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PREPINSTA PRIME

# Formulas for Simple Interest And Compound Interest

** Formulas To Find Averages In Aptitude**

### Definition of Simple Interest

- The interest calculated on the amount initially invested or loaned. It is a method for calculating the interest earned or paid on a certain balance in a specific period.
- Simple interest is a quick and easy method of calculating the interest on a sum of Amount. It is determined by
**multiplying the daily interest rate by the principal amount and the number of days .**

### Definition Of Compound Interest

- Compound interest is the addition of interest to the principal sum of a loan or deposit. Compound interest is calculated based on the principal, interest rate, and the time period involved
**.** **It is the addition of interest to the sum of Amount or Principal Amount i.e. interest on interest**. It is the result of reinvesting interest.So that interest in the next period is then earned on the principal amount and previously accumulated interest.

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### Formula for Compound Interest

- Interest Compounded Annually
**Amount =**\mathbf{P\left ( 1+\frac{R}{100n} \right )^{nT}}- Compound Interest = Total amount – Principal
- Rate of interest (R)(in %) = n[(\frac{A}{P})^{\frac{1}{nT}} – 1]

### Interest Compounded Half-Yearly

- When interest is compounded Half yearly Then, we must consider n=2, Hence, Formula for Amount = \mathbf{P\left ( 1+\frac{R}{100\times 2} \right )^{2T}}
- Compound Interest = Total amount – Principal
- Rate of interest (R) (in %)= 2(P^{\frac{1}{2T}} – 1)

### Interest Compounded Quarterly

- We have to consider n=4. So, Amount = \mathbf{P\left ( 1+\frac{R}{100\times 4} \right )^{4T}}
- Compound Interest = Total amount – Principal
- Rate of interest (R) (in %)= 4(P^{\frac{1}{4T}} – 1)

### Interest is Compound Monthly

- When the interest is compounded montly then, n=12. So, formula for Amount = \mathbf{P\left ( 1+\frac{R}{100\times 12} \right )^{12T}}

### Interest is Compounded Annually but Time is in Fraction, say 2(3/2) years

- When the Interest is Compounded Annually but Time is in Fraction. Then, the formula Amount = P (1+r/100)
^{2 }* (1+(3/2)r/100)\mathbf{(1+ \frac{r}{100})^{2}}\mathbf{(1+ \frac{\frac{3r}{2}}{100})}

### CI when Rates are Different for Different Years

- When rates are different for different years . Then, Amount = P (1+ \frac{r_{1}}{100})(1+ \frac{r_{2}}{100})(1+ \frac{r_{3}}{100})

### Formula for Simple Interest

- Simple Interest

**SI** **=** \mathbf{\frac{P * R * T}{100}}

Where,

P = money borrowed or lent out for a certain period

r = rate of interest

t = time period for which the amount is lent

Principal = \frac{100 × SI}{R × T }

Rate = \frac{100 × SI}{P × T }

Time = \frac{100 × SI}{R × P }

### Total Amount of Money

- Amount = Principal + Interest
- A = P + I

**Question 1 : Vanraj shah borrowed a sum of Rs. 10,000 at 8% per annum compounded annually. If the amount is to be paid in three equal installments, the annual installment will be**

**Solution :** Let each installment be x,

10000=\frac{x}{(1+\frac{8}{100})} + \frac{x}{(1+\frac{8}{100})}^{2}+ \frac{x}{(1+\frac{8}{100})}^{3}

10000=x{\frac{25}{27} + (\frac{25}{27}) ^{2}+ (\frac{25}{27}) ^{3}}

=x*\frac{25}{27}(1 + \frac{25}{27} + \frac{625}{729})

=\frac{25x}{27} *(\frac{2029}{729})

x =3880.335

**Question 2 : A financier lends money at simple interest, but he includes the interest every six months for calculating the principal. If he is changing an interest of 20%, the effective rate of interest becomes?**

**Solution : **Let the sum be Rs. 100. Then,

S.I. for first 6 months = \frac{100 * 20 * \frac{1}{2})}{100} = Rs.10

Next 6 months 20% of 110

S.I. for last 6 months =Rs.\frac{110 * 20 * \frac{1}{2})}{100} = Rs.11

So, amount at the end of 1 year = Rs. (100 + 10 + 11) = Rs. 121 R

= (121 – 100) = 21%

**Question 3 : Raghav singh purchases a coat for Rs.2400 cash or for Rs.1000 cash down payments and two monthly installments of Rs.800 each. Find the rate of interest.**

**Solution :** Amount as a principal for 2 month = 2400 – 1000 = 1400

At the rate of r% per annum after 2 months,

Rs.1400 will amount to Rs 1400 + (\frac{1400*r*2}{100*12})

Total amount for 2 installments at the end of second month

Rs800+(800+(\frac{800*r*1}{100*12)}))

Then 1400 + \frac{2800*r}{1200}=1600+ \frac{800*r}{1200}

R=120%

**Question 4: Mohsin khan invested Rs. 20,000 in a scheme at simple interest @ 15% per annum. After three years he withdrew the principal amount plus interest and invested the entire amount in another scheme for two years, which earned him compound interest @ 12% per annum. What would be the total interest earned by Mosses at the end of 5 years?**

**Solution:** SI= 20,000*15*3/100=9000

Amount=20,000+9000=29,000

Now CI= 29,000*(1+12/100)^{2}

= 29,000 * 28/25 * 28/25 = 36,377.6

A-P=36, 377.6-29000=7377.6

After 5yrs 7377.6+9000=16,377.6

**Question 5 : Bobby deol invested his money for a certain time. It amounts to Rs. 600 at 10% per annum. But when invested at 5% per annum, it amounts to Rs. 400. Find the time. **

**Solution :** 600-P=P*10*\frac{t}{100} —>1

===>6000-10P=Pt

400-P=P*5*\frac{t}{100}—->2

===>8000-20P=Pt

Equate 1 and 2

6000-10P=8000-20P

==>P=200

Substitute P in 1 then

400=200*10*\frac{t}{100}

==>20yrs.

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