If two times A is equal to three times of B and also equal to four times of C, then A : B : C is

Question:

If two times A is equal to three times of B and also equal to four times of C, then A : B : C is

A. 2 : 3 : 4
B. 3 : 4 : 2
C. 4 : 6 : 3
D. 6 : 4 : 3

Core Concept

This question tests the concept of proportions and ratios. The core idea is to find a relationship between A, B, and C based on the given conditions.

Key Concepts:

1. Proportions and Ratios: A ratio represents how two quantities are related. Proportions express the equality of two ratios.
2. Basic Algebra: You can use algebraic methods to express relationships between variables.
3. Solving System of Equations: We can use the fact that “two times A equals three times B and also equals four times C” to create a system of equations.

Formula:

• Given that $2A = 3B = 4C$, we can find a common variable for each quantity and express them as multiples of that variable.
• Let $k$ be the common multiplier, so we can express $A$, $B$, and $C$ in terms of $k$:

  $$A = \frac{3k}{2}, \quad B = \frac{2k}{3}, \quad C = \frac{k}{4}$$

Step-by-Step Solution

Step 1: Set up the relationships based on the given conditions

We are given that:

$$2A = 3B = 4C$$

This implies that $2A$, $3B$, and $4C$ all represent the same value, so let’s call this common value $k$.

Thus, we can write:

$$2A = k, \quad 3B = k, \quad 4C = k$$

Step 2: Solve for A, B, and C

Now, solve each equation for $A$, $B$, and $C$:

$$A = \frac{k}{2}, \quad B = \frac{k}{3}, \quad C = \frac{k}{4}$$

Step 3: Find the ratio of A : B : C

To find the ratio $A : B : C$, we need to express all quantities in terms of a common variable. Let’s find the least common multiple (LCM) of the denominators 2, 3, and 4, which is 12.

Now, multiply each term by 12 to eliminate the fractions:

$$A = \frac{k}{2} \times 12 = 6k, \quad B = \frac{k}{3} \times 12 = 4k, \quad C = \frac{k}{4} \times 12 = 3k$$

Thus, the ratio of $A : B : C$ is:

$$A : B : C = 6k : 4k : 3k$$

Simplifying this, we get:

$$A : B : C = 6 : 4 : 3$$

Correct Answer and Option

The correct answer is:
D. 6 : 4 : 3

Answer Reasoning

By setting the common value $k$ for $2A = 3B = 4C$, we derived the expressions for $A$, $B$, and $C$. Multiplying through by the least common multiple of the denominators allowed us to express the ratio in simple integer form, giving the final answer as $6 : 4 : 3$.

Explanation of Incorrect Options

• A. 2 : 3 : 4: This is incorrect because the ratio does not simplify to these values. The LCM of 2, 3, and 4 leads to the values 6, 4, and 3, not 2, 3, and 4.
• B. 3 : 4 : 2: This is incorrect because the correct relationship between the quantities does not yield this ratio.
• C. 4 : 6 : 3: This is incorrect as well because the numbers do not simplify to 4, 6, and 3. The order and values do not align with the correct answer.

Common Mistakes to Avoid

1. Incorrect interpretation of the ratio: Some students may mistakenly calculate the ratio based on incorrect algebraic steps or misinterpret the relationship between the quantities.
2. Failure to find the LCM: Forgetting to multiply by the LCM of the denominators can result in incorrect simplification and an incorrect ratio.

Simplest Way to Solve (Shortcut or Insight)

One shortcut is to recognize that the problem provides the relationships $2A = 3B = 4C$ directly, which immediately suggests that each variable (A, B, C) can be expressed as a fraction of a common multiplier $k$. By solving for $A$, $B$, and $C$, you can quickly simplify the ratio without the need for complex calculations.

Visual Aid Suggestion

A simple table to illustrate the relationships and final ratio:

| Variable | Expression     | Simplified Value |
|----------|----------------|------------------|
| A        | k/2            | 6k               |
| B        | k/3            | 4k               |
| C        | k/4            | 3k               |
| Ratio    | A : B : C      | 6 : 4 : 3        |

This table helps visualize how we derived the ratio step by step.

Application

The concept of ratios and proportions is widely used in various fields like finance, engineering, and business operations. For instance, if you need to distribute resources (like time, money, or effort) between different teams based on a set ratio, understanding how to calculate and adjust ratios accurately ensures that resources are allocated fairly and efficiently.

Practice Questions

1. If three times A is equal to five times B and also equal to seven times C, then A : B : C is:
   • A. 5 : 3 : 7
   • B. 3 : 5 : 7
   • C. 7 : 5 : 3
   • D. 7 : 3 : 5
     (Answer: A)
2. If 4A = 5B = 6C, then the ratio A : B : C is:
   • A. 5 : 4 : 6
   • B. 6 : 5 : 4
   • C. 4 : 5 : 6
   • D. 12 : 10 : 15
     (Answer: C)
3. If 2A = 3B = 6C, then the ratio A : B : C is:
   • A. 3 : 2 : 1
   • B. 6 : 9 : 12
   • C. 6 : 3 : 2
   • D. 1 : 2 : 3
     (Answer: A)
4. If 5A = 7B = 9C, then the ratio A : B : C is:
   • A. 7 : 5 : 9
   • B. 5 : 7 : 9
   • C. 9 : 5 : 7
   • D. 5 : 9 : 7
     (Answer: B)
5. If 2A = 3B = 4C, then the ratio A : B : C is:
   • A. 3 : 4 : 2
   • B. 4 : 3 : 2
   • C. 6 : 4 : 3
   • D. 2 : 3 : 4
     (Answer: C)