Question

The cost of 12 fans and 21 tables is 30000. What is the cost of 8 fans and 14 tables?

• A. 10000
• B. 15000
• C. 18000
• D. 20000

Hint:

In problems like this, always check for a proportional relationship or a common factor between the given quantities and the required quantities before trying to solve for individual prices. It's often designed to be solved this way, saving a lot of time.

Answer 1

The Correct Option is D

Step 1: Understanding the Problem
We are given the total cost of a certain number of fans and tables and asked to find the cost of a different quantity of the same items. This is a problem of proportionality.

Step 2: Key Formula or Approach
Let F be the cost of one fan and T be the cost of one table. We are given: \[ 12F + 21T = 30000 \] We need to find the value of: \[ 8F + 14T \] Notice that the coefficients in both expressions might be related by a common factor.

Step 3: Detailed Explanation
1. Analyze the relationship between the equations:
Let's look at the coefficients of F and T in both expressions.
Given: 12 fans and 21 tables. To find: 8 fans and 14 tables.
Let's see if we can find a ratio. Ratio of fans = \( \frac{8}{12} = \frac{2}{3} \)
Ratio of tables = \( \frac{14}{21} = \frac{2}{3} \)
Since the ratio is the same for both items, the cost will also be in the same proportion.
2. Calculate the required cost:
The cost of 8 fans and 14 tables will be \( \frac{2}{3} \) of the cost of 12 fans and 21 tables. \[ \text{Required Cost} = \frac{2}{3} \times (12F + 21T) \] \[ \text{Required Cost} = \frac{2}{3} \times 30000 \] \[ \text{Required Cost} = 2 \times 10000 = 20000 \] Alternative Method (Factoring):
From the given equation: \( 12F + 21T = 30000 \). We can factor out a common factor of 3. \[ 3(4F + 7T) = 30000 \] Divide by 3 to find the cost of 4 fans and 7 tables. \[ 4F + 7T = \frac{30000}{3} = 10000 \] We need to find the cost of 8 fans and 14 tables, which is \( 8F + 14T \). We can factor out a 2 from this expression: \[ 8F + 14T = 2(4F + 7T) \] Since we know that \( 4F + 7T = 10000 \), we can substitute this value. \[ \text{Required Cost} = 2 \times 10000 = 20000 \]
Step 4: Final Answer
The cost of 8 fans and 14 tables is 20,000. Therefore, option (D) is the correct answer.