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 \( b_2 \) be the number of bikes, \( c_2 \) be the number of cars, and \( t_2 \) be the number of trucks manufactured in 2022. The problem provides the following relationships: 1. \( b_1 = 0.4 \times b_2 \) (2021 bikes vs. 2022 bikes) 2. \( t_2 = 1.25 \times t_1 \) (2022 trucks vs. 2021 trucks) 3. \( b_2 + c_2 + t_2 = 120,000 \) (Total vehicles in 2022) 4. \( t_2 = \frac{b_2 + c_2}{2} \) (Average of 2022 bikes and cars equals 2022 trucks) 5. \( c_1 = 0.9 \times c_2 \) (2021 cars vs. 2022 cars) 6. \( b_1 = \frac{1}{4} \times t_1 \) (2021 bikes vs. 2021 trucks) Now, the solution proceeds step-by-step: Step 1: Total vehicles in 2022.
The total number of vehicles in 2022 is 120,000: \[b_2 + c_2 + t_2 = 120,000\] Step 2: Bikes and trucks in 2021.
Using \( b_1 = \frac{1}{4} \times t_1 \) and substituting \( b_1 = 0.4 \times b_2 \): \[0.4 \times b_2 = \frac{1}{4} \times t_1 \quad \Rightarrow \quad t_1 = 1.6 \times b_2\] Step 3: Relationships between bikes, cars, and trucks in 2022.
From \( t_2 = \frac{b_2 + c_2}{2} \), substitute into the total vehicles equation: \[b_2 + c_2 + \frac{b_2 + c_2}{2} = 120,000\] Simplifying: \[\frac{3}{2} \times (b_2 + c_2) = 120,000\] \[b_2 + c_2 = 80,000\] Substitute \( c_2 = 80,000 - b_2 \) into the equation for \( t_2 \): \[t_2 = \frac{b_2 + (80,000 - b_2)}{2} = 40,000\] Step 4: Solving for the number of cars.
Since \( t_2 = 40,000 \), and \( t_2 = 1.25 \times t_1 \): \[40,000 = 1.25 \times t_1 \quad \Rightarrow \quad t_1 = 32,000\] 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 \( \boxed{60,000} \).