SSC CGL Maths · Simple and Compound Interest

Rate Changes and Part Repayments

When the rate, the time or the principal changes partway, work out the interest stretch by stretch on what is owed in each stretch, and add.

All Simple and Compound Interest notes

Why this matters

Sixteen questions in the bank, eleven of them MODERATE, the hardest page of simple interest. Seven of them, all from the 2024 shifts, are one pattern with new numbers: part of a loan is repaid after a year and the rest is cleared a year later. The others change the rate or the time and ask for the extra interest or the sum.

Concept 1 of 3: Extra interest from a change in rate or time

If the rate goes up by 3%, the lender earns 3% more on the same sum every year. You do not need either full interest, only the extra.

Definition

Rate up by $\Delta R$ for the same sum and time: extra interest $= \frac{P \times \Delta R \times T}{100}$ .
More years $\Delta T$ at the same rate: extra interest $= \frac{P \times R \times \Delta T}{100}$ .
Rate multiplied by $k$ for the same sum and time: the interest is multiplied by $k$ .
Rate and time both changed: the new interest is the old one times the ratio of rate times time.
If only an amount is given, find the old rate first, then apply the change.

Extra interest from a change

\Delta SI = \frac{P \times \Delta R \times T}{100} \qquad \Delta SI = \frac{P \times R \times \Delta T}{100}

$\Delta R$ the rise in the rate percent
$\Delta T$ the extra years

Worked example

A sum lent for 4 years at simple interest would earn ₹480 more if the rate were 3% higher. Find the sum.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 18 Sep 2024, Shift 1 · Q65Easy

Example 1 · Simple and Compound Interest · Rate Changes and Part Repayments

A certain sum of money is given at a certain rate of simple interest for 6 years. Had it been given at 7% higher rate, it would have fetched ₹1,512 more. Find the sum (in ₹).

Applying a rate rise for one year only

A rate 2% higher on a 5-year loan earns 2% more in each of the 5 years: 10% of the sum in all, not 2%.

Drill 4 more on extra interest from a change in rate or time

Concept 2 of 3: Different rates in different years

Each stretch of years adds its own rate times its own years to the total percent. Add the stretches, and the whole loan becomes one percent of the principal.

Definition

Total percent over the loan $= R_1 T_1 + R_2 T_2 + \dots$ . The interest is that percent of the principal.
Rate rising by a fixed step each year: add the yearly rates. 6%, 7% and 8% over 3 years make 21%.
"For the period beyond 6 years" in a 9-year loan means the last 3 years.
Given the total interest, the principal is the interest divided by the total percent.

Rates that change over the years

SI = \frac{P}{100}\left(R_1 T_1 + R_2 T_2 + R_3 T_3\right)

Worked example

A loan is charged simple interest at 4% per annum for 2 years, 6% for the next 3 years and 8% for the year after that. The total interest is ₹1,360. Find the loan.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 14 Sep 2025, Shift 2 · Q69Easy

Example 2 · Simple and Compound Interest · Rate Changes and Part Repayments

Rahul borrowed a sum of money at simple interest of

5\%

per annum for the first 3 years,

7\%

per annum for the next 5 years, and

9\%

per annum for the period beyond 8 years. If he pays a total of ₹6800 as interest only at the end of 10 years, how much money did he borrow?

Averaging the rates without the years

Rates of 4% for 1 year and 10% for 3 years do not average to 7%. The total percent is

4 + 30 = 34\%

over 4 years, an average of 8.5%.

Drill 2 more on different rates in different years

Concept 3 of 3: Part repayments and interest added to the principal

Simple interest is charged on what is owed. Pay back part of the principal and next year's interest shrinks; leave the first year's interest unpaid and, under simple interest, it waits without earning anything.

Definition

Year 1: interest on the full principal $P$ . Repaying $x$ of the principal leaves $P - x$ .
Year 2: interest on $P - x$ . The year 1 interest is still owed but earns nothing.
Final payment $=$ the principal left $+$ both years' interest.
Fast route: both interests together are $R\%$ of $P + (P - x)$ , one multiplication.
Interest added to the principal at the end of a period: the amount becomes the new principal, and simple interest starts again on it.

Final payment after a part repayment

\text{Final payment} = (P - x) + \frac{R}{100}\left(P + (P - x)\right)

Pthe sum borrowed
xthe principal repaid at the end of year 1

Worked example

₹50,000 is borrowed at 12% per annum simple interest. At the end of the first year ₹10,000 of the principal is repaid. Everything left is cleared at the end of the second year. How much is paid then?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 25 Sep 2024, Shift 2 · Q62Moderate

Example 3 · Simple and Compound Interest · Rate Changes and Part Repayments

Naman borrows a sum of ₹3,48,020 at the rate of 8% per annum simple interest. At the end of the first year, he repays ₹15,090 towards return of principal amount borrowed. If Naman clears all pending dues at the end of the second year, including interest payment that accrued during the first year, how much does he pay (in ₹) at the end of the second year?

Charging the second year on the original sum

After a part of the principal is repaid, the second year's interest is on what is still owed. Charging it on the full sum overstates the payment by the rate times the amount repaid.

Leaving out the unpaid first-year interest

If the first year's interest was not paid when it fell due, it is still owed at the end. The final payment is the principal left plus both years' interest.

Drill 7 more on part repayments and interest added to the principal

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 (3)

Extra interest from a change in rate or time
Extra interest from a change
$\Delta SI = \frac{P \times \Delta R \times T}{100} \qquad \Delta SI = \frac{P \times R \times \Delta T}{100}$
Different rates in different years
Rates that change over the years
$SI = \frac{P}{100}\left(R_1 T_1 + R_2 T_2 + R_3 T_3\right)$
Part repayments and interest added to the principal
Final payment after a part repayment
$\text{Final payment} = (P - x) + \frac{R}{100}\left(P + (P - x)\right)$

Watch out for (4)

Applying a rate rise for one year only
→ Extra interest from a change in rate or time
Averaging the rates without the years
→ Different rates in different years
Charging the second year on the original sum
→ Part repayments and interest added to the principal
Leaving out the unpaid first-year interest
→ Part repayments and interest added to the principal

Test yourself on Simple and Compound Interest

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Simple and Compound Interest chapter test Drill all 16 questions