Question

Find the difference between compound interest on rupees 8000for 1 1/2 years at 10% per annum when compounded annually and semi annually.​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{Difference=21\:rupees}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given: }} \\ \tt: \implies Principal(p) = 8000\:rupees \\ \\ \tt: \implies Time(t) = 18 \:months\\ \\ \tt: \implies Rate\%(r) = 10\% \\ \\ \red{\underline \bold{To \: Find: }} \\ \tt: \implies C.I\:half\:yearly- C.I\:annualy=?$

• According to given question :

$\bold{As \: we \: know \: that(For \: 1 \: year)} \\ \tt: \implies A =p(1 + \frac{r}{100} )^{t} \\ \\ \tt: \implies A= 8000 \times (1 + \frac{10}{100} )^{1} \\ \\ \tt: \implies A= 8000 \times 1.1 \\ \\ \green{\tt: \implies A =8800\: rupees} \\ \\ \bold{For \: next \: 6 \: months : } \\ \tt: \implies Interest = \frac{8800 \times 10 \times 6}{12 \times 100} \\ \\ \green{\tt: \implies Interest = 440 \: rupees} \\ \\ \bold{For \: Compoud \: interest : } \\ \tt: \implies C.I_{1} = a + interest - p \\ \\ \tt: \implies C.I_{1}= 8800 + 440 - 8000 \\ \\ \green{\tt: \implies C.I_{1}= 1240 \: rupees}$

$\bold{As \: we \: know \: that} \\ \tt: \implies A= p(1 + \frac{ \frac{r}{2} }{100} ) ^{2t} \\ \\ \tt: \implies A = 8000 \times (1 + \frac{10}{2 \times 100} )^{3} \\ \\ \tt: \implies A= 8000 \times (1 + 0.5)^{3} \\ \\ \tt: \implies A = 8000 \times (1.05)^{3} \\ \\ \tt: \implies A= 8000 \times 1.157625 \\ \\ \green{\tt: \implies A=9261\: rupees} \\ \\ \bold{For \: compound \: interest : }\\ \tt: \implies C.I_{2}= A - p \\ \\ \tt: \implies C.I_{2}= 9261- 8000\\ \\ \green{\tt: \implies C.I_{2}= 1261\: rupees} \\ \\ \bold{For \: Difference : } \\ \tt: \implies Difference= C.I_{2}- C.I_{1}\\ \\ \tt: \implies Difference = 1261- 1240 \\ \\ \green{\tt: \implies Difference= 21 \: rupees}$

Answer 2

$\huge\sf{Answer:}$

☆ According to the question:

⇏ Find the difference between compound interest on rupees 8000for 1-1/2 years at 10% per annum when compounded annually and semi annually.

☆ Know terms:

⇏ Amount = (A)

⇏ Principal = (P)

⇏ Rate = (R)

⇏ Compound interest = (CI)

⇏ Difference = (D)

⇏ Interest = (I)

⇏ Number of years (Time) = n

☆ Using formula:

⇏ $\sf A = P ( \dfrac{1 + R}{100})^n$

☆ Finding the amount for 1 year:

⇏ $\sf A = 8000 \times ( \dfrac{1 + 10}{100})^1$

⇏ $\sf A = 8000 \times 1.1$

⇏ ${\sf{\boxed{\sf{ A = 8800}}}}$

Therefore, 8800 is the amount for a year.

☆ Finding the amount for next 6 months:

⇏ $\sf I = ( \dfrac{8800 \times 10 \times 6}{12 \times 100})$

⇏ ${\sf{\boxed{\sf{I = 440}}}}$

Therefore, 440 is the amount for next 6 months.

☆ Note:

⇏ Let us assume CI_1 as compound interest half yearly and CI_2 as the Compound interest yearly.

☆ Now, for CI half-yearly:

⇏ $\sf CI_1 = [(A + I) - P]$

⇏ $\sf CI_{1} = [(8800 + 440) - 8000]$

⇏ ${\sf{\boxed{\sf{ CI_1 = 1240}}}}$

Therefore, 1240 is CI half-yearly.

☆ Using formula:

⇏ $\sf A = P ( \dfrac{1 + \frac{R}{2} }{100} ) {}^{2n}$

⇏ $\sf A = 8000 \times ( 1+\dfrac{10}{2 \times 100} )^3$

⇏ $\sf A = 8000 \times (1 + 0.5)^3$

⇏ $\sf A = 8000 \times 1.157625$

⇏ ${\sf{\boxed{\sf{A = 6261}}}}$

☆ Now, for CI yearly:

⇏ $\sf CI = (A - P)$

⇏ $\sf CI = (9261 - 8000)$

⇏ ${\sf{\boxed{\sf{CI = 1261}}}}$

☆ Finding difference between CI_1 and CI_2:

⇏ $\sf D = CI_1 - CI_2$

⇏ $\sf D = 1261 - 1240$

⇏ ${\sf{\boxed{\sf{ D = 21}}}}$

Therefore, 21 is the difference.