Question

d. 2,500
19. The present value of an annuity of 500, paid at the end of each quarter for 1 year at
the rate of interest 9% compounded quarterly, is approximately.
a. 1000​

Answer 1

${\underline{\large{\pmb{\frak{Appropriate \; question : }}}}}$

⋆ The amount paid when the principal of Rs. 500 when compounded quarterly for 1 year at the rate of interest 9% per annum approximately is

${\underline{\large{\pmb{\frak{Given\; that : }}}}}$

★ Principal = Rs.500

★ Time = 1 year

★ Rate = 9% per annum

${\underline{\large{\pmb{\frak{To\; Find : }}}}}$

★ The final amount when the given principal amount compounded quarterly - yearly.

${\underline{\large{\pmb{\frak{Let's \; understand \; the \;concept : }}}}}$

❍ Concept : Here, we have  the principal, rate of interest and the time let's use the formula to find the amount when compounded quarterly and find the required answer.

✦As we have been given the values of thee principal, time , rate of interest respectively let's substitute the values in the formula and find the amount.

${\underline{\large{\pmb{\frak{Using \; the \; concept : }}}}}$

✪ Formula to find the amount when compounded quarterly :

$\tt A = P \bigg [ 1 + \frac{r}{400} \bigg]^{4n}$

${\underline{\large{\pmb{\sf{ RequirEd \; Solution : }}}}}$

★ The amount to be paid at the end is Rs. 545 when compounded

${\underline{\large{\pmb{\frak{Full \; solution : }}}}}$

~ Using the above mentioned formula let's substitute the values of the principal which is Rs.500, the rate which is 9% per annum and the time which is 1 year

→ Formula :

$\tt A = P \bigg[ 1 + \frac{r}{400} \bigg]^{4n}$

→ Here,

• P denotes Principal
• R denotes rate of interest
• N denotes no. of years
• A denotes amount

~ Substituting the values we get

➟ Amount = P [ 1 + R/400 ] ^ 4n

➟ Amount = 500 [ 1 + 9/400 ] ^ 4*1

➟ Amount = 500 [ 400/400 + 9/400 ] ^ 4

➟ Amount = 500 [ 409/400 ] ^ 4

➟ Amount = 500 × 409/400 × 409/400 × 409/400 × 409/400

➟ Amount = 5 × 409/4 × 409/400 × 409/400 × 409/400

➟ Amount = 5 × 409 × 409 × 409 × 409 / 4 × 400 × 400 × 400

➟ Amount = 1399146648905/ 256000000

➟ Amount = 546.541  ≈ Rs.545

• Henceforth the amount to be paid is Rs.545 approx

${\underline{\large{\pmb{\frak{Additional \; Information : }}}}}$

• The formula to find the amount when compounded per annum

${:\implies}\bf A = P \bigg[ 1 + \frac{r}{100} \bigg]^{n}$

• The formula to find the amount when compounded half yearly

${:\implies}\bf A = P \bigg[ 1 + \frac{r}{200} \bigg]^{2n}$

• The formula to find the amount when compounded at different rate of interests

${:\implies}\bf A = P \bigg[ 1 + \frac{r_1}{100} \bigg] \bigg[ 1 + \frac{r_2}{100} \bigg] \bigg[ 1 + \frac{r_3}{100} \bigg]$

*Note : you can further subtract the principal from the amount to find the compound interest if needed

Answer 2

Answer:

{\underline{\large{\pmb{\frak{Appropriate \; question : }}}}}

Appropriatequestion:

Appropriatequestion:

⋆ The amount paid when the principal of Rs. 500 when compounded quarterly for 1 year at the rate of interest 9% per annum approximately is

{\underline{\large{\pmb{\frak{Given\; that : }}}}}

Giventhat:

Giventhat:

★ Principal = Rs.500

★ Time = 1 year

★ Rate = 9% per annum

{\underline{\large{\pmb{\frak{To\; Find : }}}}}

ToFind:

ToFind:

★ The final amount when the given principal amount compounded quarterly - yearly.

{\underline{\large{\pmb{\frak{Let's \; understand \; the \;concept : }}}}}

Let

′

sunderstandtheconcept:

Let

′

sunderstandtheconcept:

❍ Concept : Here, we have the principal, rate of interest and the time let's use the formula to find the amount when compounded quarterly and find the required answer.

✦As we have been given the values of thee principal, time , rate of interest respectively let's substitute the values in the formula and find the amount.

{\underline{\large{\pmb{\frak{Using \; the \; concept : }}}}}

Usingtheconcept:

Usingtheconcept:

✪ Formula to find the amount when compounded quarterly :

\tt A = P \bigg [ 1 + \frac{r}{400} \bigg]^{4n}A=P[1+

400

r

]

4n

{\underline{\large{\pmb{\sf{ RequirEd \; Solution : }}}}}

RequirEdSolution:

RequirEdSolution:

★ The amount to be paid at the end is Rs. 545 when compounded

{\underline{\large{\pmb{\frak{Full \; solution : }}}}}

Fullsolution:

Fullsolution:

~ Using the above mentioned formula let's substitute the values of the principal which is Rs.500, the rate which is 9% per annum and the time which is 1 year

→ Formula :

\tt A = P \bigg[ 1 + \frac{r}{400} \bigg]^{4n}A=P[1+

400

r

]

4n

→ Here,

P denotes Principal

R denotes rate of interest

N denotes no. of years

A denotes amount

~ Substituting the values we get

➟ Amount = P [ 1 + R/400 ] ^ 4n

➟ Amount = 500 [ 1 + 9/400 ] ^ 4*1

➟ Amount = 500 [ 400/400 + 9/400 ] ^ 4

➟ Amount = 500 [ 409/400 ] ^ 4

➟ Amount = 500 × 409/400 × 409/400 × 409/400 × 409/400

➟ Amount = 5 × 409/4 × 409/400 × 409/400 × 409/400

➟ Amount = 5 × 409 × 409 × 409 × 409 / 4 × 400 × 400 × 400

➟ Amount = 1399146648905/ 256000000

➟ Amount = 546.541 ≈ Rs.545

Henceforth the amount to be paid is Rs.545 approx

{\underline{\large{\pmb{\frak{Additional \; Information : }}}}}

AdditionalInformation:

AdditionalInformation:

The formula to find the amount when compounded per annum

{:\implies}\bf A = P \bigg[ 1 + \frac{r}{100} \bigg]^{n}:⟹A=P[1+

100

r

]

n

The formula to find the amount when compounded half yearly

{:\implies}\bf A = P \bigg[ 1 + \frac{r}{200} \bigg]^{2n}:⟹A=P[1+

200

r

]

2n

The formula to find the amount when compounded at different rate of interests

{:\implies}\bf A = P \bigg[ 1 + \frac{r_1}{100} \bigg] \bigg[ 1 + \frac{r_2}{100} \bigg] \bigg[ 1 + \frac{r_3}{100} \bigg]:⟹A=P[1+

100

r

1

][1+

100

r

2

][1+

100

r

3

]

*Note : you can further subtract the principal from the amount to find the compound interest if needed