# Explain this shortcut C.I solution.

#### Ganesh Ujwal

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
The sum becomes 3Â³ 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.

#### Ganesh Ujwal

The sum becomes 3Â³ times itself in 3*5 years.
the sum becomes 3Â³ 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
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.

#### Ganesh Ujwal

Is my interpretation is wrong or right: the sum becomes 3Â³ times itself means C.I is 3^3 bigger than Principal in 15 years?

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#### skipjack

Forum Staff
Wrong - it means that the total of the original amount and all the interest earned (including interest on interest) over the 15 years is 3Â³ times the original amount.

#### Ganesh Ujwal

It means that the total of the original amount and all the interest earned (including interest on interest) = C.I?

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#### Ganesh Ujwal

Is there any single word for "total of the original amount and all the interest earned" ?

#### skipjack

Forum Staff
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.

#### Ganesh Ujwal

You mean total amount i.e A = CI + P?