Question:

If A and B are in the ratio 3 : 4, and B and C in the ratio 12 : 13, then A and C will be in the ratio

A. 3 : 13
B. 9 : 13
C. 36 : 13
D. 13 : 9

Core Concept

This question involves the concept of composite ratios, where two ratios share a common term (in this case, $B$ ), and we need to combine them into a single ratio involving $A$ and $C$ . The principle to solve this type of problem is based on finding a common value for the shared term and scaling the ratios to make the values of the common term equal.

Key Concepts:

Ratios: A ratio represents a relationship between two quantities, showing how many times one value is contained within another.
Composite Ratios: When two ratios share a common term, they can be combined by making the common term equal in both ratios.
Scaling the Ratios: To combine two ratios involving a common term, we must find the least common multiple (LCM) or adjust the terms so the common term is the same in both.

Step-by-Step Solution

Step 1: Write the given ratios

The given ratios are:

$A : B = 3 : 4$ $B : C = 12 : 13$

Step 2: Find a common value for $B$

We observe that $B$ is the common term between the two ratios. In the first ratio, $B = 4$ , and in the second ratio, $B = 12$ . To make the values of $B$ the same in both ratios, we need to scale the first ratio by multiplying both parts by 3 (since the LCM of 4 and 12 is 12).

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

For $A : B = 3 : 4$ , multiply both parts by 3 to get:
$A : B = 3 \times 3 : 4 \times 3 = 9 : 12$
For $B : C = 12 : 13$ , no multiplication is needed because $B = 12$ already.

Step 4: Combine the ratios

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

$A : B : C = 9 : 12 : 13$

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

$A : C = 9 : 13$

Correct Answer and Option

The correct answer is:
B. 9 : 13

Answer Reasoning

To solve this problem, we first adjusted the ratios so that the common term $B$ had the same value in both ratios. After scaling the first ratio by multiplying both parts by 3, we obtained $A : B = 9 : 12$ . Combining it with the second ratio $B : C = 12 : 13$ , we arrived at the final combined ratio $A : C = 9 : 13$ .

Explanation of Incorrect Options

A. 3 : 13: This is incorrect because the process of scaling and combining the ratios wasn’t followed correctly. After adjusting $B$ , the correct ratio of $A : C$ is 9 : 13, not 3 : 13.
C. 36 : 13: This is incorrect because it suggests that the value of $A$ is scaled by a factor of 4, which doesn’t align with the correct ratio calculation.
D. 13 : 9: This is incorrect because the order of the ratio is reversed. The correct ratio of $A : C$ is 9 : 13, not the reverse.

Common Mistakes to Avoid

Not scaling the ratios correctly: Students may forget to scale the first ratio properly, leading to an incorrect common value for $B$ in both ratios.
Reversing the ratio: Some students might mistakenly reverse the order of terms when combining the ratios, which leads to an incorrect result.

Simplest Way to Solve (Shortcut or Insight)

The simplest way to solve this problem is to first focus on the common term $B$ and scale the ratios accordingly. Once both ratios have the same value for $B$ , simply combine them and express the final ratio. This ensures that all steps are handled efficiently and leads directly to the correct answer.

Visual Aid Suggestion

Here’s a table that helps visualize the scaling and combination process:

| Step                     | A : B   | B : C   | Scaled A : B | Combined A : B : C |
|--------------------------|---------|---------|--------------|--------------------|
| Given Ratios             | 3 : 4   | 12 : 13 | 9 : 12       | 9 : 12 : 13        |
| Scaling Factor           | × 3     | × 1     |              |                    |
| Final Combined Ratio     |         |         |              | 9 : 13             |

Explanation:

Given Ratios: We start with the initial ratios $A : B = 3 : 4$ and $B : C = 12 : 13$ .
Scaling Factor: The first ratio is scaled by multiplying both terms by 3 so that $B = 12$ .
Final Combined Ratio: After the scaling step, we combine the ratios to get $A : B : C = 9 : 12 : 13$ , and the ratio of $A : C = 9 : 13$ .

This table visually breaks down the steps of the problem, making it easier to understand how the ratios are combined.

Application

The concept of composite ratios is often used in various applications such as investment distributions, mixture problems, and resource allocation. For instance, in a project where different departments are contributing resources at different rates (like time, money, or effort), understanding how to combine ratios helps ensure that resources are allocated efficiently. This is particularly useful in budgeting, production planning, and financial modeling, where different elements need to be balanced in proportion to each other.

Practice Questions

If A and B are in the ratio 5 : 6, and B and C are in the ratio 8 : 9, then A and C will be in the ratio:
- A. 5 : 9
- B. 40 : 54
- C. 25 : 54
- D. 30 : 45
  (Answer: B)
If A and B are in the ratio 2 : 3, and B and C are in the ratio 4 : 5, then A and C will be in the ratio:
- A. 8 : 15
- B. 6 : 15
- C. 8 : 10
- D. 6 : 10
  (Answer: A)
If the ratio of A : B is 4 : 5, and B : C is 5 : 6, then the ratio of A : C will be:
- A. 20 : 30
- B. 4 : 6
- C. 20 : 24
- D. 24 : 30
  (Answer: C)
If A and B are in the ratio 7 : 8, and B and C are in the ratio 9 : 10, then A and C will be in the ratio:
- A. 63 : 80
- B. 63 : 90
- C. 63 : 100
- D. 63 : 70
  (Answer: A)
If the ratio of A : B is 3 : 4, and B : C is 6 : 7, then the ratio of A : C is:
- A. 18 : 28
- B. 3 : 7
- C. 9 : 14
- D. 18 : 28
  (Answer: C)

Categories: Question and Answer