The difference between compound interest and simple interest on a sum for 2 years at 20 per annum

The difference between the compound interest and simple interest on a certain sum of money at 10% per annum for 2 years is Rs.500. Find the sum when the interest is compounded annually.

Answer

Hint: Let the sum be \[x\] rupees. We know the compound interest \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] , we need to find P. we know simple interest formula \[S.I = P \times \dfrac{r}{{100}} \times T\] . We know compound interest is the difference between amount and principal amount. Since the difference between compound and simple interest is given we can find the value of \[x\] .

Complete step-by-step answer:
We know,
 \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] , where A is amount, R is rate of interest, n is number of times the interest is compounded per year.
 \[P = x\] , \[n = 2\] \[r = 10\] , substituting we get,
 \[ \Rightarrow A = x{\left( {1 + \dfrac{{10}}{{100}}} \right)^2}\]
 \[ \Rightarrow A = x{\left( {1 + \dfrac{1}{{10}}} \right)^2}\]
Taking L.C.M and simplifying we get,
 \[ \Rightarrow A = x{\left( {\dfrac{{10 + 1}}{{10}}} \right)^2}\]
 \[ \Rightarrow A = x{\left( {\dfrac{{11}}{{10}}} \right)^2}\]
We know that compound interest is the difference between the amount of money accumulated after n years and the principal amount.
 \[C.I = A - P\]
 \[ \Rightarrow C.I = x{\left( {\dfrac{{11}}{{10}}} \right)^2} - x\] .
Now to find the simple interest we have, \[S.I = P \times \dfrac{r}{{100}} \times T\]
Substituting the known values,
 \[ \Rightarrow S.I = x \times \dfrac{{10}}{{100}} \times 2\]
 \[ \Rightarrow S.I = x \times \dfrac{1}{{10}} \times 2\]
 \[ \Rightarrow S.I = \dfrac{x}{5}\]
Given the difference between compound and simple interest is 500
 \[ \Rightarrow C.I - S,I = 500\]
Substituting C.I and S.I we get
 \[ \Rightarrow x{\left( {\dfrac{{11}}{{10}}} \right)^2} - x - \dfrac{x}{5} = 500\]
Simple division \[\dfrac{{11}}{{10}} = 1.1\] and \[\dfrac{1}{5} = 0.2\] we get,
 \[ \Rightarrow x{(1.1)^2} - x - 0.2x = 500\]
 \[ \Rightarrow 1.21x - 1x - 0.20x = 500\]
 \[ \Rightarrow 0.21x - 0.20x = 500\]
Taking x as common,
 \[ \Rightarrow (0.21 - 0.20)x = 500\]
 \[ \Rightarrow 0.01x = 500\]
 \[ \Rightarrow x = \dfrac{{500}}{{0.01}}\]
Multiply numerator and denominator by 100.
 \[ \Rightarrow x = 50,000\]
That is \[P = 50,000\] .
 \[50,000\] Rupees is the sum when the interest is compounded annually.
So, the correct answer is “\[P = 50,000\]”.

Note: Here we used three formulas. Remember the formula for simple interest, compound interest and amount formula. We can also take P as P as it is, and solve for P. Above all we did is substituting the given data in the formula and simplifying. Principal amount is the initial amount you borrow or deposit.

Q.1.The difference in simple interest and compound interest on a certain sum of money in 2 years at 10 % p.a. is Rs. 50. The sum is

a) Rs. 10000

b) Rs. 6000

c) Rs. 5000

d) Rs. 2000

e) None of these

Q.2. The difference in simple interest and compound interest on a certain sum of money in 2 years at 18 % p.a. is Rs. 162. The sum is

a) Rs. 4000

b) Rs. 5200

c) Rs. 4250

d) Rs. 5000

e) None of these

Q.3. The compound interest on a certain sum of money for 2 years is Rs. 208 and the simple interest for the same time at the same rate is Rs. 200. Find the rate %.

a) 5 %

b) 6 %

c) 7 %

d) 4 %

e) 8 %

Q.4.The difference between compound interest and simple interest on a certain sum for 2 years at 10 % is Rs. 25. The sum is

a) Rs. 1200

b) Rs. 2500

c) Rs. 750

d) Rs. 1250

e) Rs. 2000

Q.5.The simple interest on a certain sum for 3 years in Rs. 225 and the compound interest on the same sum for 2 years is Rs. 165. Find the rate percent per annum.

a) 20 %

b) 2.5 %

c) 5 %

d) 15 %

e) 7.5%

Q.6.The simple interest on a sum of money for 2 years is Rs. 150 and the compound interest on the same sum at same rate for 2 years is Rs. 155. The rate % p.a. is

a) 16 %

b) 20/3 %

c) 12 %

d) 10 %

e) None of these

Q7.Mihir’s capital is 5/4 times more than Tulsi’s capital. Tulsi invested her capital at 50 % per annum for 3 years (compounded annually). At what rate % p.a. simple interest should Mihir invest his capital so that after 3 years, they both have the same amount of capital?

a) 20/3 %

b) 10 %

c) 50/3 %

d) 1.728 %

e) None of these

Q8.The difference in simple interest and compound interest on a certain sum of money in 3 years at 10 % p.a. is Rs. 372. The sum is

a) Rs. 8000

b) Rs.9000

c) Rs. 10000

d) Rs. 12000

e) None of these

Q9.Sahil’s capital is 1/6 times more than Chaya’s capital. Chaya invested her capital at 20 % per annum for 2 years (compounded annually). At what rate % p.a. simple interest should Sahil invest his capital so that after 2 years, they both have the same amount of capital?

a) 10%

b) 11 5/7%

c) 20%

d) 13 5/7%

e) None of these

Q10.The difference in simple interest and compound interest on a certain sum of money in 3 years at 20 % p.a. is Rs. 640. The sum is

a) Rs. 5000

b) Rs. 8500

c) Rs. 8250

d) Rs. 6000

e) None of these

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