Mixing
Value of the last installment.
$begingroup$
The price of a TV set is ₹20,000 to be paid in 20 installments of ₹1,000 each. Rate of interest=6% p.a
The first installment is to be paid at the time of purchase, then what will be the value of the last installment covering the interest as well?
There are two explanations I found:
- Money paid in cash = ₹ 1000
Balance payment = (20000 - 1000) = ₹ 19000
(the above is from R.S. Aggarwal book and doesn't seem complete) - These links explain and conclude the answer to be 1950 but I didn't exactly understand: link 1 and link 2
I thought something like below (assuming monthly installments, hence taking 6/12 as rate of interest):
- Value after $20$ installments $= 20000 + (20000cdot 20cdot 6)/(12cdot 100) = 22000$
- So, value after 19 installments $=
1000+(1000cdot 20cdot 6)/(12cdot 100) +....+ 1000+(1000cdot 2cdot 6)/(12cdot 100)
= 19000 + (1000cdot 6/(12cdot 100))cdot(19cdot 20/2) = 19950$
- Thus, value of last installment $= 22000-19950=2050$
Can someone explain what am I doing wrong?
finance
edited Jan 21 at 20:54
alexjo
12.5k1430
asked Jan 20 at 11:13
ANANYA AKHOURIANANYA AKHOURI
81
$endgroup$
add a comment |
$begingroup$
The price of a TV set is ₹20,000 to be paid in 20 installments of ₹1,000 each. Rate of interest=6% p.a
The first installment is to be paid at the time of purchase, then what will be the value of the last installment covering the interest as well?
There are two explanations I found:
- Money paid in cash = ₹ 1000
Balance payment = (20000 - 1000) = ₹ 19000
(the above is from R.S. Aggarwal book and doesn't seem complete) - These links explain and conclude the answer to be 1950 but I didn't exactly understand: link 1 and link 2
I thought something like below (assuming monthly installments, hence taking 6/12 as rate of interest):
- Value after $20$ installments $= 20000 + (20000cdot 20cdot 6)/(12cdot 100) = 22000$
- So, value after 19 installments $=
1000+(1000cdot 20cdot 6)/(12cdot 100) +....+ 1000+(1000cdot 2cdot 6)/(12cdot 100)
= 19000 + (1000cdot 6/(12cdot 100))cdot(19cdot 20/2) = 19950$
- Thus, value of last installment $= 22000-19950=2050$
Can someone explain what am I doing wrong?
finance
edited Jan 21 at 20:54
alexjo
12.5k1430
asked Jan 20 at 11:13
ANANYA AKHOURIANANYA AKHOURI
81
$endgroup$
add a comment |
$begingroup$
The price of a TV set is ₹20,000 to be paid in 20 installments of ₹1,000 each. Rate of interest=6% p.a
The first installment is to be paid at the time of purchase, then what will be the value of the last installment covering the interest as well?
There are two explanations I found:
- Money paid in cash = ₹ 1000
Balance payment = (20000 - 1000) = ₹ 19000
(the above is from R.S. Aggarwal book and doesn't seem complete) - These links explain and conclude the answer to be 1950 but I didn't exactly understand: link 1 and link 2
I thought something like below (assuming monthly installments, hence taking 6/12 as rate of interest):
- Value after $20$ installments $= 20000 + (20000cdot 20cdot 6)/(12cdot 100) = 22000$
- So, value after 19 installments $=
1000+(1000cdot 20cdot 6)/(12cdot 100) +....+ 1000+(1000cdot 2cdot 6)/(12cdot 100)
= 19000 + (1000cdot 6/(12cdot 100))cdot(19cdot 20/2) = 19950$
- Thus, value of last installment $= 22000-19950=2050$
Can someone explain what am I doing wrong?
finance
edited Jan 21 at 20:54
alexjo
12.5k1430
asked Jan 20 at 11:13
ANANYA AKHOURIANANYA AKHOURI
81
$endgroup$
The price of a TV set is ₹20,000 to be paid in 20 installments of ₹1,000 each. Rate of interest=6% p.a
The first installment is to be paid at the time of purchase, then what will be the value of the last installment covering the interest as well?
There are two explanations I found:
- Money paid in cash = ₹ 1000
Balance payment = (20000 - 1000) = ₹ 19000
(the above is from R.S. Aggarwal book and doesn't seem complete) - These links explain and conclude the answer to be 1950 but I didn't exactly understand: link 1 and link 2
I thought something like below (assuming monthly installments, hence taking 6/12 as rate of interest):
- Value after $20$ installments $= 20000 + (20000cdot 20cdot 6)/(12cdot 100) = 22000$
- So, value after 19 installments $=
1000+(1000cdot 20cdot 6)/(12cdot 100) +....+ 1000+(1000cdot 2cdot 6)/(12cdot 100)
= 19000 + (1000cdot 6/(12cdot 100))cdot(19cdot 20/2) = 19950$
- Thus, value of last installment $= 22000-19950=2050$
Can someone explain what am I doing wrong?
finance
finance
edited Jan 21 at 20:54
alexjo
12.5k1430
asked Jan 20 at 11:13
ANANYA AKHOURIANANYA AKHOURI
81
edited Jan 21 at 20:54
alexjo
12.5k1430
asked Jan 20 at 11:13
ANANYA AKHOURIANANYA AKHOURI
81
edited Jan 21 at 20:54
alexjo
12.5k1430
edited Jan 21 at 20:54
alexjo
12.5k1430
edited Jan 21 at 20:54
alexjo
12.5k1430
12.5k1430
asked Jan 20 at 11:13
ANANYA AKHOURIANANYA AKHOURI
81
asked Jan 20 at 11:13
ANANYA AKHOURIANANYA AKHOURI
81
asked Jan 20 at 11:13
ANANYA AKHOURIANANYA AKHOURI
81
81
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let be $P=1,000$ the installment, $n=20$ number of monthly installments, $S=20,000$ the value of the TV.
Observing that $Ptimes n=1,000times 20=20,000=S$, we have that installments are for the principal amount $S$ only and interest is yet to be paid.
The interest $I$ is to be paid with the amount of last installment $L$ and hence last installment will be
$$
L=P+I
$$
Let be $i=6%$ the annual interest rate. the duration of each month is $1/12$ of a year. With the simple interest we have
$$
I=text{principal}times text{interest rate}times text{time}
$$
At month $1$, the principal is $S_1=S-P=19,000$ and the interest to be paid at month $20$ is $$I_1=S_1times itimes 1=19,000times0.06times 1/12$$
At month $2$, the principal is $S_2=S_1-P=18,000$ and the interest to be paid at month $20$ is $$I_2=S_2times itimes 1=18,000times0.06times 1/12$$
At month $19$, the principal is $S_{19}=S_{18}-P=1,000$ and the interest $$I_{19}=S_{19}times itimes 1=1,000times0.06times 1/12$$
At month $20$, the principal is $S_{20}=S_{19}-P=0$ and the interest is zero.
Summing up all the interests we have
$$
I=sum_{k=1}^{20}I_k=sum_{k=1}^{19}I_k=sum_{k=1}^{19} S_{k}times itimes 1/12
$$
Observing $S_k=1000times k$ we have
$$
I=frac{1}{12}sum_{k=1}^{19} S_{k}times i=frac{1}{12}sum_{k=1}^{19} 1000ktimes i=frac{1000times 0.06}{12}sum_{k=1}^{19} k=5sum_{k=1}^{19} k=5frac{19times 18}{2}=950
$$
beacause $sum_{k=1}^n k=frac{n(n-1)}{2}$.
so we have that the last installmenet is
$$
L=P+I=1,000+950=1,950
$$
answered Jan 21 at 20:54
alexjoalexjo
12.5k1430
$endgroup$
$begingroup$
I understood! The interest is being calculated on the principal that is left. Thanks a ton! :) However, one thing that still confuses me is that when we paid our first installment of Rs. 1000 at the time of purchase, why wasn't interest calculated on the whole amount of 20,000? That is 2000*0.06*1/12=100. Why wasn't it added to the last installment?
$endgroup$
– ANANYA AKHOURI
Jan 22 at 15:14
$begingroup$
Because you payed 1000 immediately and you left 19000 on which you have to pay interest. Equivalently you pay on 20, 0000 the interest $I_0=20,000times 6% times 0/12=0$.
$endgroup$
– alexjo
Jan 22 at 15:19
$begingroup$
Oh..understood now! Thanks a lot!! :)
$endgroup$
– ANANYA AKHOURI
Jan 22 at 18:35
add a comment |
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$begingroup$
Let be $P=1,000$ the installment, $n=20$ number of monthly installments, $S=20,000$ the value of the TV.
Observing that $Ptimes n=1,000times 20=20,000=S$, we have that installments are for the principal amount $S$ only and interest is yet to be paid.
The interest $I$ is to be paid with the amount of last installment $L$ and hence last installment will be
$$
L=P+I
$$
Let be $i=6%$ the annual interest rate. the duration of each month is $1/12$ of a year. With the simple interest we have
$$
I=text{principal}times text{interest rate}times text{time}
$$
At month $1$, the principal is $S_1=S-P=19,000$ and the interest to be paid at month $20$ is $$I_1=S_1times itimes 1=19,000times0.06times 1/12$$
At month $2$, the principal is $S_2=S_1-P=18,000$ and the interest to be paid at month $20$ is $$I_2=S_2times itimes 1=18,000times0.06times 1/12$$
At month $19$, the principal is $S_{19}=S_{18}-P=1,000$ and the interest $$I_{19}=S_{19}times itimes 1=1,000times0.06times 1/12$$
At month $20$, the principal is $S_{20}=S_{19}-P=0$ and the interest is zero.
Summing up all the interests we have
$$
I=sum_{k=1}^{20}I_k=sum_{k=1}^{19}I_k=sum_{k=1}^{19} S_{k}times itimes 1/12
$$
Observing $S_k=1000times k$ we have
$$
I=frac{1}{12}sum_{k=1}^{19} S_{k}times i=frac{1}{12}sum_{k=1}^{19} 1000ktimes i=frac{1000times 0.06}{12}sum_{k=1}^{19} k=5sum_{k=1}^{19} k=5frac{19times 18}{2}=950
$$
beacause $sum_{k=1}^n k=frac{n(n-1)}{2}$.
so we have that the last installmenet is
$$
L=P+I=1,000+950=1,950
$$
answered Jan 21 at 20:54
alexjoalexjo
12.5k1430
$endgroup$
$begingroup$
I understood! The interest is being calculated on the principal that is left. Thanks a ton! :) However, one thing that still confuses me is that when we paid our first installment of Rs. 1000 at the time of purchase, why wasn't interest calculated on the whole amount of 20,000? That is 2000*0.06*1/12=100. Why wasn't it added to the last installment?
$endgroup$
– ANANYA AKHOURI
Jan 22 at 15:14
$begingroup$
Because you payed 1000 immediately and you left 19000 on which you have to pay interest. Equivalently you pay on 20, 0000 the interest $I_0=20,000times 6% times 0/12=0$.
$endgroup$
– alexjo
Jan 22 at 15:19
$begingroup$
Oh..understood now! Thanks a lot!! :)
$endgroup$
– ANANYA AKHOURI
Jan 22 at 18:35
add a comment |
$begingroup$
Let be $P=1,000$ the installment, $n=20$ number of monthly installments, $S=20,000$ the value of the TV.
Observing that $Ptimes n=1,000times 20=20,000=S$, we have that installments are for the principal amount $S$ only and interest is yet to be paid.
The interest $I$ is to be paid with the amount of last installment $L$ and hence last installment will be
$$
L=P+I
$$
Let be $i=6%$ the annual interest rate. the duration of each month is $1/12$ of a year. With the simple interest we have
$$
I=text{principal}times text{interest rate}times text{time}
$$
At month $1$, the principal is $S_1=S-P=19,000$ and the interest to be paid at month $20$ is $$I_1=S_1times itimes 1=19,000times0.06times 1/12$$
At month $2$, the principal is $S_2=S_1-P=18,000$ and the interest to be paid at month $20$ is $$I_2=S_2times itimes 1=18,000times0.06times 1/12$$
At month $19$, the principal is $S_{19}=S_{18}-P=1,000$ and the interest $$I_{19}=S_{19}times itimes 1=1,000times0.06times 1/12$$
At month $20$, the principal is $S_{20}=S_{19}-P=0$ and the interest is zero.
Summing up all the interests we have
$$
I=sum_{k=1}^{20}I_k=sum_{k=1}^{19}I_k=sum_{k=1}^{19} S_{k}times itimes 1/12
$$
Observing $S_k=1000times k$ we have
$$
I=frac{1}{12}sum_{k=1}^{19} S_{k}times i=frac{1}{12}sum_{k=1}^{19} 1000ktimes i=frac{1000times 0.06}{12}sum_{k=1}^{19} k=5sum_{k=1}^{19} k=5frac{19times 18}{2}=950
$$
beacause $sum_{k=1}^n k=frac{n(n-1)}{2}$.
so we have that the last installmenet is
$$
L=P+I=1,000+950=1,950
$$
answered Jan 21 at 20:54
alexjoalexjo
12.5k1430
$endgroup$
$begingroup$
I understood! The interest is being calculated on the principal that is left. Thanks a ton! :) However, one thing that still confuses me is that when we paid our first installment of Rs. 1000 at the time of purchase, why wasn't interest calculated on the whole amount of 20,000? That is 2000*0.06*1/12=100. Why wasn't it added to the last installment?
$endgroup$
– ANANYA AKHOURI
Jan 22 at 15:14
$begingroup$
Because you payed 1000 immediately and you left 19000 on which you have to pay interest. Equivalently you pay on 20, 0000 the interest $I_0=20,000times 6% times 0/12=0$.
$endgroup$
– alexjo
Jan 22 at 15:19
$begingroup$
Oh..understood now! Thanks a lot!! :)
$endgroup$
– ANANYA AKHOURI
Jan 22 at 18:35
add a comment |
$begingroup$
Let be $P=1,000$ the installment, $n=20$ number of monthly installments, $S=20,000$ the value of the TV.
Observing that $Ptimes n=1,000times 20=20,000=S$, we have that installments are for the principal amount $S$ only and interest is yet to be paid.
The interest $I$ is to be paid with the amount of last installment $L$ and hence last installment will be
$$
L=P+I
$$
Let be $i=6%$ the annual interest rate. the duration of each month is $1/12$ of a year. With the simple interest we have
$$
I=text{principal}times text{interest rate}times text{time}
$$
At month $1$, the principal is $S_1=S-P=19,000$ and the interest to be paid at month $20$ is $$I_1=S_1times itimes 1=19,000times0.06times 1/12$$
At month $2$, the principal is $S_2=S_1-P=18,000$ and the interest to be paid at month $20$ is $$I_2=S_2times itimes 1=18,000times0.06times 1/12$$
At month $19$, the principal is $S_{19}=S_{18}-P=1,000$ and the interest $$I_{19}=S_{19}times itimes 1=1,000times0.06times 1/12$$
At month $20$, the principal is $S_{20}=S_{19}-P=0$ and the interest is zero.
Summing up all the interests we have
$$
I=sum_{k=1}^{20}I_k=sum_{k=1}^{19}I_k=sum_{k=1}^{19} S_{k}times itimes 1/12
$$
Observing $S_k=1000times k$ we have
$$
I=frac{1}{12}sum_{k=1}^{19} S_{k}times i=frac{1}{12}sum_{k=1}^{19} 1000ktimes i=frac{1000times 0.06}{12}sum_{k=1}^{19} k=5sum_{k=1}^{19} k=5frac{19times 18}{2}=950
$$
beacause $sum_{k=1}^n k=frac{n(n-1)}{2}$.
so we have that the last installmenet is
$$
L=P+I=1,000+950=1,950
$$
answered Jan 21 at 20:54
alexjoalexjo
12.5k1430
$endgroup$
Let be $P=1,000$ the installment, $n=20$ number of monthly installments, $S=20,000$ the value of the TV.
Observing that $Ptimes n=1,000times 20=20,000=S$, we have that installments are for the principal amount $S$ only and interest is yet to be paid.
The interest $I$ is to be paid with the amount of last installment $L$ and hence last installment will be
$$
L=P+I
$$
Let be $i=6%$ the annual interest rate. the duration of each month is $1/12$ of a year. With the simple interest we have
$$
I=text{principal}times text{interest rate}times text{time}
$$
At month $1$, the principal is $S_1=S-P=19,000$ and the interest to be paid at month $20$ is $$I_1=S_1times itimes 1=19,000times0.06times 1/12$$
At month $2$, the principal is $S_2=S_1-P=18,000$ and the interest to be paid at month $20$ is $$I_2=S_2times itimes 1=18,000times0.06times 1/12$$
At month $19$, the principal is $S_{19}=S_{18}-P=1,000$ and the interest $$I_{19}=S_{19}times itimes 1=1,000times0.06times 1/12$$
At month $20$, the principal is $S_{20}=S_{19}-P=0$ and the interest is zero.
Summing up all the interests we have
$$
I=sum_{k=1}^{20}I_k=sum_{k=1}^{19}I_k=sum_{k=1}^{19} S_{k}times itimes 1/12
$$
Observing $S_k=1000times k$ we have
$$
I=frac{1}{12}sum_{k=1}^{19} S_{k}times i=frac{1}{12}sum_{k=1}^{19} 1000ktimes i=frac{1000times 0.06}{12}sum_{k=1}^{19} k=5sum_{k=1}^{19} k=5frac{19times 18}{2}=950
$$
beacause $sum_{k=1}^n k=frac{n(n-1)}{2}$.
so we have that the last installmenet is
$$
L=P+I=1,000+950=1,950
$$
answered Jan 21 at 20:54
alexjoalexjo
12.5k1430
answered Jan 21 at 20:54
alexjoalexjo
12.5k1430
answered Jan 21 at 20:54
alexjoalexjo
12.5k1430
answered Jan 21 at 20:54
alexjoalexjo
12.5k1430
12.5k1430
$begingroup$
I understood! The interest is being calculated on the principal that is left. Thanks a ton! :) However, one thing that still confuses me is that when we paid our first installment of Rs. 1000 at the time of purchase, why wasn't interest calculated on the whole amount of 20,000? That is 2000*0.06*1/12=100. Why wasn't it added to the last installment?
$endgroup$
– ANANYA AKHOURI
Jan 22 at 15:14
$begingroup$
Because you payed 1000 immediately and you left 19000 on which you have to pay interest. Equivalently you pay on 20, 0000 the interest $I_0=20,000times 6% times 0/12=0$.
$endgroup$
– alexjo
Jan 22 at 15:19
$begingroup$
Oh..understood now! Thanks a lot!! :)
$endgroup$
– ANANYA AKHOURI
Jan 22 at 18:35
add a comment |
$begingroup$
I understood! The interest is being calculated on the principal that is left. Thanks a ton! :) However, one thing that still confuses me is that when we paid our first installment of Rs. 1000 at the time of purchase, why wasn't interest calculated on the whole amount of 20,000? That is 2000*0.06*1/12=100. Why wasn't it added to the last installment?
$endgroup$
– ANANYA AKHOURI
Jan 22 at 15:14
$begingroup$
Because you payed 1000 immediately and you left 19000 on which you have to pay interest. Equivalently you pay on 20, 0000 the interest $I_0=20,000times 6% times 0/12=0$.
$endgroup$
– alexjo
Jan 22 at 15:19
$begingroup$
Oh..understood now! Thanks a lot!! :)
$endgroup$
– ANANYA AKHOURI
Jan 22 at 18:35
$begingroup$
I understood! The interest is being calculated on the principal that is left. Thanks a ton! :) However, one thing that still confuses me is that when we paid our first installment of Rs. 1000 at the time of purchase, why wasn't interest calculated on the whole amount of 20,000? That is 2000*0.06*1/12=100. Why wasn't it added to the last installment?
$endgroup$
– ANANYA AKHOURI
Jan 22 at 15:14
$begingroup$
I understood! The interest is being calculated on the principal that is left. Thanks a ton! :) However, one thing that still confuses me is that when we paid our first installment of Rs. 1000 at the time of purchase, why wasn't interest calculated on the whole amount of 20,000? That is 2000*0.06*1/12=100. Why wasn't it added to the last installment?
$endgroup$
– ANANYA AKHOURI
Jan 22 at 15:14
$begingroup$
Because you payed 1000 immediately and you left 19000 on which you have to pay interest. Equivalently you pay on 20, 0000 the interest $I_0=20,000times 6% times 0/12=0$.
$endgroup$
– alexjo
Jan 22 at 15:19
$begingroup$
Because you payed 1000 immediately and you left 19000 on which you have to pay interest. Equivalently you pay on 20, 0000 the interest $I_0=20,000times 6% times 0/12=0$.
$endgroup$
– alexjo
Jan 22 at 15:19
$begingroup$
Oh..understood now! Thanks a lot!! :)
$endgroup$
– ANANYA AKHOURI
Jan 22 at 18:35
$begingroup$
Oh..understood now! Thanks a lot!! :)
$endgroup$
– ANANYA AKHOURI
Jan 22 at 18:35
add a comment |
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