Question

What will be the ratio of the simple interest earned by a certain amount at the same rate of interest for \[6\] years and that for \[9\] years?
a). \[1:3\]
b). \[1:4\]
c). \[2:3\]
d). None of these

Answer 1

Hint: Here we are asked to find the ratio of simple interest for \[6\] years to that of \[9\] years with the same principal and rate. Since the principal and the rate are the same, we don’t need to bother about them, we will keep them as a variable itself. First, we will find the simple interest with \[6\] years, and then we will find the simple interest with \[9\] years. Then, we will find the ratio.
Formula Used: Formula that we need to know before solving this problem:
The simple interest with a principal amount \[P\] for \[n\] years at the rate \[r\] is \[SI = \dfrac{{Pnr}}{{100}}\]
The ratio a to b is nothing but the \[\dfrac{a}{b}\] which is represented as \[a:b\]

Complete step-by-step solution:
It is given that the same principal amount is invested at the same rate for \[6\] & \[9\] years separately. We aim to find the ratio of the interest earned in one investment (\[6\] years) to the other (\[9\] years).
For that, we first need to find the simple interest of both investments separately.
First investment:
Let \[P\] be the principal amount and \[r\] be the rate. Let the year be \[6\] for this investment.
We know that the simple interest with a principal amount \[P\] for \[n\] years at the rate \[r\] is \[SI = \dfrac{{Pnr}}{{100}}\]
Let \[S{I_1}\] be the simple interest of this investment.
Substituting the values in the SI formula we get
\[S{I_1} = \dfrac{{P \times 6 \times r}}{{100}}\]
Thus, we now have the simple interest for the first investment. Let us find the simple interest for the second investment.
Second investment:
Since it is given that the principal amount and the rate are the same, let \[P\] be the principal amount and \[r\] be the rate for this investment also. And the year for this investment is \[9\] years.
Let \[S{I_2}\] be the simple interest for this investment.
Again, by the formula of simple interest, we get
\[\Rightarrow S{I_2} = \dfrac{{P \times 9 \times r}}{{100}}\]
Now we have the simple interest of both given cases. Here we aim to find the ratio of simple interest of \[6\] years to the simple interest of \[9\] years. That is \[S{I_1}:S{I_2}\].
\[\Rightarrow \dfrac{{S{I_1}}}{{S{I_2}}} = \dfrac{{\dfrac{{P \times 6 \times r}}{{100}}}}{{\dfrac{{P \times 9 \times r}}{{100}}}}\]
Let us rewrite the above expression
\[\Rightarrow \dfrac{{S{I_1}}}{{S{I_2}}} = \dfrac{{P \times 6 \times r}}{{100}} \times \dfrac{{100}}{{P \times 9 \times r}}\]
On simplifying this we get
\[\Rightarrow \dfrac{{S{I_1}}}{{S{I_2}}} = \dfrac{6}{9}\]
On further simplification we get
\[\Rightarrow \dfrac{{S{I_1}}}{{S{I_2}}} = \dfrac{2}{3}\]
We know that the ratio \[S{I_1}:S{I_2}\] can be written as \[\dfrac{{S{I_1}}}{{S{I_2}}}\]
Thus, \[S{I_1}:S{I_2} = 2:3\]
Therefore, the ratio of the simple interest earned by a certain amount at the same rate of interest for \[6\] years and that for \[9\] years is \[2:3\].
Now let us see the options to find the correct answer.
Option (a) \[1:3\] is an incorrect answer as we got that the ratio \[S{I_1}:S{I_2} = 2:3\].
Option (b) \[1:4\] is an incorrect answer as we got that the ratio \[S{I_1}:S{I_2} = 2:3\].
Option (c) \[2:3\] is the correct option as we got the same value in our calculation above.
Option (d) None of these is an incorrect answer as we got the option (c) as the correct answer.
Hence, option (c) \[2:3\] is the correct option.

Note: The method of calculating the interest amount earned from a principal amount at a particular rate and years. A ratio is nothing but comparing two quantities of the same kind by division and it is denoted by the symbol \[':'\].