Compound Interest Questions and Answers

Answer: B

The compound interest is Rs 66,351. Compounded half-yearly for 2 years at 8% a year means 4 periods at 4%.

Rs 456976 is the amount, not the interest. Simple interest would be Rs 62,500, and the other options do not equal the compound interest.

The step of halving the rate and doubling the periods is the key to half-yearly compounding: 8% a year becomes 4% for each of the four half-years. The amount is 390625 × 1.04 four times over, which is Rs 4,56,976, and the compound interest is the amount minus the principal. Rs 66,351 is therefore the interest and Rs 4,56,976 is not.

Answer: A

The compound interest is Rs 512.50. The amount is 5000 x 1.05 x 1.05 = 5512.50 so the interest is 5512.50 - 5000 = 512.50. Rs 500 is simple interest for one year, and Rs 510.50, 500.50 and 511.50 do not match the compounding of 5 per cent for two years. The square of 1.05 is 1.1025, so the compound interest rate for two years is 10.25 per cent of Rs 5,000, which is Rs 512.50. Simple interest would be only Rs 500, so the compound interest is larger by the interest earned on the first year's interest, Rs 12.50.

Answer: A

With compound interest the amount grows by the same factor in each equal period. It becomes 3 times in 5 years, so after the next 5 years it is 3 x 3 = 9 times, and after another 5 years it is 9 x 3 = 27 times. So it takes 3 periods of 5 years, which is 15 years, option (A). The options 20, 25 and 30 years would give 81 times, 243 times and 729 times, which are more than 27 times. In general, if a sum becomes k times in n years, it becomes k to the power m times in m x n years, which here gives 3 cubed, 27, in 3 x 5 years.

Answer: D

The compound interest is 4,634. In one year at 10 % compounded half-yearly, the rate is 5 % per half-year for two half-years, so the amount is the principal times 1.1025.

\[ P = \frac{15435}{1.1025} = 14000 \]

Compounded yearly for 3 years at 10 %:

\[ CI = 14000 \times (1.1^3 - 1) \]

\[ = 14000 \times 0.331 = 4634 \]

The options 4,524, 4,554 and 4,624 do not equal 14000 times 0.331, so the fourth option is correct. As a check, 14000 x 1.1025 = 15435, the amount given in the question, and the amount after 3 years at yearly compounding is 14000 x 1.331 = 18634.

Answer: B

The total interest charged is ₹ 616. The rate per half-year is 4%. Let each instalment be \(x\); the present value of the two instalments equals the loan: \[ \frac{x}{1.04} + \frac{x}{1.04^{2}} = 10200 \] \[ x\times\frac{1.04 + 1}{1.0816} = 10200 \] \[ x = \frac{10200\times 1.0816}{2.04} = 5408 \] Total paid is \(2\times 5408 = 10816\), so the interest is \(10816 - 10200 = 616\). Simple interest at 8% for one year on 10,200 would be 816, which is not the compounded two-instalment figure.

Answer: C

The sum of money is Rs. 4460.

Under compound interest, each further 3 years multiplies the amount by the same factor. From year 3 to year 6 the amount grows by 10035 / 6690.

\[ \begin{aligned} (1 + r)^3 &= \frac{10035}{6690} = 1.5 \\ P &= \frac{6690}{1.5} = 4460 \end{aligned} \]

The first 3 years multiply the principal by the same 1.5, so the principal is 6690 divided by 1.5. Check: \(4460 \times 1.5 = 6690\) and \(6690 \times 1.5 = 10035\).

4260, 4360 and 4560 do not grow to 6690 by this factor; for instance \(4560 \times 1.5 = 6840\).

Answer: A

The compound interest is Rs. 15,121.20.

With a different rate each year, the principal is multiplied by each year's growth factor in turn:

\[ \begin{aligned} A &= 75000 \times 1.04 \times 1.06 \times 1.09 \\ &= 78000 \times 1.06 \times 1.09 \\ &= 82680 \times 1.09 = 90121.20 \end{aligned} \]

\[ \text{CI} = 90121.20 - 75000 = 15121.20 \]

Simple interest at the three rates would give \(75000 \times 0.19 = 14250\); the extra comes from interest earned on interest. The other options differ by whole thousands and come from slips in the multiplication.

Answer: B

The depreciated value after 2 years is Rs. 59,850.

Depreciation at a rate on the reduced value works like compound interest in reverse:

\[ \begin{aligned} V &= 70000 \times 0.95 \times 0.90 \\ &= 66500 \times 0.90 = 59850 \end{aligned} \]

After the first year the machine is worth Rs. 66,500 (a fall of Rs. 3,500), and the second year's 10% is taken on that, a further Rs. 6,650.

Taking both rates on the original cost would give \(70000 - 3500 - 7000 = 59500\), which is not offered; 59,750, 59,950 and 59,805 do not come from either method.

Answer: C

The correct relation is R > S.

Compounding more often makes money grow faster at the same nominal rate. Bi-monthly compounding (six times a year) at S% must produce the same amount Q in one year as annual compounding at R%, so S has to be the smaller rate.

\[ \begin{aligned} \left(1 + \frac{S}{600}\right)^{6} &= 1 + \frac{R}{100} \\ \Rightarrow S &\lt R \end{aligned} \]

For example, 10% compounded bi-monthly gives about 10.43% in a year. R = S would hold only if there were no compounding within the year, so \(R \le S\) and \(R \lt S\) are wrong, and the data are sufficient because the relation holds for every positive rate.

Answer: D

N is greater than 400.

Compounded quarterly, 4% a year means 1% per quarter for 4 quarters:

\[ \begin{aligned} A &= 10000\,(1.01)^{4} \\ &= 10406.04 \\ N &= 406.04 \end{aligned} \]

Simple interest would be exactly 400; compounding adds interest on interest, so the compound interest is a little more than 400. That rules out the three lower ranges.