 3: The simplified ratio is calculated by eliminating fractions and finding the LCM of the denominators.

This table clearly illustrates how the ratio of $A : B : C$ is derived by expressing each term in terms of a common variable and simplifying the relationship.

Application

The concept of ratios and proportions is widely applied in business, finance, and resource management. For example, when calculating profit sharing, companies often use ratios to distribute earnings among different partners. Understanding how to combine ratios effectively ensures that the resources are allocated fairly based on the pre-established conditions.

Practice Questions

1. If $\frac{1}{2}$ of A = 60% of B = 0.5 of C, then A : B : C is:
   • A. 2 : 3 : 4
   • B. 3 : 4 : 5
   • C. 4 : 5 : 6
   • D. 9 : 8 : 10
     (Answer: B)
2. If $\frac{3}{4}$ of A = 50% of B = 0.75 of C, then A : B : C is:
   • A. 3 : 4 : 5
   • B. 4 : 5 : 6
   • C. 9 : 8 : 10
   • D. 5 : 6 : 7
     (Answer: A)
3. If $\frac{1}{3}$ of A = 80% of B = 0.5 of C, then A : B : C is:
   • A. 3 : 5 : 6
   • B. 4 : 6 : 8
   • C. 9 : 8 : 7
   • D. 5 : 7 : 9
     (Answer: A)
4. If 30% of A = 50% of B = 0.4 of C, then A : B : C is:
   • A. 3 : 5 : 6
   • B. 2 : 3 : 4
   • C. 4 : 6 : 5
   • D. 5 : 7 : 9
     (Answer: C)
5. If 2/5 of A = 40% of B = 0.6 of C, then A : B : C is:
   • A. 3 : 5 : 7
   • B. 2 : 3 : 4
   • C. 9 : 8 : 10
   • D. 6 : 4 : 5
     (Answer: B)