# Variable Rate of Compound Interest

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

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