Monthly deposits and the first crossing day
Chloe plans to invest dollars on day 1 of each month from 1 January 2021. The bank offers:
Plan A: Each investment earns a fixed 5-dollar bonus at the end of every month it has been in the account. The bonus is added to the account. Accumulated bonuses earn no further bonuses.
(a) For Plan A, find if all investments including bonuses are worth 22200 dollars at the end of 31 December 2022. marks
Plan B: Interest is 0.2% monthly, so on the last day of each month the amount increases by 0.2%.
(b) Assume Plan B for the rest.
(i) Find in terms of the interest on the first invested on 1 January 2021 at the end of 31 December 2022. mark
(ii) Find minimum (nearest integer) for total at least 22200 dollars at that date. marks
(iii) Let . Find the month the total first exceeds 30000 dollars. Explain whether it happens on the first or last day of the month. marks
[2020/TMJC/Prelim/I/10]
Solution
Plan A monthly deposit
For Plan A, find giving a total of 22200 dollars after 24 months.
Hint
The deposits grow by a fixed 5-dollar bonus each month, not by compound interest.
Answer · (a)
dollars.
Count bonus events
Deposit 1 receives 24 bonuses and deposit 24 receives one.
Add the principal
The fixed bonus is independent of the deposit amount.
Interest on the first deposit
For Plan B, find the interest earned by the first deposit after 24 months.
Hint
Interest earned is accumulated value minus the original k.
Answer · (b)(i)
dollars.
Separate interest from final value
Subtract the initial principal from its accumulated value.
Minimum integer monthly deposit
Find the minimum integer monthly deposit reaching at least 22200 dollars after 24 months.
Hint
Why does the last deposit have exponent 1 rather than 0?
Answer · (b)(ii)
Minimum integer dollars.
Form the end-of-month total
Every deposit earns a different number of credits. Equivalently, let and use : the new deposit arrives at the start of the month, then the whole balance earns interest at its end. This reproduces the source month-by-month table.
Solve and round upwards
A minimum integer satisfying an inequality requires a ceiling, not nearest rounding.
First crossing of 30000
For , find the first date when the balance exceeds 30000 dollars.
Hint
Do not stop after finding the first qualifying month-end balance.
Answer · (b)(iii)
At the month-start deposit on September .
Write the month-end total for k=900
The index counts January 2021 as month 1.
Bracket the crossing month
Both balances are month-end balances.
Inspect the intervening deposit
The month 33 deposit arrives before its month-end interest.
Convert count to date
Month 32 is August 2023, so the crossing event is the next deposit.
Explore the idea
Put deposits and interest on a timeline
An end-of-month formula identifies the month; the deposit at its start may cross the target earlier.