how to solve compound interest word problems

Compound Interest Word Problems

How To Solve Compound And Simple Interest Problems Quickly

February 14, 2023

How To solve Compound Interest And Simple Interest Problems Quickly

The Basic Difference Between Simple Interest (SI) and Compound Interest(CI) is that the Simple Interest is the interest calculated on the Principal or sum of amount while Compound Interest is calculated on the principal amount and the previous earned interest also i.e.

SI = {\frac{P × R × T}{100}} and CI = {P (1+ \frac{R}{100\times n})^{nT} – P}

How to solve Compound Interest and Simple interest Problem (2)

Definition of Simple and Compound Interest

Type 2: Solve Simple Interest and Compound Interest Quickly. Find the amount/time/rate of interest when CI or SI or their difference is given

Question 1. The difference between the CI and SI on a certain amount is at 10% p.a. for 3 years is Rs. 31. Find the principal?

Solution The difference between compound interest and simple interest for three years is 31.

On solving further, we get

Correct option: A

Question 2. If the SI on a sum of money for 2 years at 5% p.a. is Rs. 500, what is the CI on the same sum at the same rate and for the same time?

Solution Sum = \frac{500 × 100 }{2 × 5}

Sum = \frac{50000 }{10}

Amount = 5000 (1+ \frac{5}{100})^{2}

5000 × 1.05 × 1.05 = 5512.5

CI = 5512.5 – 5000

Question 3. The difference between CI and SI on a principal of Rs. 15,000 for two years is Rs. 24. What is the annual rate of interest?

Solution CI – SI = \frac{P × (R)^2}{(100)^2}

On solving further we get,

24 × (100)^{2} = 15000 × R 2

Correct option: B

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COMPOUND INTEREST PROBLEMS

$800 is invested in compound interest where the rate of interest is 20% per year. If interest is compounded half yearly, what will be the accumulated value and compound interest after 2 years?

The formula to find accumulated value in compound interest is

A = P(1 + r/n) nt

A ----> Accumulated value (final value)

P ----> Principal (initial value of an investment)

r ----> Annual interest rate (in decimal)

n ----> Number of times interest compounded per year

t ----> Time (in years)

Then, the accumulated value is

A = 800(1 + 0.1) 4

A = 800(1.1 ) 4

A = 800 ⋅ 1.4641

Compound interest is

= 1171.28 - 800

A sum of money placed at compound interest doubles itself in 3 years. In how many years will it amount to four times itself?

Let P be the amount invested initially. From the given information,

P becomes 2P in 3 years

Because the investment is in compound interest, the principal in the 4th year will be 2P. And 2P becomes 4P (it doubles itself) in the next 3 years.

Therefore, at the end of 6 years accumulated value will be 4P. So, the amount deposited will amount to 4 times itself in 6 years.

The compound interest and simple interest on a certain sum for 2 years is $ 1230 and $ 1200 respectively. The rate of interest is same for both compound interest and simple interest and it is compounded annually. What is the principle?

Simple interest for two years is 1200 and interest for one year is 600.

So, compound interest for 1st year is 600 and for 2nd year is 630.

(Since it is compounded annually, simple interest and compound interest for 1st year would be same)

When we compare the compound interest for 1st year and 2nd year, it is clear that the interest earned in 2nd year is 30 more than the first year.

Because, interest 600 earned in 1st year earned this 30 in 2nd year.

It can be considered as simple interest for one year.

That is, principal = 600, interest = 30.

30 = (600 ⋅ r ⋅ 1)/100

In the given problem, simple interest earned in two years is 1200.

1200 = (P ⋅ 5 ⋅ 2)/100

Multiply each side by 10.

So, the principal is $ 12,000.

Mr. David borrowed $15,000 at 12% per year compounded annually. He repaid $7000 at the end of 1st year. What amount should he pay at the end of second year to completely discharge the load?

A = P(1 + r/n) nt

To find the accumulated value for the first year,

Substitute P = 15000, r = 0.12, n = 1 and t = 1 in the above formula.

A = 15000(1 + 0.12/1) 1x1

= 15000(1 + 0.12) 1

Amount paid at the end of 1st year is 7000.

Balance to be repaid :

This 9800 is going to be the principal for the 2 nd year.

Now we need the accumulated value for the principal 9800 in one year. (That is, at the end of 2nd year)

A = 9800(1 + 0.12/1) 1x1

= 9800(1 + 0.12) 1

= 9800 ⋅ 1.12

So, to completely discharge the loan, at the end of 2nd year, Mr. David has to pay $ 10,976.

There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of $12,000 after 3 years at the same rate?

Let the principal in simple interest be $100.

Since there is 60% increase, simple interest = 60

We already know the formula for S.I.

Here, I = 60, P = 100, t = 6.

60 = (100 ⋅ r ⋅ 6)/100

Divide each side by 6.

Because the rate of interest is same for both S.I and C.I, we can use the rate of interest 10% in C.I.

To know compound interest for 3 years,

Substitute P = 12000, r = 0.1, n = 1 and t = 3 in the formula of C.I.

A = 12000(1 + 0.1/1) 1x3

= 12000(1 + 0.1) 3

Compound interest = A - P

= 15972 - 12000

So, the compound interest after 3 years at the same rate of interest is $3972.

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Question Video: Solving Word Problems Involving Compound Interest Mathematics • 9th Grade

how to solve compound interest word problems

Bank A offers depositors 4% annual interest compounded once per year. Bank B offers 3.93% per year, compounded monthly. Write an explicit formula for the return 𝑅 after 𝑛 years on a deposit of 𝑅₀ dollars with both offers. Which bank’s offer is better?

Video Transcript

Bank A offers depositors four percent annual interest compounded once per year. Bank B offers 3.93 percent per year compounded monthly. Write an explicit formula for the return 𝑅 after 𝑛 years on a deposit of 𝑅 sub zero dollars with both offers. Which bank’s offer is better?

So in this question, we’re given information about the interest rates for bank A and bank B. Bank A offers a higher interest rate of four percent, and that’s compounded only once per year, whereas bank B offers an interest rate of 3.93 percent, but that’s compounded monthly, We’re asked to do two things in this question. The first thing is to write a formula for bank A and then for bank B. And secondly, we need to work out which of the banks’ offers would be better.

So let’s start by looking at how we would write a formula. And we’ll start with bank A. Both of these banks will apply compound interest. And in bank A, that compound interest is applied once per year, so that means that we can use this simpler compound interest formula: 𝑉 equals 𝑃 times one plus 𝑟 to the power of 𝑦. In this formula, 𝑉 represents the value of the investment, 𝑃 is the principal or starting amount, 𝑟 is the interest rate, and 𝑦 is the number of years the money is invested for.

So let’s use the information in the question to work out what the values of 𝑉, 𝑃, 𝑟, and 𝑦 will be for bank A. Given that we’re asked to write a formula for the return 𝑅, then that means that our value of 𝑉 can be given as 𝑅. Notice that that’s a capital 𝑅 to make it different from the interest rate, which is a lowercase 𝑟. Next, we’re told that this formula applies for a deposit of 𝑅 sub zero dollars. So that means that our principal amount will be 𝑅 sub zero.

Then we have the annual interest rate as four percent. So that means that our lowercase 𝑟 is four percent. Finally, we’re given that this money will be invested for 𝑛 years, so that means that our number of years 𝑦 is equal to 𝑛.

When we fill these values into the formula, we need to take care with the interest rate, remembering that four percent is four over 100 or 0.04. So here, we have a formula in terms of the return 𝑅. It’s 𝑅 equals 𝑅 sub zero times one plus 0.04 to the power of 𝑛. And of course, we could further simplify within the parentheses to give us a simple formula if we wished.

But now let’s have a look at a formula for bank B. Because we’re told that bank B’s interest is compounded more than once a year, then we need to use a slightly different compound interest formula. This formula is often written as 𝑉 equals 𝑃 times one plus 𝑟 over 𝑛 to the power of 𝑛𝑦, where 𝑉 is the value of the investment, 𝑃 is the principal amount, 𝑟 is the interest rate, 𝑛 is the number of times per year that the interest is compounded, and 𝑦 is the number of years that the money is invested for.

A handy tip at this point is that we might notice that we could get confused with this given value of 𝑛 in the question. We have an 𝑛 in the formula, so let’s change this to a different letter. We can choose the letter 𝑥 for example. So now in the formula, we have this value of 𝑥 in the parentheses and in the exponent. And that 𝑥 just represents the number of times per year that the interest is compounded.

And so just like we did with bank A, let’s find the values for each of these letters using the information given in the question. The value of the investment is simply the return 𝑅. Then we have the same starting amount of 𝑅 sub zero dollars. Next, the interest rate is 3.93 percent. For the value of 𝑥, that’s the number of times per year that the interest is compounded. And while we’re not specifically told that, we are given that the interest is compounded monthly. And since there are 12 months in a year, then that means that our value of 𝑥 must be equal to 12. And finally then, just like for bank A, the number of years is given as 𝑛, so 𝑦 is equal to 𝑛.

Before we put these values into the formula, it can be helpful to remember that the interest rate is 3.93 percent, which is 3.93 over 100. Or as a decimal, it’s 0.0393. When we fill these values into the formula, we get 𝑅 equals 𝑅 sub zero times one plus 0.0393 over 12 to the power of 12𝑛. And so that’s the first part of the question answered. We have two formulas which give the return 𝑅 after 𝑛 years on a deposit of 𝑅 sub zero dollars.

Before we look at the second part of the question, there’s one other thing that we can point out. In this second formula, which we use for bank B, we could have also used this to create a formula for the annual interest in bank A. In bank A, the number of times per year that the interest was compounded is one. So that would make each value here of 𝑥 equal to one. When 𝑥 is equal to one, that would give us the same formula that we used for bank A.

But now let’s clear some space to work out which of these banks has the better offer. In order to identify the bank with the better offer, that is, the one which gives us the higher return on the investment, then let’s see what happens after one year.

The value of 𝑛 represents the number of years, so let’s substitute 𝑛 equals one into both formulas. For bank A, we have 𝑅 equals 𝑅 sub zero times 1.04 to the power of one. This simplifies to 𝑅 equals 1.04𝑅 sub zero. For bank B, when we substitute 𝑛 equals one, we get 𝑅 equals 𝑅 sub zero times 1.003275 to the power of 12. When we simplify it and round this coefficient of 𝑅 sub zero to five decimal places, we get 𝑅 equals 1.04002𝑅 sub zero. And so when we compare what we get for bank A and bank B, we can see that the coefficient of 𝑅 sub zero in bank B is a little bit higher.

It may be helpful here to illustrate this with an example for 𝑅 sub zero. Let’s say that the starting amount, 𝑅 sub zero, is 10,000 dollars. That’s what’s invested in both of the banks. That would mean that the return of the investment 𝑅 for bank A would be 10,400 dollars. Then if we invested the same amount of money in bank B, the return on the investment 𝑅 would be 10,400 dollars and 20 cents. So even though it’s just by a small amount, bank B would be better.

Remember that we substituted the number of years to be equal to one in order to compare both banks after one year. Each year that money is invested in either bank, that will mean that the bank B return on investment will get proportionally larger each year. And so we can give the answer for the second part of the question that bank B has the better offer.

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How to solve Compound Interest Problems

How to find compound interest / how to calculate compound interest using formula, sharing buttons:.

00:00 hello everyone today in this video we

00:02 are going to learn how to find compound

00:05 using formula so the formulas we are

00:08 going to use

00:09 i will be solving two examples in this

00:13 and the formulas we are going to use are

00:15 the first one

00:16 is when the interest is compounded

00:19 annually or

00:20 yearly the second one is when the

00:23 interest is compounded half yearly

00:25 we are going to solve one example of

00:27 each so the first formula is

00:30 amount is equal to principal time

00:33 1 plus r over 100 whole raised to power

00:37 n where a is the amount p

00:40 is the principle r is the rate of

00:43 and n is the number of years so when we

00:47 get the amount

00:48 we are going to use the formula for

00:50 compound interest which is

00:52 amount minus principle so we will get

00:55 compound interest

00:58 the next one is when the interest is

01:01 compounded half yearly

01:03 so when the interest is compounded half

01:05 yearly half yearly means

01:06 so there will be two half years in one

01:09 year so that

01:10 the n will become 2 n and the rate of

01:13 interest will become

01:14 half which is r over 2 so our formula

01:18 will get changed to

01:19 a is equal to p times 1 plus

01:23 r over 200 whole raised to power

01:27 to n with the use of the formula

01:30 amount minus principle we can get the

01:34 interest so let's get started with our

01:37 example where the interest is compounded

01:41 yearly let's solve our first example

01:44 where our principal amount is 6000

01:48 and it is been kept or it is being

01:50 compounded for two years

01:52 with the rate of interest of 9 annually

01:56 so we need to find the compound interest

01:59 so we'll use the formula for amount

02:02 which is a is equal to p

02:05 times 1 plus r over 100

02:09 raised to power n

02:12 and after finding the amount we can find

02:14 the compound interest

02:16 using the formula compound interest is

02:19 equal to amount minus principle

02:22 so our principle is 6000 rupees

02:26 so we'll put 6000 p is 6000

02:29 now we'll put the values in the formula

02:33 is 6000 1 plus 9

02:36 over 100 raised to power

02:39 2 so amount is

02:42 equal to 6000

02:46 this will give us 100 plus 9

02:50 over 100 raised to power 2

02:56 then this is 6 000 so this is 109

03:00 over 100 square

03:05 and after solving this gets multiplied

03:08 two times

03:13 and hundred times

03:16 hundred so our amount will come out to

03:22 this gets cancelled three zeros gets

03:26 cancelled

03:27 so we are left we are left with six

03:32 times 109 times 109

03:36 over 10. so amount comes out to be

03:41 after solving this comes out to be

03:45 rupees 7128.6

03:52 so this is our amount now we need to

03:55 compound interest so compound interest

04:00 is equal to amount minus principle

04:06 so amount is 7128.6

04:12 minus the principal amount is 6000

04:16 so our compound interest comes out to be

04:20 rupees 1128.6

04:26 so this is compound interest when sixth

04:30 house principal amount of six

04:31 thousand is compounded annually for two

04:34 years with a rate of interest of nine so

04:36 these are the steps you need to follow

04:39 to find the compound interest using the

04:43 when the amount is compounded annually

04:46 let's take one more example where the

04:48 amount is compounded

04:50 semi-annually or half yearly so this is

04:53 our second example

04:54 where the we have to find the compound

04:56 interest when the principal of 8000

04:59 rupees at a rate of 10 percent per annum

05:02 and for one year is compounded half

05:06 so let's use our formula which we have

05:09 discussed earlier

05:10 amount will become a is equal to

05:13 p times 1 plus r

05:17 over 200 200 because it is

05:21 compounded half yearly raise to power

05:24 2 n so this is our formula for

05:28 amount is compounded half yearly so

05:31 we'll substitute or put the values here

05:38 one plus r is ten percent

05:43 over two hundred raised to power two

05:46 one so this will become eight

05:50 thousand 1 plus

05:54 1 over 20 raised to power 2

05:59 so this is going to be 8 000

06:02 20 plus 1 over

06:05 twenty square

06:09 or eight thousand times

06:12 twenty one over twenty square

06:21 so this will be 8 000

06:25 times 21 times 21

06:29 over 20 times 20

06:33 this is our amount so amount is equal to

06:37 the zeros gets cancelled

06:40 2 times 4 and 2 times

06:44 2 so this will be 20

06:47 times 21 times 21 amount will become

06:52 8 8 to 0 rupees

06:55 so this is our amount and we need to

06:58 find the compound

07:00 interest which will be amount minus the

07:03 principal amount

07:05 so our amount 8820

07:08 minus the principal is 8000

07:13 so compound interest will be 820

07:19 rupees so this is our final answer

07:23 when the interest is compounded half

07:27 so these are the steps you need to

07:28 follow to find the compound interest

07:31 when the interest is compounded half

07:34 i hope this is helpful to you thanks for

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Compound Interest - Corbettmaths

Interesting topics

Compound interest word problems

We will use the compound interest formula to solve these compound interest word problems.

Example #1 A deposit of $3000 earns 2% interest compounded semiannually. How much money is in the bank after for 4 years? Solution B = P( 1 + r) n P = $3000 r = 2% annual interest rate / 2 interest periods = 1% semiannual interest rate n = number of payment periods = number of interest periods times number of years

n = 2 times 4 = 8

B = 3000( 1 + 1%) 8 = 3000(1 + 0.01) 8 = 3000(1.01) 8 B = 3000(1.082856) B = 3248.57

After four years, there will be 3248.57 dollars in the bank account.

Example #2 A deposit of $2150 earns 6% interest compounded quarterly. How much money is in the bank after for 6 years? Solution B = P( 1 + r) n P = $2150 r = 6% annual interest rate / 4 interest periods = 1.5% quarterly interest rate n = number of payment periods = number of interest periods times number of years

n = 4 times 6 = 24

B = 2150( 1 + 1.5%) 24 = 2150(1 + 0.015) 24 = 2150(1.015) 24 B = 2150(1.4295) B = 3073.425

After 6 years, there will be 3073.425 dollars in the bank account.

Example #3 A deposit of $495 earns 3% interest compounded annually. How much money is in the bank after for 3 years? Solution B = P( 1 + r) n P = $495 r = 3% annual interest rate / 1 interest period = 3% annual interest rate n = number of payment periods = number of interest periods times number of years

n = 1 times 3 = 3

B = 495( 1 + 3%) 3 = 495(1 + 0.03) 3 = 495(1.03) 3 B = 495(1.092727) B = 540.89

After 3 years, there will be 540.89 dollars in the bank account.

Compound interest

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COMMENTS

How to Solve Compound Interest Word Problems
Compound Interest Word Problems:Videos:YouTube - Video. Text Lessons:CK-12 - Text Lesson. Purple Math - Text Lesson. Worksheets:Khan Academy - Practice
How To Solve Compound and Simple Interest Problems Quickly
The Basic Difference Between Simple Interest (SI) and Compound Interest(CI) is that the Simple Interest is the interest calculated on the Principal or sum of amount while Compound Interest is calculated on the
Compound Interest Problems
Because the investment is in compound interest, the principal in the 4th year will be 2P.And 2P becomes 4P (it doubles itself) in the next 3 years. The compound interest and simple interest on a certain sum for 2 years is $ 1230 and $ 1200 respectively
Question Video: Solving Word Problems Involving Compound Interest
Bank A offers depositors 4% annual interest compounded once per year. And in bank A, that compound interest is applied once per year, so that means that we can use this simpler compound interest formula: 𝑉
How to solve compound interest problems
3 Answers. use the formula (and its variations) relating P_0 the beginning principal, “r” the percent interest rate, and P_c the current principal after time T in terms of compounding period (typically in months or years)
Compound Interest Word Problems and Solutions
We will use the compound interest formula to solve these compound interest word problems. Example #1 A deposit of $3000 earns 2% interest compounded semiannually