**Compound Interest Tricks: Results related to Multiple Compounding in a year:**

In each of the following results, we use the following denotations:

A = future value

P = principal amount (initial investment)

r = annual nominal interest rate

n = number of times the interest is compounded per year

t = number of years for which the money is borrowed

*Compound Interest Trick 1: If interest is not compounded yearly then*

where

n= number of times compounding is done

if compounding is done half yearly then n = 2

if compounding is done quarterly then n = 4

when compounded monthly then n=12

**Amount for Half Yearly Compounding, A = P {1+(R/2)/ 100} ^{2T }**(compound interest applied two times in a year)

Like Half Yearly Compound Interest, we can calculate the amount for Quarterly Compounding.

**Amount for Quarterly Compounding,A = P {1+(R/4)/ 100} ^{4T}**

**Example 1: **Sona deposited Rs. 4000 in a bank for 2 years at 5% p.a.rate. Find the amount received by her from the bank if interest is compounded half yearly.

**Solution: **

Principal value = Rs. 4000

Rate = 5%

Time = 2 years

Since the interest is compounded half yearly so 2 years = 4 times in two years

So, we have

**A = P {1+(R/2)/ 100} ^{2T}**

A= 4000{1+ (5/2)/100}

^{4}

A = 4000 x 41/40 x 41/40 x 41/40 x 41/40

A = Rs. 4415.25

So, Sona received Rs. 4415.25 from the bank after two years

**Example 2:** Manpreet lent Rs 5000 to Richa at 10% rate for 1 year. But she told her that she will take her money on compound interest. Find the amount of interest received by Manpreet when compounded quarterly.

**Solution: **

Principal value = Rs. 5000

Rate = 10%

Time = 1 year

The interest is compounded quarterly, that is 4 times in 1 year.

We use the formula=**A = P {1+(R/4)/ 100} ^{4T}**

A= 5000{1+ (10/4)/100}

^{4}

A = 5000 x 41/40 x 41/40 x 41/40 x 41/40

A = Rs. 5519.064

So, Manpreet received Rs. 5519.064 from Richa and the total amount of interest received by her is 5519.064 – 5000 = Rs. 519.06

**Compound Interest Tricks: When the rate % is not same for every year and interest is taken on yearly basis then amount can be calculated as:**

**Example 3:** Find the compound interest on 16000 in 2 years , while rate of interest at first year is 3% and for the second year is 4%.

**Solution: **

**Compound Interest Tricks: When interest is compounded yearly and time is given in fraction**

If time is given in mixed fraction

Use formula,

**Example 4: **The compound interest on Rs. 8,000 at 15% per annum for 2 years 4 months, compounded annually is:

**Solution:**

**Compound Interest Tricks: Practice Questions**

**Question 1:** A certain sum, invested at 4% per annum compound interest, compounded half – yearly, amounts to Rs 7,803 at the end of one year. The sum is

A. Rs 7,000

B. Rs 7,200

C. Rs 7,500

D. Rs 7,700

**Answer and Explanation**

**Option C**Let the sum be Rs. P.

As, the interest is compounded half-yearly.

R = 2%, T = 2 half years

**Question 2:** If the rate of interest be 4% per annum for first year, 5% per annum for second year and 6% per annum for third year, then the compound interest of Rs. 10,000 for 3 years will be

A. Rs. 1,600

B. Rs. 1,625.80

C. Rs. 1,575.20

D. Rs. 2,000

**Answers and Explanations**

**Option C**When the rate of interest is different for the different years, we can use the following formula to find the amount.

**Question 3:** The compound interest on Rs. 16,000 for 9 months at 20% per annum, interest being compounded quarterly, is

A. Rs. 2,520

B. Rs. 2,524

C. Rs. 2,522

D. Rs. 2,518

**Answers and Explanation**

**Option C**The interest is compounded quarterly. So

**Question 4:** The compound interest on Rs. 6,000 at 10% per annum for 3/2 years, when the interest be¬ing compounded annually, is

A. Rs. 930

B. Rs. 870

C. Rs. 910

D. Rs. 900

**Answers and Explanations**

**Option A**

**Compound Interest Shortcuts: Tooltip-1**

**A sum of money placed at compound interest becomes x time in ‘a’ years and y times in ‘b’ years. These two sums can be related by the following formula:**

**Derivation for this result:**

We use the basic formula for calculating Compound Interest:

For condition 1, a sum of money becomes x times in “a” years.

Therefore, using the formula for calculating Compound Interest:

**Example-1:** A sum of money placed at compound interest doubles itself in 4 years.In how many years will it amount to 16 times itself?

**Compound Interest Shortcuts: ****Tooltip 2**

**If an amount of money grows up to Rs x in t years and up to Rs y in (t+1) years on compound interest, then**

**Derivation for this result:**

Principal + CI for t years = x …… (1)

Principal + CI for (t+1) years= y ……. (2)

(2) – (1) =>CI for last year = y-x

Which is basically the simple interest upon x

**Example-2: **An amount of money grows upto Rs 3000 in 3 years and upto Rs 4000 in 4 years on compound interest. What will be the rate percent?

**Solution:**

Principal + CI for 3 years = 3000 …… (1)

Principal + CI for 4 years= 4000 ……. (2)

Hence (2) – (1) =>CI for 4th year = 4000-3000= Rs 1000

Which is basically the simple interest upon 3000

**Compound Interest Shortcuts: Tooltip 3**

**A sum at a rate of interest compounded yearly becomes Rs. A _{1} in n years and Rs. A_{2} in (n + 1) years, then **

**Example-3: **A sum of money invested at compound interest amounts to Rs. 100 at the end of first year and Rs. 120 at the end of second year. The sum of money is :

**Solution:**

**Simple Interest for one year = compound interest for one year**

Interest on Rs. 100 for 1 year = 120-100= Rs. 20

**Compound Interest Shortcut: Tooltip 4**

**If a certain sum becomes x times of itself in t years, the rate of compound interest will be equal to**

**Derivation for this result:**

Use the formula for Compound Interest Calculation:

Sum becomes x times of itself in t years so

**Example 4: ** If a certain sum becomes 16 times in 2 years ,what will be the rate of compound interest?

**Solution:**

Using the formula derived above:

**Compound Interest Shortcut: Tooltip 5**

**If the compound interest on a certain sum for 2 years is CI and simple interest for two years is SI ,then rate of interest per annum is**

**Derivation for this result:**

**Example 5: **If the compound interest on a certain sum for 2 years is 20rs and simple interest for two years is 10rs ,then what wil be the rate of interest per annum ?

**Solution:**

Using the formula derived above:

**Compound Interest Solved Problems using Compound Interest Shortcuts:**

Let’s go through some compound interest solved problems and learn how to use and implement compound interest shortcuts in actual problem solving. Remember, till the time you actually solve questions using these tricks, you won’t be able to memorize and understand them. Go through compound interest solved problems and hone your skills for the topic.

**Question 1:** What sum of money at compound interest will amount to Rs. 650 at the end of the first year and Rs. 676 at the end of the second year?

A. Rs. 600

B. Rs. 600.25

C. Rs. 625

D. Rs. 625.25

**Answers and Explanations**

**Option C**Using the formula we have derived in the article for this exercise:

**If an amount of money grows up to Rs x in t years and up to Rs y in (t+1) years on compound interest, then**

**Question 2:** If the amount is 2.25 times of the sum after 2 years at compound interest (compound annually) , the rate of interest per annum is :

A. 25%

B. 30%

C. 45%

D. 50%

**Answers and Explanations**

**Option D**Using the formula we have derived in the article for this exercise:

**If a certain sum becomes x times of itself in t years, the rate of compound interest will be equal to**

**Question 3:** A sum of money doubles itself in 4 years at compound interest. It will amount to 8 times itself at the same rate of interest in:

A. 18 years

B. 12 years

C. 16 years

D. 24 years

#### Answes and Explanations

**Option B**Using the formula we have derived in the article for this exercise:

**A sum of money placed at compound interest becomes x time in ‘a’ years and y times in ‘b’ years. These two sums can be related by the following formula:**

**Question 4:** A sum borrowed under compound interest doubles itself in 10 years. When will it become fourfold of itself at the same rate of interest?

A. 15 years

B. 20 years

C. 24 years

D. 40 years

#### Answers and Explanations

**Option B**Using the formula we have derived in the article for this exercise:

**A sum of money placed at compound interest becomes x time in ‘a’ years and y times in ‘b’ years. These two sums can be related by the following formula:**

**Question 5:** A sum of money becomes eight times of itself in 3 years at compound interest. The rate of interest per annum is

A. 100%

B. 80%

C. 20%

D. 10%

**Answers and Explanations**

**Option A**Let the principal be Rs. x and the rate of compound interest be r% per annum.

Then,

**Ques. A sum of money lent at compound interest for 2 years at 20% per annum would fetch Rs.723 more, if the interest was payable half yearly than if it was payable annually. The sum is ____**

A.Rs. 20000

B.Rs. 15000

C.Rs. 30000

D.Rs. 45000

E.None of these

Answer & Explanation

Answer – C.Rs. 30000

Explanation :

sum – Rs.x

C.I. compounded half yearly = (4641/10000)x

C.I. compounded annually = (11/25)x

(4641/10000)x – (11/25)x = 723

x = 30000

**Ques. A sum of Rs.7140 is to be divided between Anita and Bala who are respectively 18 and 19 yr old, in such a way that if their shares will be invested at 4% per annum at compound interest, they will receive equal amounts on attaining the age of 21 year. The present share of Anita is**

A. 4225

B. 4352

C. 3500

D. 4000

E. None of these

Answer & Explanation

Answer – C. 3500

Explanation :

Amount got by Anita after 3 yr = Amount got by Bala after 2 yr

x*(26/25)³ = (7140 – x)*(26/25)

26/25 = 7140 – x / x

x = 3500

**Ques. Suresh borrows Rs.6375 to be paid back with compound interest at the rate of 4 % pa by the end of 2 year in two equal yearly installments. How much will each installment will be?**

A.3840

B.3380

C.4800

D.Data inadequate

E.None of these

Answer & Explanation

Answer – B.3380

Explanation :

25x/26 + 625/676x = 6375

x = (6375 * 676)/1275 = 3380

**Ques. A sum of Rs. 8400 was taken as loan. This is to be paid in two equal annual installments. If the rate of interest be 20% compounded annually, then the value of each installment is**

A. 5400

B. 5700

C. 5100

D. 5200

E. None of these

Answer & Explanation

Answer – A. 5400

Explanation :

Let value of each installment be X.

X/(1 + 20/100) + X/(1 + 20/100)² = 8400

⇒ X(5/6 + 25/36) = 8400

⇒ X(56/36) = 8400

X = 5400

**Ques. During the first year the population of a village is increased by 5% and the second year it is diminished by 5%. At the end of the second year its population was 31500. What was the population at the beginning of the first year?**

A. 35500

B. 31578

C. 33500

D. 33000

E. None of these

Answer & Explanation

Answer – B. 31578

Explanation :

x * 105/100 * 95/100 = 31500

x = 31500 * 100/105 * 100/95

D = 31578

**Ques. If Rs. 7200 amounts to Rs.10368 at compound interest in a certain time , then Rs. 7200 amounts to what in half of the time?**

A. 8640

B. 8600

C. 8800

D. 8520

E. None of these

Answer & Explanation

Answer – A. 8640

Explanation :

Let rate = R% and time = n year

Then, 10368 =7200(1+R/100)n

⇒ (1+R/100)n = 10368/7200 = 1.44

∴ (1 + R/100)n/2 = √1.44 = 1.2

∴ Required amount for n/2 yr

= 7200(1+ R/100)n/2

= 7200 x 1.2 = Rs. 8640

**Ques. A part of 70000 is lent out at 10% annum. The rest of the amount is lent out at 5% per annum after one year. The ratio of interest after 3 years from the time when first amount was lent out is 1:2. Find the second part that was lent out at 5%.**

A.40000

B.50000

C.60000

D.48000

E.55000

Answer & Explanation

Answer – C.60000

Explanation :

10*3*x/5*2*y = 1/2

x/y = 1/6

6/7*70000 = 60000

**Ques. There is 50% increase in an amount in 5 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?**

A.Rs. 2255

B.Rs. 2792

C.Rs. 3580

D.Rs. 3972

E.None of these

Answer & Explanation

Answer – D.Rs. 3972

Explanation :

In S.I,

Let P=100, I=50, T=5 yrs

R = 50*100/100*5 = 10%

In C.I,

P = 12000, T=3 yrs, R= 10%

C.I = [12000*(1 + 10/100)^3 – 1 ] C.I = 3972.

**Ques. Karthik lends a certain amount to Vignesh on simple interest for two years at 20%. Vignesh gives this entire amount to Kamal on compound interest for two years at the same rate annually. Find the percentage earning of Vignesh at the end of two years on the entire amount.**

A.3%

B.3(1/7)%

C.4%

D.5(6/7)%

E.None of these

Answer & Explanation

Answer – C.4%

Explanation :

SI=20*2=40%

CI=20+20+(400/100)=44%

Diff = 44-40=4%

**Ques. A man borrows 3000 rupees at 10% compound interest. At the end every year he pays rupees 1000 back. How much amount should he pay at the end of the fourth Year to clear all his debt?**

A.Rs. 680.5

B.Rs. 651.3

C.Rs. 751.3

D.Rs. 790.3

E.None of these

Answer & Explanation

Answer – C.Rs. 751.3

Explanation :

After one year amount = 3000 *110/100 = 3300

He pays 1000 back, so remaining = 3300-1000 = 2300

After two year amount = 2300 *110/100 = 2530

He pays 1000 back, so remaining = 2530-1000 = 1530

After three year amount = 1530*110/100 = 1683

He pays 1000 back, so remaining = 1683-1000= 683

After fouth year = 683 * 110/100 = 751.3

**Ques. Rahul saves an amount of 800 every year and then lent that amount at an interest of 10 percent compounded annually. Find the amount after 3 years.**

A.Rs. 1822.8

B.Rs. 2252

C.Rs. 2550.50

D.Rs. 2912.8

E.None of these

Answer & Explanation

Answer – D.Rs. 2912.8

Explanation :

800*(11/10)³=1064.8

800*(11/10)²=968

800*(11/10)=880

Total amount =2912.8