Question

If the number of registered cars were to increase yearly through the year 2000 at the same average annual rate shown for the period 1981-1985, for which of the following years would the number of registered cars be closest to 76 million?

• A. 1995
• B. 1996
• C. 1997
• D. 1998
• E. 1999

Hint:

When a problem's data seems contradictory (like projecting growth from a period of decline), look for a reasonable assumption that makes the problem solvable. Clearly stating your assumption (e.g., "using the fastest growth period instead") is a good practice.

Answer 1

Answer: (E)

Note: The period 1981-1985 shown on the graph involves a net decrease, making a projection of future increase based on this rate impossible. A logical interpretation is that the question intends for us to use the period of fastest sustained growth shown on the graph, which is 1975-1980, as the basis for the projection.
Step 1: Understanding the Concept:
This problem requires us to calculate a rate of change from a historical period, and then use this rate to project when a future milestone will be reached.
Step 2: Key Formula or Approach:
1. Calculate the average annual rate of change for the chosen period (1975-1980). Rate = (Change in Value) / (Number of Years). 2. Determine the total increase needed from the starting point (1985) to the target value (76 million). 3. Calculate the number of years required to achieve this increase. Years = (Total Increase Needed) / (Annual Rate). 4. Add the calculated number of years to the start year (1985).
Step 3: Detailed Explanation:
1. Calculate the Annual Rate (using 1975-1980 data): - Cars in 1975: 30 million. - Cars in 1980: 40 million. - Change in Value: 40 - 30 = 10 million. - Number of Years: 1980 - 1975 = 5 years. - Average Annual Rate: \(\frac{10 \text{ million}}{5 \text{ years}} = 2\) million cars/year. 2. Project the Future Date: - Starting Value (at year-end 1985): 43.1 million. - Target Value: 76 million. - Increase Needed: 76 - 43.1 = 32.9 million. - Number of Years to Reach Target: \(\frac{32.9 \text{ million}}{2 \text{ million/year}} = 16.45\) years. 3. Determine the Target Year: - Target Year = 1985 + 16.45 years = 2001.45. The number of registered cars would reach 76 million during the year 2001. We must find which of the options is "closest" to this value. Given the options are all in the late 1990s, this suggests the intended rate might have been slightly higher, or there is an issue with the question's data. However, among the choices provided, 1999 is the latest year and therefore mathematically closest to 2001.45.
Step 4: Final Answer:
Based on a logical interpretation of the graph's growth rates, the projection leads to a target date of late 2001. Of the available choices, 1999 is the closest.