Exam practice

Interest worked out at the end of each year and added to the principal, so that the next year’s interest is worked out on the principal plus the interest so far.

Interest worked out every six months at half the yearly rate and added to the principal, so that each half-year’s interest is on the principal plus all earlier interest.

Interest worked out every three months at a quarter of the yearly rate and added to the principal, so that each quarter’s interest is on the principal plus all earlier interest.

$\text{CA} = P\left(1 + \frac{R}{100}\right)^T$

$\text{CI} = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]$

$$\text{CA} = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right)$$

Year 1: interest $\frac{10000 \times 6}{100} = 600$; principal for year 2: 10,600.
Year 2: interest $\frac{10600 \times 6}{100} = 636$.
CI = 600 + 636 = Rs 1,236; CA = 10,000 + 1,236 = Rs 11,236.

Year 1: interest 3,840; principal for year 2: 67,840.
Year 2: interest $\frac{67840 \times 6}{100} = 4070.40$; principal for year 3: 71,910.40.
Year 3: interest $\frac{71910.40 \times 6}{100} \approx 4314.62$.
CI = 3,840 + 4,070.40 + 4,314.62 = Rs 12,225.02; CA = Rs 76,225.02.

Year 1: interest 2,000; principal for year 2: 22,000.
Year 2: interest $\frac{22000 \times 12}{100} = 2640$.
CI = 2,000 + 2,640 = Rs 4,640; CA = Rs 24,640.

In one year compound and simple interest are the same: Rs 12 on Rs 100 is 12%.

$200\left(1 + \frac{R}{100}\right)^2 = 242$, so $\left(1 + \frac{R}{100}\right)^2 = 1.21 = 1.1^2$.
$1 + \frac{R}{100} = 1.1$: 10%.

$\text{CA} = 20000 \times 1.15^3 = 20000 \times 1.520875 = 30417.50$.
CA = Rs 30,417.50; CI = 30,417.50 − 20,000 = Rs 10,417.50.

$\text{CA} = 50000 \times 1.1^3 = 50000 \times 1.331 = 66550$.
CA = Rs 66,550; CI = Rs 16,550.

T = 2, M = 5: $\text{CA} = 150000\left(1 + \frac{6}{100}\right)^2\left(1 + \frac{5 \times 6}{1200}\right)$.

• i.CA = 150000 × 1.1236 × 1.025 = 172753.50: Rs 1,72,753.50.
• ii.CI = 172753.50 − 150000 = Rs 22,753.50.

4% a half-year, 4 half-years: $\text{CA} = 50000 \times 1.04^4 = 50000 \times 1.16985856 \approx 58492.93$.
CA = Rs 58,492.93; CI = Rs 8,492.93.

3% a quarter, 4 quarters: $\text{CA} = 5000000 \times 1.03^4 = 5000000 \times 1.12550881 = 5627544.05$.
CA = Rs 56,27,544.05; CI = Rs 6,27,544.05.

SI = $\frac{80000 \times 2 \times 8}{100} = 12800$. CI = $80000\,(1.08^2 - 1) = 80000 \times 0.1664 = 13312$.
Difference: 13,312 − 12,800 = Rs 512. (Or $P\left(\frac{R}{100}\right)^2 = 80000 \times 0.0064 = 512$.)

SI = $\frac{7500 \times 3 \times 12}{100} = 2700$. CI = $7500\,(1.12^3 - 1) = 7500 \times 0.404928 = 3036.96$.
Difference: Rs 336.96.

Yearly: $40000\,(1.06^2 - 1) = 40000 \times 0.1236 = 4944$.
Half-yearly: $40000\,(1.03^4 - 1) = 40000 \times 0.12550881 \approx 5020.35$.
Difference: Rs 76.35 (half-yearly more).

Half-yearly: $18000\,(1.06^2 - 1) = 18000 \times 0.1236 = 2224.80$.
Quarterly: $18000\,(1.03^4 - 1) = 18000 \times 0.12550881 \approx 2259.16$.
Difference: Rs 34.36 (quarterly more).

• i.P: $60000\,(1.05^4 - 1) = 60000 \times 0.21550625 \approx 12930.38$: Rs 12,930.38.
• ii.Q: $60000\,(1.12^2 - 1) = 60000 \times 0.2544 = 15264$: Rs 15,264.
• iii.Q: it gives Rs 15,264 − 12,930.38 = Rs 2,333.62 more. Adding interest more often can’t make up for a rate 2% lower.

For 2 years, CI − SI = $P\left(\frac{R}{100}\right)^2$: $P \times 0.15^2 = 180$, so $P = \frac{180}{0.0225} =$ Rs 8,000.

Half-yearly: $P\,(1.05^2 - 1) = 0.1025P$. Yearly: 0.10P.
0.1025P − 0.10P = 40, so 0.0025P = 40 and P = Rs 16,000.

He receives $P\,(1.05^4 - 1) = 0.21550625P$ and pays $P\,(1.1^2 - 1) = 0.21P$.
0.00550625P = 2019.24, so $P = \frac{2019.24}{0.00550625} \approx 366717.82$.
He borrowed about Rs 3,66,718 (Rs 3,66,717.82).

$1.1^T = \frac{121000}{100000} = 1.21 = 1.1^2$: 2 years.

CA = 8,00,000 + 1,26,100 = 9,26,100, so $1.05^T = \frac{926100}{800000} = 1.157625$.
$1.05^2 = 1.1025$, $1.05^3 = 1.157625$: 3 years.

$\left(1 + \frac{R}{100}\right)^2 = \frac{847}{700} = 1.21$, so $1 + \frac{R}{100} = 1.1$: 10%.

CA = 3,43,000 + 1,13,533 = 4,56,533. $\left(1 + \frac{R}{100}\right)^3 = \frac{456533}{343000} = 1.331 = 1.1^3$: 10%.

• i.$1 + \frac{R}{100} = \frac{6655}{6050} = 1.1$: 10%.
• ii.$P \times 1.1^2 = 6050$, so $P = \frac{6050}{1.21} =$ Rs 5,000.

• i.$1 + \frac{R}{100} = \frac{12167}{10580} = 1.15$: 15%.
• ii.$P = \frac{10580}{1.15^2} = \frac{10580}{1.3225} =$ Rs 8,000.

Amounts: P + 1800 after 1 year and P + 3816 after 2 years. The second year’s interest, 3816 − 1800 = 2016, is earned on P + 1800; the first year’s, 1,800, on P.
So 2016 − 1800 = 216 is the interest on Rs 1,800 for a year: $R = \frac{216}{1800} \times 100 = 12$.
$P = \frac{1800 \times 100}{12} = 15000$. 12% and Rs 15,000.

• i.$500000\,(1.05^2 - 1) = 51250$; after 5% tax: 51250 × 0.95 = Rs 48,687.50.
• ii.Principal for year 2: 5,48,687.50. Quarterly: $548687.50 \times (1.025^4 - 1) \approx 548687.50 \times 0.1038128906 \approx 56960.84$; after tax: $\times 0.95 \approx$ Rs 54,112.79.
• iii.54112.79 − 48687.50 = Rs 5,425.29.
• iv.$\frac{5425.29}{48687.50} \times 100\% \approx$ 11.14% more.

Year 1: 80000 × 0.15 = 12000; after tax 12000 × 0.95 = 11400.
Year 2: principal 91,400; half-yearly $91400\,(1.075^2 - 1) = 91400 \times 0.155625 \approx 14224.13$; after tax $\approx 13512.92$.
Difference: 13512.92 − 11400 = 2112.92; $\frac{2112.92}{11400} \times 100\% \approx$ 18.53% more.

Let the 2-year part be x. Then $x \times 1.05^2 = (41000 - x) \times 1.05^3$, so x = 1.05 (41000 − x).
2.05x = 43050, x = 21000.
Rs 21,000 for 2 years and Rs 20,000 for 3 years. (Each grows to Rs 23,152.50.)

Let the first part be x: $x \times 1.1^3 = (21000 - x) \times 1.1^2$, so 1.1x = 21000 − x.
2.1x = 21000, x = 10000.
First part Rs 10,000, second part Rs 11,000. (Each grows to Rs 13,310.)

Second year’s interest: 945 − 450 = 495. The extra 495 − 450 = 45 is the interest on Rs 450 for a year: $R = \frac{45}{450} \times 100 = 10$.
$P = \frac{450 \times 100}{10} = 4500$. 10% and Rs 4,500.

This time by the formula. $P \times \frac{R}{100} = 1800$ and $P\left[\left(1 + \frac{R}{100}\right)^2 - 1\right] = 3816$.
The second is $P \times \frac{R}{100} \times \left(2 + \frac{R}{100}\right) = 3816$, so dividing by the first, $2 + \frac{R}{100} = \frac{3816}{1800} = 2.12$ and R = 12.
$P = \frac{1800}{0.12} = 15000$. 12% and Rs 15,000.

For whole years, never: in the first year the two are equal, and after it compound interest is always more.
They are also equal for any time under a year when interest is compounded yearly, since nothing is added before the year ends.
Simple interest comes out more only for part of a period: if compound interest for six months is worked out with a fraction as the power, $P\left(1 + \frac{R}{100}\right)^{1/2}$, it gives a little less than the simple interest’s R/2%. ($1.1^{1/2} \approx 1.0488$, under 1.05.)

A bank uses the money it holds: interest it owes a saver but hasn’t paid out stays in the bank, and interest a borrower hasn’t paid is money the bank hasn’t got back. Adding it to the principal treats it as money deposited or lent, which it is.
Savers gain: their interest earns interest, more so the longer they save and the more often it is added.
Borrowers lose if they don’t pay: unpaid interest earns interest too, so a debt left alone grows faster and faster (doubling in about 6 years at 12%). Paying the interest as it falls due stops it compounding.

Largely yes. A bank loan charges interest every month or quarter on what is still owed. If each instalment covers that interest, nothing is left to compound; but any interest not paid is added to what is owed and earns interest itself, so the loan is not simple interest.
Example: Rs 1,00,000 at 12% a year, interest added quarterly, nothing paid for a year: $100000 \times 1.03^4 \approx 112551$, Rs 551 more than the Rs 12,000 of simple interest.

Interest is income, and income is taxed by law, like a salary. Taking it from the interest before paying it out (withholding) is the simplest way to collect it: the government gets it from a few hundred banks instead of millions of savers, and a saver doesn’t have to file anything for it.