CDS Mathematics · Simple and Compound Interest

CI Minus SI, and Instalments

Over two years compound interest exceeds simple interest by the interest on the first year's interest, P(r/100)²; instalments repay a loan when their values today add up to it.

All Simple and Compound Interest notes CDS Mathematics strategy

Why this matters

Eight PYQs, one of them HARD. Six use the CI − SI gap, which for two years is P(r/100)², so the sum is read off in one line. The two instalment items discount each payment back to today and add.

Concept 1 of 2: The gap between CI and SI

In year one both methods earn the same. In year two, compound interest also earns interest on year one's interest:

P \cdot \tfrac r{100} \cdot \tfrac r{100}

. That extra is the whole gap for two years.

Definition

Two years: $\text{CI} - \text{SI} = P\left(\dfrac{r}{100}\right)^2$ .
Three years: $\text{CI} - \text{SI} = P\left(\dfrac{r}{100}\right)^2\left(3 + \dfrac{r}{100}\right)$ .
Half-yearly compounding: work out CI with the adjusted rate and periods, then subtract the SI directly.

Two-year gap

\text{CI} - \text{SI} = P\left(\dfrac{r}{100}\right)^2

Worked example

The CI and SI on a sum for

2

years at

8\%

differ by Rs.

32

. Find the sum.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q43Moderate

Example 1 · Simple and Compound Interest · Instalments and Difference of SI and CI

The difference between the compound interest (compounded annually) and simple interest on a sum of money deposited for 2 years at 5% per annum is Rs. 15. What is the sum of money deposited ?

The two-year formula is for two years only

For three years the gap is

P(r/100)^2(3 + r/100)

, not

P(r/100)^3

and not

P(r/100)^2

. Use the formula for the years asked, or compute both interests.

Drill 5 more on the gap between ci and si

Concept 2 of 2: Equal instalments

A payment of

X

made after

k

years is worth only

\dfrac{X}{(1 + r)^k}

today. Equal instalments repay the loan when those present values add up to the sum borrowed.

Definition

Present value of $X$ due in $k$ years: $\dfrac{X}{\left(1 + \frac r{100}\right)^k}$ .
$n$ equal yearly instalments: $X\sum_{k=1}^{n}\dfrac{1}{\left(1 + \frac r{100}\right)^k} = P$ .
Two instalments at $10\%$ : $X\left(\dfrac{10}{11} + \dfrac{100}{121}\right) = P$ .

Two instalments

\dfrac{X}{1 + r} + \dfrac{X}{(1 + r)^2} = P

Worked example

A loan is repaid in two equal yearly instalments of Rs.

1210

10\%

compound interest. How much was borrowed?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q26Moderate

Example 2 · Simple and Compound Interest · Instalments and Difference of SI and CI

A sum of Rs. 8,400 was taken as a loan. This is to be paid in two equal instalments. If the rate of interest is 10% per annum, compounded annually, then the value of each instalment is

Discount each payment separately

The second instalment is discounted over TWO years, not one. Dividing the loan in half and adding a year's interest gives an option that is always offered and always wrong.

Drill 1 more on equal instalments

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The gap between CI and SI
Two-year gap
$\text{CI} - \text{SI} = P\left(\dfrac{r}{100}\right)^2$
Equal instalments
Two instalments
$\dfrac{X}{1 + r} + \dfrac{X}{(1 + r)^2} = P$

Watch out for (2)

The two-year formula is for two years only
→ The gap between CI and SI
Discount each payment separately
→ Equal instalments

Test yourself on Simple and Compound Interest

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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