We will discuss here how to use the formula for variable rate of compound interest.
When the rate of compound interests for successive/consecutive years are different (r \(_{1}\)%, r \(_{2}\)%, r \(_{3}\)%, r \(_{4}\)%, .................. ) then:
A = P( 1 + \(\frac{r_{1}}{100}\))(1 + \(\frac{r_{2}}{100}\))(1 + \(\frac{r_{3}}{100}\)) .............
Where,
A = amount;
P = principal;
r \(_{1}\), r \(_{2}\), r \(_{3}\), r \(_{4}\).......... = rates for successive years.
Word problems on variable rate of compound interest:
1. If the rate of compound interest for the first, second and third year be 8%, 10% and 15% respectively, find the amount and the compound interest on $ 12,000 in 3 years.
Solution:
The man will receive an interest of 8% in the first year, 10% in the second year and 15% in the third year.
Therefore,
Amount = P( 1 + \(\frac{r_{1}}{100}\))(1 + \(\frac{r_{2}}{100}\))(1 + \(\frac{r_{3}}{100}\))
⟹ A = $ 12,000(1 + \(\frac{8}{100}\))(1 + \(\frac{10}{100}\))(1 + \(\frac{15}{100}\))
⟹ A = $ 12,000 (1 + 8/100)(1 + 10/100)(1 + 15/100)
⟹ A = $ 12,000 × 267/25 × 11/10 × 23/20
⟹ A = $ 12,000 × \(\frac{6831}{5000}\)
⟹ A = $ 16,394.40
Therefore, the required amount = $ 16,394.40
Therefore, the compound interest = Final amount  Initial principal
= $ 16,394.40  $ 12,000
= $ 4,394.40
2. Find the compound interest accrued by Aaron from a bank on $ 16000 in 3 years, when the rates of interest for successive years are 10%, 12% and 15% respectively.
Solution:
For the first year:
Principal = $ 16,000;
Rate of interest = 10% and
Time = 1 years.
Therefore, interest for the first year = \(\frac{P × R × T}{100}\)
= $ \(\frac{16000 × 10 × 1}{100}\)
= $ \(\frac{160000}{100}\)
= $ 1,600
Therefore, the amount after 1 year = Principal + Interest
= $16,000 + $ 1,600
= $ 17,600
For the second year, the new principal is $ 17,600
Rate of interest = 12% and
Time = 1 years.
Therefore, the interest for the second year = \(\frac{P × R × T}{100}\)
= $ \(\frac{17600 × 12 × 1}{100}\)
= $ \(\frac{211200}{100}\)
= $ 2,112
Therefore, the amount after 2 year = Principal + Interest
= $ 17,600 + $ 2,112
= $ 19,712
For the third year, the new principal is $ 19,712
Rate of interest = 15% and
Time = 1 years.
Therefore, the interest for the third year = \(\frac{P × R × T}{100}\)
= $ \(\frac{19712 × 15 × 1}{100}\)
= $ \(\frac{295680}{100}\)
= $ 2,956.80
Therefore, the amount after 3 year = Principal + Interest
= $ 19,712 + $ 2,956.80
= $ 22,668.80
Therefore, the compound interest accrued = Final amount  Initial principal
= $ 22,668.80  $ 16,000
= $ 6,668.80
3. A company offers the following growing rates of compound interest annually to the investors on successive years of investment.
4%, 5% and 6%
(i) A man invests $ 31,250 for 2 years. What amount will he receive after 2 years?
(ii) A man invests $ 25,000 for 3 years. What will be his gain?
Solution:
The man will get 4% for the first year, which will be compounded at the end of the first year. Again for the second year, he will get 5%. So,
A = P( 1 + \(\frac{r_{1}}{100}\))(1 + \(\frac{r_{2}}{100}\))
⟹ A = $ 31250(1 + \(\frac{4}{100}\))(1 + \(\frac{5}{100}\))
⟹ A = $ 31250 × 26/25 × 21/20
⟹ A = $ 34,125
Therefore, at the end of 2 years he will receive $ 34125.
(ii) The man will receive an interest of 4% in the first year, 5% in the second year and 6% in the third year.
Therefore,
Amount = P( 1 + \(\frac{r_{1}}{100}\))(1 + \(\frac{r_{2}}{100}\))(1 + \(\frac{r_{3}}{100}\))
⟹ A = $ 25000(1 + \(\frac{4}{100}\))(1 + \(\frac{5}{100}\))(1 + \(\frac{6}{100}\))
⟹ A = $ 25000 × 26/25 × 21/20 × 53/50
⟹ A = $ 28,938
Therefore, he gain = Final amount  Initial principal
= $ 28,938  $ 25000
= $ 3,938
● Compound Interest
Compound Interest with Growing Principal
Compound Interest with Periodic Deductions
Compound Interest by Using Formula
Practice Test on Compound Interest
● Compound Interest  Worksheet
Worksheet on Compound Interest
Worksheet on Compound Interest with Growing Principal
Worksheet on Compound Interest with Periodic Deductions8th Grade Math Practice
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