Explain this shortcut C.I solution.

Aug 2014
507
1
India
A sum of money placed at Compound interest becomes 27 times of itself in 15 years. In 25 years, it will becomes how many times?

Shortcut Sol: 3^3 = 3*5
?=5*5

? = 3^5 = 243 times.

Actual CI formulae is \(\displaystyle CI = P(1+\large\frac{r}{100})\normalsize^{n}\)

But solution looks totally different. Please explain.
 
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skipjack

Forum Staff
Dec 2006
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The sum becomes 3³ times itself in 3*5 years.

Hence in 5*5 years, it becomes 3$^{\large3(5*5)/(3*5)}$ times itself, i.e. 3$^{\large5}$ times itself.

Alternatively, one can see that the sum becomes 3 times itself in 5 years, so it becomes 3$^{\large n}$ times itself in $5n$ years. For $n$ = 5, this becomes 243 times itself.
 
Aug 2014
507
1
India
The sum becomes 3³ times itself in 3*5 years.
the sum becomes 3³ times itself means C.I is \(\displaystyle 3^3\) bigger than Principal in 15 years?

I didn't understand clearly.
 
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skipjack

Forum Staff
Dec 2006
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It seems easy to understand "A sum of money placed at Compound interest becomes 27 times of itself in 15 years." However, "3^3 = 3*5" in the shortcut solution could mislead if interpreted too literally.
 
Aug 2014
507
1
India
Is my interpretation is wrong or right: the sum becomes 3³ times itself means C.I is 3^3 bigger than Principal in 15 years?
 
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skipjack

Forum Staff
Dec 2006
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Wrong - it means that the total of the original amount and all the interest earned (including interest on interest) over the 15 years is 3³ times the original amount.
 
Aug 2014
507
1
India
It means that the total of the original amount and all the interest earned (including interest on interest) = C.I?
 
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Aug 2014
507
1
India
Is there any single word for "total of the original amount and all the interest earned" ?
 

skipjack

Forum Staff
Dec 2006
21,482
2,472
Amount. Your first post used the wording "sum of money", possibly so that you could use that phrase instead of "initial amount" to refer to or define the principal, thus leaving the word "amount" available for the total that you asked about. Sometimes, the principal is defined as a function of time and denoted by, say, P(t), so that P(0) is the initial sum and P(t) is the total amount (including all interest) after time t. This can be done for simple interest just as easily as for compound interest.

I've occasionally seen "C.I." used instead of P(t), but I don't recommend doing that.