Compound Interest and Compound Growth or Decrease is an important chapter of Madhyamik Mathematics. In this chapter, students learn how an amount changes when interest is added to the principal at regular intervals and how a quantity increases or decreases by a fixed percentage over time.
The chapter covers Compound Interest, Amount, changing annual rates, half-yearly and quarterly compounding, growth, depreciation and related numerical problems.
Compound Interest – Basic Concept
In simple interest, interest is calculated only on the original principal. In compound interest, the interest earned during one period is added to the principal, and the next period's interest is calculated on the new amount.
Here, P represents the principal, R is the rate of interest per period and n is the number of periods.
Compound Interest and Simple Interest
| Point | Simple Interest | Compound Interest |
|---|---|---|
| Interest calculation | On the original principal | On the accumulated amount |
| Previous interest | Not added to principal | Added to principal |
| Principal for next period | Normally unchanged | Changes after each compounding period |
| Growth over time | Linear | Compound growth |
MCQ – Multiple Choice Questions
1. Compound Interest on ₹5,000
```At an average annual rate of 5%, what will be the compound interest on ₹5,000 for 2 years?
= ₹5,512.50
Compound Interest = ₹5,512.50 − ₹5,000
= ₹512.50
2. Annual Rate in Compound Interest
```In compound interest, the rate of interest for every year is:
3. Finding the Rate of Interest
```If the compound interest on a certain principal for 2 years is ₹105 and the simple interest is ₹100, find the rate of interest.
4. Change in Principal
```In compound interest:
5. Simple Interest and Compound Interest
```The source includes a fifth MCQ comparing the simple interest for 2 years with compound interest compounded annually for 2 years. The exact options and answer are provided in the source as an image.
True or False
1. In compound interest, interest is also earned on the previously accumulated interest.
2. In compound interest, interest is added to the principal at regular intervals, causing the amount to increase progressively.
3. As the number of interest periods increases, compound interest becomes smaller.
4. Banks generally provide both simple interest and compound interest in the same manner.
5. For a fixed period, simple interest is greater than compound interest.
6. In compound interest, the rate of interest for every compounding period must always be the same.
Fill in the Blanks
1. When a machine becomes older, its value generally decreases. This decrease in value is called ________.
2. At what annual rate will ₹1,000 become ₹1,210 in 2 years?
3. In compound interest, the annual rate for different years may be ________.
4. According to the source's fill-in-the-blank question, as the rate of interest increases, compound interest ________.
5. A quantity decreasing at a fixed rate over time is called ________.
6. A quantity increasing at a fixed rate over time represents ________.
Short Answer Questions
1. Different Interest Rates for Two Years
```The compound interest rate is 6% in the first year and 4% in the second year. Find the amount after 2 years on a principal of ₹22,000.
₹22,000 × 1.06 = ₹23,320
Second-year amount:
₹23,320 × 1.04 = ₹24,252.80
2. Compound Interest on ₹50,000
```Find the compound interest on ₹50,000 at an annual rate of 10% for 22 years.
Compound Interest = Amount − 50,000
3. Half-Yearly Compound Interest
```The source contains a problem involving ₹20,000, a half-yearly compounding period and an annual interest rate. The exact numerical rate and duration are contained in the source image.
4. Depreciation of a Machine
```A machine loses 10% of its value every year. If its present value is ₹1,62,000, find its value 2 years ago.
After one year:
Value = P × 90/100
After two years:
1,62,000 = P × (90/100)2
1,62,000 = P × 0.81
P = 1,62,000 ÷ 0.81
= ₹2,00,000
Long Answer Questions
1. From Double to Four Times
```If a certain amount becomes double in n years at a fixed annual compound rate, determine how many years it will take to become four times the original amount.
After n years:
Amount = 2P
After another n years, the same compound growth factor applies again:
Amount = 2 × 2P
= 4P
2. Difference Between Compound Interest and Simple Interest
```If the difference between compound interest and simple interest on a certain principal for 3 years at 10% per annum is ₹30, find the principal.
Simple Interest = 30% of P
= 0.30P
Compound Interest = P[(1.10)3 − 1]
= 0.331P
Difference = 0.331P − 0.30P
= 0.031P
0.031P = 30
P = 30 ÷ 0.031
≈ ₹967.74
3. Quarterly Compound Interest for 9 Months
```Find the compound interest on ₹10,000 for 9 months at an annual compound interest rate of 8%, compounded every 3 months.
Rate per quarter = 8% ÷ 4 = 2%
Number of quarters in 9 months = 9 ÷ 3 = 3
Therefore:
Compound Interest = ₹10,612.08 − ₹10,000
= ₹612.08 approximately
4. Different Compound Rates in Three Years
```Find the compound interest on ₹40,000 for 3 years when the annual rates of interest for the first, second and third years are 4%, 5% and 6% respectively.
₹40,000 × 1.04 = ₹41,600
Second year:
₹41,600 × 1.05 = ₹43,680
Third year:
₹43,680 × 1.06 = ₹46,300.80
= ₹6,300.80
5. Half-Yearly Compounding
```The source contains a problem involving half-yearly compounding, an annual rate of 4% and a target amount of ₹6,632.55. The exact wording and duration are presented in the source image.
Number of Half-Yearly Periods = Number of Years × 2
6. Finding the Number of Years
```At an annual compound rate of 8%, find the number of years required for ₹40,000 to become ₹46,656.
46,656 = 40,000 × (1.08)n
Therefore:
(1.08)n = 46,656 ÷ 40,000
= 1.1664
1.08 × 1.08 × 1.08 = 1.259712
and
1.08 × 1.08 = 1.1664
Therefore, n = 2 years.
7. Compound Interest at the End of Two and Three Years
```The source gives a problem in which the compound-interest values at the end of two years and three years are stated as ₹880 and ₹968 respectively, and asks for the original principal.
Compound Growth and Depreciation
Compound growth and depreciation use the same basic mathematical idea as compound interest. The difference is whether the quantity increases or decreases over time.
| Situation | Formula |
|---|---|
| Growth at R% | New Value = Original Value × (1 + R/100)n |
| Decrease at R% | New Value = Original Value × (1 − R/100)n |
| Depreciation | Value decreases by a fixed percentage over successive periods |
Important Formulas for Chapter 6
Quick Revision Chart
Exam Preparation for Madhyamik Mathematics Chapter 6
While preparing this chapter, students should practise problems involving compound interest, amount, different annual rates, half-yearly compounding, quarterly compounding, compound growth and depreciation.
Pay special attention to the compounding period. When interest is compounded half-yearly or quarterly, both the rate per period and the number of periods must be adjusted before applying the formula.
For broader WBBSE Class 10 preparation, students can also visit Cademy's Madhyamik Suggestion 2027 All Subjects resource, which includes Mathematics among its subject-wise preparation resources.
Final Revision Tip: Before solving a compound-interest problem, identify four things first — principal, rate, time and compounding period. Once these are correctly identified, choose the appropriate formula and calculate the amount step by step.