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Difficulty: HardLinear and Exponential Growth

A financial technology platform offers two merchant rewards programs, Program PP and Program EE, based on a merchant's monthly transaction volume. The table below shows the monthly reward for both programs at a transaction volume of $50,000\$50,000.

ProgramGrowth ModelReward at $50,000\$50,000 Volume
Program PPLinear$800\$800
Program EEExponential$800\$800

For every increase of $10,000\$10,000 in monthly transaction volume:
- Under Program PP, the monthly reward increases by $150\$150.
- Under Program EE, the monthly reward increases by 5%5\%.

If a merchant's monthly transaction volume increases from $50,000\$50,000 to $90,000\$90,000, what is the closest dollar amount to the difference between the monthly rewards the merchant would receive under the two programs?

  1. A
    $159
  2. B
    $440
  3. $428Answer
  4. D
    $1,178

Answer

$428
The correct answer of $428\$428 represents the difference between the rewards under the two models at a monthly transaction volume of $90,000\$90,000. The number of $10,000\$10,000 increments from $50,000\$50,000 to $90,000\$90,000 is 44. The linear reward under Program P increases by $150\$150 per increment, resulting in a reward of 800+4(150)=1400800 + 4(150) = 1400. The exponential reward under Program E increases by 5%5\% per increment, resulting in a reward of 800×(1.05)4≈972.41800 \times (1.05)^4 \approx 972.41. The difference is 1400−972.41=427.591400 - 972.41 = 427.59, which is closest to $428\$428.

Step-by-Step Solution

1
Determine the number of $10,000\$10,000 increments from the initial transaction volume to the target volume.
44 increments
The transaction volume increases from $50,000\$50,000 to $90,000\$90,000, a change of $40,000\$40,000. Since the growth rates are defined per $10,000\$10,000 increase, the number of increments is 40,00010,000=4\frac{40,000}{10,000} = 4.
2
Calculate the monthly reward under the linear model (Program P) after 44 increments.
14001400 dollars
Program P adds a constant $150\$150 per increment. The reward after 44 increments is 800+4(150)=800+600=1400800 + 4(150) = 800 + 600 = 1400.
3
Calculate the monthly reward under the exponential model (Program E) after 44 increments.
972.41972.41 dollars
Program E increases by a constant percent (5%5\%) per increment, which corresponds to a growth factor of 1.051.05. The reward after 44 increments is 800×(1.05)4≈800×1.2155=972.41800 \times (1.05)^4 \approx 800 \times 1.2155 = 972.41.
4
Find the difference between the two monthly rewards.
427.59427.59 dollars, which rounds to 428428 dollars
Subtract the reward under Program E from the reward under Program P: 1400−972.41=427.591400 - 972.41 = 427.59.

Key Concept

Distinguishing between linear and exponential growth models and evaluating them at specific values in a real-world context.
Estimated Time:2m 30s
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Topics

Linear and Exponential Growth

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