If A : B = 2 : 3 and B : C = 4 : 5 then A : B : C is

Question:

If A : B = 2 : 3 and B : C = 4 : 5 then A : B : C is

A. 2 : 3 : 5
B. 5 : 4 : 6
C. 8 : 12 : 15
D. 6 : 4 : 5

Core Concept

This question is based on the concept of composite ratios. Composite ratios are used when there are two or more given ratios involving common terms. To find a combined ratio involving all the terms (A, B, and C), you must express the ratios in a way that the common terms are the same.

Key Concepts:

1. Ratios: A ratio represents a relationship between two quantities, showing how many times one value contains or is contained by another.
2. Composite Ratio: When two ratios share a common term, they can be combined to form a single ratio involving all the terms.
3. Common Multiple: To combine two ratios involving a common term, you need to find a common multiple for that term.

Step-by-Step Solution

Step 1: Write the given ratios

We are given two ratios:

$$A : B = 2 : 3$$ $$B : C = 4 : 5$$

Step 2: Find a common value for $B$

Since $B$ is the common term in both ratios, we need to make sure the value of $B$ is the same in both ratios. In the first ratio, $B = 3$, and in the second ratio, $B = 4$. To make them the same, we find the least common multiple (LCM) of 3 and 4, which is 12.

Step 3: Scale the ratios to have the same value for $B$

To make $B = 12$ in both ratios:

• For $A : B = 2 : 3$, multiply both terms by 4:

  $$A : B = 2 \times 4 : 3 \times 4 = 8 : 12$$

• For $B : C = 4 : 5$, multiply both terms by 3:

  $$B : C = 4 \times 3 : 5 \times 3 = 12 : 15$$

Step 4: Combine the ratios

Now that both ratios are expressed with the same value for $B$, we can combine them:

$$A : B : C = 8 : 12 : 15$$

Correct Answer and Option

The correct answer is:
C. 8 : 12 : 15

Answer Reasoning

To solve the problem, we first recognize that the common term between the two ratios is $B$. By finding the least common multiple (LCM) of 3 and 4, we made the values of $B$ the same in both ratios. Once both ratios had the same value for $B$, we simply combined them to form the final ratio of $A : B : C = 8 : 12 : 15$.

Explanation of Incorrect Options

• A. 2 : 3 : 5: This is incorrect because the ratios given in the question cannot simplify to this combination. The ratios must first be adjusted to have a common value for $B$, which leads to a different set of numbers.
• B. 5 : 4 : 6: This is incorrect because it does not align with the values of $A$, $B$, and $C$ once the ratios are combined correctly.
• D. 6 : 4 : 5: This is incorrect because it represents an incorrect simplification or misunderstanding of the process of combining the given ratios.

Common Mistakes to Avoid

1. Ignoring the need to make $B$ the same: A common mistake is to directly combine the ratios without adjusting for the common term $B$, which can lead to incorrect results.
2. Simplifying ratios too early: Some students may simplify the ratios before making sure that the common term has been adjusted. This leads to confusion when the final result does not match the expected answer.

Simplest Way to Solve (Shortcut or Insight)

The key to solving this problem quickly is to focus on making the common term (in this case, $B$) the same in both ratios. Multiply the terms of each ratio by appropriate factors to achieve this, then combine the ratios directly. Using the least common multiple (LCM) of the common term will always guide you to the correct combined ratio.

Visual Aid Suggestion

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

| Step                          | A  : B  | B  : C  | Combined A : B : C |
|-------------------------------|---------|---------|--------------------|
| Original Ratios               | 2 : 3   | 4 : 5   |                    |
| Scaled to make B the same     | 8 : 12  | 12 : 15 | 8 : 12 : 15        |

This table clearly shows the step-by-step adjustment of the ratios and their final combined form.

Application

The concept of composite ratios is applied in many practical situations, such as business, where products might be mixed in certain proportions, or engineering, where multiple materials are combined according to specific ratios. Understanding how to combine ratios ensures that you can accurately measure, mix, or allocate resources in various contexts, whether in chemistry, construction, or even financial budgeting.

Practice Questions

1. If A : B = 5 : 7 and B : C = 6 : 9, then A : B : C is:
   • A. 5 : 6 : 9
   • B. 10 : 14 : 18
   • C. 15 : 21 : 27
   • D. 25 : 35 : 45
     (Answer: B)
2. If A : B = 3 : 4 and B : C = 5 : 6, then A : B : C is:
   • A. 15 : 20 : 24
   • B. 10 : 12 : 15
   • C. 6 : 8 : 10
   • D. 9 : 12 : 15
     (Answer: B)
3. If A : B = 8 : 12 and B : C = 10 : 15, then A : B : C is:
   • A. 8 : 12 : 18
   • B. 16 : 24 : 30
   • C. 12 : 16 : 20
   • D. 24 : 36 : 45
     (Answer: B)
4. If A : B = 2 : 5 and B : C = 4 : 7, then A : B : C is:
   • A. 4 : 10 : 7
   • B. 6 : 15 : 7
   • C. 8 : 20 : 7
   • D. 10 : 25 : 7
     (Answer: C)
5. If A : B = 3 : 5 and B : C = 7 : 11, then A : B : C is:
   • A. 21 : 35 : 55
   • B. 9 : 15 : 22
   • C. 6 : 10 : 15
   • D. 14 : 25 : 33
     (Answer: B)