Question

The difference between the compound interest
and the simple interest on a certain sum of
money at 10% per annum for 3 years is Rs 496.
Find the sum when the interest is compounded
annually.
Sum of money = Rs​

Answer 1

ANSWER :

★ If the difference between the compound interest and the simple interest on a certain sum of money at 10% per annum for 3 years is Rs. 496, then the sum is Rs. 16000 when the interest is compounded annually.

$\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

SOLUTION :

❖ Given :-

• Difference between C.I. and S.I. = Rs. 496

• Rate of Interest, r = 10% p.a.

• No. of years, n = 3 years

❖ To Find :-

• The sum of money, P = ?

❖ Required Formulas :-

$\bigstar \: \: \: \boxed {\rm{ \: S.I. = \frac{ \: P \times r \times n \: }{100} \: }} \: \: \: \: \\ \\ \bigstar \: \: \: \boxed {\rm{ \: A = P \: (1 + \dfrac{r}{100}) {}^{n} }} \: \: \: \: \: \\ \\ \: \: \: \: \: \: \bigstar \: \: \: \boxed {\rm{ \: C.I. = P \: [(1 + \dfrac{r}{100} ) {}^{n} - 1 ]}}$

❖ Calculation of Simple Interest :-

$\bigstar \: \: \: \sf{ S.I. = \dfrac{ \: P \times r \times n \: }{100} } \\ \\ \sf{ \implies \: S.I. = \dfrac{P \times 10 \times 3}{100} } \\ \\ \sf{ \implies \: S.I. = \dfrac{ \: 30 \: P \: }{100} } \: \: \: \: \: \: \: \: \\ \\ \sf{ \therefore \: \: \underline{ \: S.I. = \dfrac{3P}{10} \: } }$

❖ Calculation of Compound Interest :-

$\bigstar \: \: \sf{ C.I. = P \: [(1 + \dfrac{r}{100} ) {}^{n} - 1 ]} \\ \\ \sf{\implies \: C.I. = P \: [(1 + \dfrac{10}{100} ) {}^{3} - 1 ]} \\ \\ \sf{\implies \: C.I. = P \: [(1 + \dfrac{1}{10} ) {}^{3} - 1 ]} \\ \\ \sf{\implies \: C.I. = P \: [( \dfrac{10 + 1}{10} ) {}^{3} - 1 ]} \\ \\ \sf{\implies \: C.I. = P \: [( \dfrac{11}{10} ) {}^{3} - 1 ]} \: \: \: \: \: \: \: \\ \\ \sf{\implies \: C.I. = P \: [( \dfrac{1331}{1000} )- 1 ]} \: \: \: \: \: \\ \\ \sf{\implies \: C.I. = P \: ( \dfrac{1331 - 1000}{1000})} \\ \\ \sf{\implies \: C.I. = P \: ( \dfrac{331}{1000})} \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{\therefore \: \: \underline{ \: C.I. = \frac{ \: 331 \: P \: }{1000 \: }} }$

According to Question,

$\: \: \: \: \bigstar \: \: \: \sf{C.I. - S.I. = 496} \\ \\ \sf{ \implies \: \dfrac{331P}{1000} - \frac{3P}{10} = 496} \: \: \\ \\ \sf{ \implies \: \dfrac{331P - 300P}{1000} = 496} \\ \\ \sf{ \implies \: \dfrac{31P}{1000} = 496} \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \implies \: 31P = 496 \times 1000} \: \: \\ \\ \sf{\implies \: 31P = 496000} \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \implies \: P= \frac{496000}{31} } \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf{ \: \therefore \: \: \underline{ \: P = 16000 \: }} \: \: \: \: \:$

• Hence, the sum of money, P = Rs. 16000.

$\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

VERIFICATION :

❖ We have :-

• Sum of money, P = Rs. 16000

• Rate of Interest, r = 10%

• No. of years, n = 3 years.

❖ To Verify :-

• Difference between C.I. and S.I. = Rs. 496

Now,

$\: \: \bigstar \: \: \sf{\: S.I. = \dfrac{3P}{10} \: } \: \: \: \: \: \: \: \: \: \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ = \dfrac{3 \times 16000}{10} } \\ \\ \: \: \: \: \: \: \: \: \: \: \sf{ = \dfrac{48000}{10} } \\ \\ \: \: \: \: \: \: \: \: \sf{ = 4800}$

And,

$\: \: \bigstar \: \: \sf{\: C.I.= \dfrac{331P}{1000} \: } \: \: \: \: \: \: \: \: \: \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ = \dfrac{331 \times 16000}{1000} } \\ \\ \: \: \: \: \: \: \: \: \: \: \sf{ = \dfrac{5296000}{1000} } \\ \\ \: \: \: \: \: \: \: \: \sf{ = 5296}$

$\sf{ \therefore \: \: Difference \: \: between \: \: C.I. \: \: and \: \: S.I. = C.I. - S.I.} \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ = Rs. \: 5296 - Rs. \: 4800 } \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{= Rs. \: 496}$

• Hence, Verified.