Question

A company manufactures only three types of vehicles: bikes, cars, and trucks. In 2021, the number of bikes manufactured was 40% of the number of bikes manufactured in 2022. The number of trucks manufactured in 2022 was 25% more than in 2021. In 2022, the number of trucks manufactured was the average of the number of bikes and cars manufactured that year. The company manufactured 10% fewer cars in 2021 as compared to 2022. The total number of vehicles produced in 2022 was 120,000. In 2021, the ratio of bikes and trucks manufactured was 1:4. How many cars were produced in 2022?

• A. 50000
• B. 54000
• C. 55000
• D. 60000
• E. 63000

Hint:

Work through systems of equations step by step, using relationships between variables to simplify calculations.

Answer 1

The Correct Option is D

Let the number of bikes manufactured in 2022 be \( b_2 \), the number of cars manufactured in 2022 be \( c_2 \), and the number of trucks manufactured in 2022 be \( t_2 \). From the problem, we can extract the following relationships: 1. In 2021, the number of bikes manufactured was 40% of the number of bikes manufactured in 2022: \[ b_1 = 0.4 \times b_2 \] 2. The number of trucks manufactured in 2022 was 25% more than in 2021: \[ t_2 = 1.25 \times t_1 \] 3. The total number of vehicles produced in 2022 is 120,000: \[ b_2 + c_2 + t_2 = 120,000 \] 4. The average number of bikes and cars manufactured in 2022 equals the number of trucks manufactured: \[ t_2 = \frac{b_2 + c_2}{2} \] 5. The company manufactured 10% fewer cars in 2021 compared to 2022: \[ c_1 = 0.9 \times c_2 \] 6. In 2021, the ratio of bikes and trucks manufactured was 1:4: \[ b_1 = \frac{1}{4} \times t_1 \] Now, we solve these equations step by step: Step 1: Use the total number of vehicles in 2022.
We know the total number of vehicles produced in 2022 is 120,000, so: \[ b_2 + c_2 + t_2 = 120,000 \] Step 2: Use the relation between bikes and trucks in 2021.
We have: \[ b_1 = \frac{1}{4} \times t_1 \] Substitute \( b_1 = 0.4 \times b_2 \), we have: \[ 0.4 \times b_2 = \frac{1}{4} \times t_1 \quad \Rightarrow \quad t_1 = 1.6 \times b_2 \] Step 3: Use the relation between bikes, cars, and trucks in 2022.
From the average relation, we know: \[ t_2 = \frac{b_2 + c_2}{2} \] Substitute this into the total vehicles equation: \[ b_2 + c_2 + \frac{b_2 + c_2}{2} = 120,000 \] Simplify: \[ \frac{3}{2} \times (b_2 + c_2) = 120,000 \] \[ b_2 + c_2 = 80,000 \] Now, substitute \( c_2 = 80,000 - b_2 \) into the relation for \( t_2 \): \[ t_2 = \frac{b_2 + (80,000 - b_2)}{2} = 40,000 \] Step 4: Solve for the number of cars.
We now have \( t_2 = 40,000 \). Using \( t_2 = 1.25 \times t_1 \), we can find \( t_1 \): \[ 40,000 = 1.25 \times t_1 \quad \Rightarrow \quad t_1 = 32,000 \] Now, using \( t_1 = 1.6 \times b_2 \): \[ 32,000 = 1.6 \times b_2 \quad \Rightarrow \quad b_2 = 20,000 \] Finally, substitute \( b_2 = 20,000 \) into \( b_2 + c_2 = 80,000 \) to find \( c_2 \): \[ 20,000 + c_2 = 80,000 \quad \Rightarrow \quad c_2 = 60,000 \] Step 5: Conclusion.
The number of cars produced in 2022 is \( \boxed{60,000} \).