Finding the compound interest. Task from the Elements of Algebra by L. Euler The 2019 Stack Overflow Developer Survey Results Are InCompound interest for retirementCompound Interest -Confirming Answer-[solved]Compound interest problemCompound Interest CalculationGeometric progression (compound interest)Compound Interest vs Continuous Interestcompound interest amountCompound Interest Problems:Compound Interest - Derive compound frequeny ($n$) and interest rate ($r$), rate from $A$, $P$ and $T$.Compound Interest Rates
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Finding the compound interest. Task from the Elements of Algebra by L. Euler
The 2019 Stack Overflow Developer Survey Results Are InCompound interest for retirementCompound Interest -Confirming Answer-[solved]Compound interest problemCompound Interest CalculationGeometric progression (compound interest)Compound Interest vs Continuous Interestcompound interest amountCompound Interest Problems:Compound Interest - Derive compound frequeny ($n$) and interest rate ($r$), rate from $A$, $P$ and $T$.Compound Interest Rates
$begingroup$
A trader allows 100$.per annum for the expenses of his family, and yearly augments that part of his stock which is not so expended, by a third part of it; at the end of three years, his original stock was doubled: what had he at first?
I came up with equation $(1/3)^3 - (1/3)^2*100 - 1/3*100 - 100 = 2a$. But this equation doesn't work when I try to calculate this thing. I think I misunderstood the exercise. I'd appreciate some help
algebra-precalculus
asked Mar 30 at 18:04
JohnyJohny
134
$endgroup$
add a comment |
$begingroup$
A trader allows 100$.per annum for the expenses of his family, and yearly augments that part of his stock which is not so expended, by a third part of it; at the end of three years, his original stock was doubled: what had he at first?
I came up with equation $(1/3)^3 - (1/3)^2*100 - 1/3*100 - 100 = 2a$. But this equation doesn't work when I try to calculate this thing. I think I misunderstood the exercise. I'd appreciate some help
algebra-precalculus
asked Mar 30 at 18:04
JohnyJohny
134
$endgroup$
$begingroup$
Welcome to MSE! Please add some context to your question. As it is now, all that is in the post is Euler's question, but it's unclear what your question is. Are you asking what the question means when it is translated out of archaic English? Are you asking how to solve this question? If you are asking how to solve the question, where are you getting stuck? It is difficult for us to help you without knowing where you are having problems.
$endgroup$
– jgon
Mar 30 at 18:12
$begingroup$
Sorry. I edited my question. Hope it's more clear now
$endgroup$
– Johny
Mar 30 at 18:16
add a comment |
$begingroup$
A trader allows 100$.per annum for the expenses of his family, and yearly augments that part of his stock which is not so expended, by a third part of it; at the end of three years, his original stock was doubled: what had he at first?
I came up with equation $(1/3)^3 - (1/3)^2*100 - 1/3*100 - 100 = 2a$. But this equation doesn't work when I try to calculate this thing. I think I misunderstood the exercise. I'd appreciate some help
algebra-precalculus
asked Mar 30 at 18:04
JohnyJohny
134
$endgroup$
A trader allows 100$.per annum for the expenses of his family, and yearly augments that part of his stock which is not so expended, by a third part of it; at the end of three years, his original stock was doubled: what had he at first?
I came up with equation $(1/3)^3 - (1/3)^2*100 - 1/3*100 - 100 = 2a$. But this equation doesn't work when I try to calculate this thing. I think I misunderstood the exercise. I'd appreciate some help
algebra-precalculus
algebra-precalculus
asked Mar 30 at 18:04
JohnyJohny
134
asked Mar 30 at 18:04
JohnyJohny
134
edited Mar 30 at 18:14
Johny
asked Mar 30 at 18:04
JohnyJohny
134
asked Mar 30 at 18:04
JohnyJohny
134
asked Mar 30 at 18:04
JohnyJohny
134
134
$begingroup$
Welcome to MSE! Please add some context to your question. As it is now, all that is in the post is Euler's question, but it's unclear what your question is. Are you asking what the question means when it is translated out of archaic English? Are you asking how to solve this question? If you are asking how to solve the question, where are you getting stuck? It is difficult for us to help you without knowing where you are having problems.
$endgroup$
– jgon
Mar 30 at 18:12
$begingroup$
Sorry. I edited my question. Hope it's more clear now
$endgroup$
– Johny
Mar 30 at 18:16
add a comment |
$begingroup$
Welcome to MSE! Please add some context to your question. As it is now, all that is in the post is Euler's question, but it's unclear what your question is. Are you asking what the question means when it is translated out of archaic English? Are you asking how to solve this question? If you are asking how to solve the question, where are you getting stuck? It is difficult for us to help you without knowing where you are having problems.
$endgroup$
– jgon
Mar 30 at 18:12
$begingroup$
Sorry. I edited my question. Hope it's more clear now
$endgroup$
– Johny
Mar 30 at 18:16
$begingroup$
Welcome to MSE! Please add some context to your question. As it is now, all that is in the post is Euler's question, but it's unclear what your question is. Are you asking what the question means when it is translated out of archaic English? Are you asking how to solve this question? If you are asking how to solve the question, where are you getting stuck? It is difficult for us to help you without knowing where you are having problems.
$endgroup$
– jgon
Mar 30 at 18:12
$begingroup$
Welcome to MSE! Please add some context to your question. As it is now, all that is in the post is Euler's question, but it's unclear what your question is. Are you asking what the question means when it is translated out of archaic English? Are you asking how to solve this question? If you are asking how to solve the question, where are you getting stuck? It is difficult for us to help you without knowing where you are having problems.
$endgroup$
– jgon
Mar 30 at 18:12
$begingroup$
Sorry. I edited my question. Hope it's more clear now
$endgroup$
– Johny
Mar 30 at 18:16
$begingroup$
Sorry. I edited my question. Hope it's more clear now
$endgroup$
– Johny
Mar 30 at 18:16
add a comment |
1 Answer
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$begingroup$
Let $S_n$ be the trader's starting stock in year $n$. As I understand the archaic English, it says that every year, the trader's stock at the end of the year is given by the equation
$$S_n+1=frac43(S_n-100).$$
Thus what we are being asked is if $S_3=2S_0$, what is $S_0$?
Then expanding $S_3$ out, we get
$$S_3=frac43(S_2-100)=frac43left(frac43(S_1-100)-100right)=frac43left(frac43left(frac43(S_0-100)-100right)-100right),$$
or $$27S_3 = 64S_0-64cdot 100 - 16cdot 3cdot 100 - 4cdot 9cdot 100$$
$$=64S_0-6400-4800-3600$$
$$=64S_0-14800.$$
Substituting $S_3=2S_0$, we get
$$54S_0=64S_0-14800,$$
or $10S_0=14800$. Thus $S_0=1480$.
Indeed, this yields $S_1=frac43cdot 1380 = 1840$, $S_2=frac43cdot 1740=2320$,
and $S_3= frac43 cdot 2220=2960=2cdot 1480$, as desired.
answered Mar 30 at 18:30
jgonjgon
16.5k32143
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add a comment |
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1 Answer
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$begingroup$
Let $S_n$ be the trader's starting stock in year $n$. As I understand the archaic English, it says that every year, the trader's stock at the end of the year is given by the equation
$$S_n+1=frac43(S_n-100).$$
Thus what we are being asked is if $S_3=2S_0$, what is $S_0$?
Then expanding $S_3$ out, we get
$$S_3=frac43(S_2-100)=frac43left(frac43(S_1-100)-100right)=frac43left(frac43left(frac43(S_0-100)-100right)-100right),$$
or $$27S_3 = 64S_0-64cdot 100 - 16cdot 3cdot 100 - 4cdot 9cdot 100$$
$$=64S_0-6400-4800-3600$$
$$=64S_0-14800.$$
Substituting $S_3=2S_0$, we get
$$54S_0=64S_0-14800,$$
or $10S_0=14800$. Thus $S_0=1480$.
Indeed, this yields $S_1=frac43cdot 1380 = 1840$, $S_2=frac43cdot 1740=2320$,
and $S_3= frac43 cdot 2220=2960=2cdot 1480$, as desired.
answered Mar 30 at 18:30
jgonjgon
16.5k32143
$endgroup$
add a comment |
$begingroup$
Let $S_n$ be the trader's starting stock in year $n$. As I understand the archaic English, it says that every year, the trader's stock at the end of the year is given by the equation
$$S_n+1=frac43(S_n-100).$$
Thus what we are being asked is if $S_3=2S_0$, what is $S_0$?
Then expanding $S_3$ out, we get
$$S_3=frac43(S_2-100)=frac43left(frac43(S_1-100)-100right)=frac43left(frac43left(frac43(S_0-100)-100right)-100right),$$
or $$27S_3 = 64S_0-64cdot 100 - 16cdot 3cdot 100 - 4cdot 9cdot 100$$
$$=64S_0-6400-4800-3600$$
$$=64S_0-14800.$$
Substituting $S_3=2S_0$, we get
$$54S_0=64S_0-14800,$$
or $10S_0=14800$. Thus $S_0=1480$.
Indeed, this yields $S_1=frac43cdot 1380 = 1840$, $S_2=frac43cdot 1740=2320$,
and $S_3= frac43 cdot 2220=2960=2cdot 1480$, as desired.
answered Mar 30 at 18:30
jgonjgon
16.5k32143
$endgroup$
add a comment |
$begingroup$
Let $S_n$ be the trader's starting stock in year $n$. As I understand the archaic English, it says that every year, the trader's stock at the end of the year is given by the equation
$$S_n+1=frac43(S_n-100).$$
Thus what we are being asked is if $S_3=2S_0$, what is $S_0$?
Then expanding $S_3$ out, we get
$$S_3=frac43(S_2-100)=frac43left(frac43(S_1-100)-100right)=frac43left(frac43left(frac43(S_0-100)-100right)-100right),$$
or $$27S_3 = 64S_0-64cdot 100 - 16cdot 3cdot 100 - 4cdot 9cdot 100$$
$$=64S_0-6400-4800-3600$$
$$=64S_0-14800.$$
Substituting $S_3=2S_0$, we get
$$54S_0=64S_0-14800,$$
or $10S_0=14800$. Thus $S_0=1480$.
Indeed, this yields $S_1=frac43cdot 1380 = 1840$, $S_2=frac43cdot 1740=2320$,
and $S_3= frac43 cdot 2220=2960=2cdot 1480$, as desired.
answered Mar 30 at 18:30
jgonjgon
16.5k32143
$endgroup$
Let $S_n$ be the trader's starting stock in year $n$. As I understand the archaic English, it says that every year, the trader's stock at the end of the year is given by the equation
$$S_n+1=frac43(S_n-100).$$
Thus what we are being asked is if $S_3=2S_0$, what is $S_0$?
Then expanding $S_3$ out, we get
$$S_3=frac43(S_2-100)=frac43left(frac43(S_1-100)-100right)=frac43left(frac43left(frac43(S_0-100)-100right)-100right),$$
or $$27S_3 = 64S_0-64cdot 100 - 16cdot 3cdot 100 - 4cdot 9cdot 100$$
$$=64S_0-6400-4800-3600$$
$$=64S_0-14800.$$
Substituting $S_3=2S_0$, we get
$$54S_0=64S_0-14800,$$
or $10S_0=14800$. Thus $S_0=1480$.
Indeed, this yields $S_1=frac43cdot 1380 = 1840$, $S_2=frac43cdot 1740=2320$,
and $S_3= frac43 cdot 2220=2960=2cdot 1480$, as desired.
answered Mar 30 at 18:30
jgonjgon
16.5k32143
answered Mar 30 at 18:30
jgonjgon
16.5k32143
answered Mar 30 at 18:30
jgonjgon
16.5k32143
answered Mar 30 at 18:30
jgonjgon
16.5k32143
16.5k32143
add a comment |
add a comment |
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$begingroup$
Welcome to MSE! Please add some context to your question. As it is now, all that is in the post is Euler's question, but it's unclear what your question is. Are you asking what the question means when it is translated out of archaic English? Are you asking how to solve this question? If you are asking how to solve the question, where are you getting stuck? It is difficult for us to help you without knowing where you are having problems.
$endgroup$
– jgon
Mar 30 at 18:12
$begingroup$
Sorry. I edited my question. Hope it's more clear now
$endgroup$
– Johny
Mar 30 at 18:16