Laptop Savings Plan with a Conditional Bonus and Rising Price
A student plans to buy a laptop that costs 420 KD at the start of September. She deposits 40 KD in September. Each following month she deposits 5 KD more than the previous month, so October is 45 KD, November 50 KD, and so on. Her parents promise a one-time bonus of 60 KD if she reaches at least 300 KD by the end of February. The bonus is added immediately after the February deposit if the threshold is met. The laptop price rises by 2% at the end of each month, compounded monthly, starting with September. She buys immediately after making a monthly deposit and after that month's price rise. She wants to know whether the bonus is earned, how much she has by the end of February, and the first month in which she can afford the laptop.
1. Find the total of the student's own deposits from September through February and determine whether the bonus is earned. What is her total savings after the February bonus?
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The deposits form the arithmetic sequence 40, 45, 50, 55, 60, 65. There are 6 terms, so the sum is
(6/2)(40 + 65) = 3(105) = 315 KD.
Since 315 >= 300, the 60 KD bonus is earned. Total savings after the February bonus is 315 + 60 = 375 KD. The bonus applies because the threshold is checked after the February deposit and before the purchase decision.
2. Model the laptop price after n monthly rises. Using the savings path, find the first month in which she can afford the laptop. Check February and March explicitly.
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After n monthly rises, the price is 420(1.02)^n.
At the end of February, n = 6, so the price is 420(1.02)^6 = 420(1.12616) = 472.99 KD. Her savings are 375 KD, so she cannot buy it.
In March she deposits 65 + 5 = 70 KD, giving savings 375 + 70 = 445 KD. After 7 rises, the price is 420(1.02)^7 = 482.45 KD, so March is still not enough.
In April she deposits 75 KD, giving savings 445 + 75 = 520 KD. After 8 rises, the price is 420(1.02)^8 = 492.10 KD. Since 520 >= 492.10, April is the first affordable month.
3. Suppose the parents change the rule: the bonus is paid only if she reaches 300 KD by the end of January. Does she still get it, and what is the first affordable month then?
Read the worked answer
By the end of January there are 5 deposits: 40, 45, 50, 55, 60. Their sum is
(5/2)(40 + 60) = 2.5(100) = 250 KD.
Since 250 < 300, she does not earn the bonus under the stricter rule. Her savings are then 315 KD at the end of February, 315 + 70 = 385 KD at the end of March, 385 + 75 = 460 KD at the end of April, and 460 + 80 = 535 KD at the end of May.
The April price is 492.10 KD, so 460 KD is not enough. The May price is 420(1.02)^9 = 420(1.19509) = 501.94 KD, and 535 >= 501.94. Therefore the first affordable month becomes May, one month later than under the original bonus rule.