Question

Problem 1: The # of Stuclents attending summer school at a local commmity college has been increasing each year by 2.5% If 984 stolent currenty atlead immer School & this rate continues, finct the # of Stoelenti attending Summer School in 5 year.

Answer 1

Given that, the number of students attending summer school at a local community college has been increasing each year by '2.5%'. Currently, '984' students attend summer school and the rate continues for every year.

We have to find the number of students attending summer school in '5' years.

If we observe clearly, this problem can be related to a compound interest problem where the principal amount is compounded annually at a certain rate of interest for certain number of years.

Let's 'compare' the given terms with a compound interest problem terms. That is, current number of students (984) becomes the Principal,'P' and the students' increasing rate (2.5%) becomes the rate of interest,'r'. We have to find the "Amount (Principal + Interest)" after 5 years when compounded annually.

The formula for calculating amount when compounded annually is given by

$"\underline{A= P(1+r)^t} \;"$

where A = Amount (Principal + Interest),

P = Principal,

r = rate of interest

& t = time period in years

Now, on comparison with these terms, we have "P = 984 ; r = 2.5% = 0.025 ; t = 5".

Let's substitute these values in the above formula.

$\Rightarrow A = P(1+r)^t$

$\Rightarrow A = 984 \;(1+0.025)^5$

→ A= 984 (1.025

= A= 984 (1.13141

$\Rightarrow \mathbf{A = 1113.306 \simeq \underline{1113}}$

This amount is nothing but the number of students attending summer school in 5 years.

$\therefore$ The number of students attending summer school in 5 years is $\mathbf{\underline{"1113"}}$ .

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Just for knowledge and concept understanding purpose:

We can also do this in an alternative way i.e., by directly calculating the number of students for every successive year using the increasing rate %. We need five calculations for 5 years. They are given below.

Let the number of students be 'n'.

After 1^st year :

The number of students increased by 2.5% after one year, i.e.,

$\Rightarrow n = 984 + \left (984\times\frac{2.5}{100} \right )$

$\Rightarrow n = 984 + (24.6 ) = \mathbf{1008.6}$

After 2^nd year :

The number of students again increased by 2.5%, i.e.,

$\Rightarrow n = 1008.6 + \left (1008.6\times\frac{2.5}{100} \right )$

$\Rightarrow n = 1008.6 + (25.215 ) = \mathbf{1033.815}$

After 3^rd year :

The number of students again increased by 2.5%, i.e.,

$\Rightarrow n = 1033.815 + \left (1033.815\times\frac{2.5}{100} \right )$

$\Rightarrow n = 1033.815 + (25.8454 ) = \mathbf{1059.6604}$

After 4^th year :

The number of students again increased by 2.5%, i.e.,

$\Rightarrow n = 1059.6604 + \left (1059.6604\times\frac{2.5}{100} \right )$

$\Rightarrow n = 1059.6604 + (26.4915) = \mathbf{1086.152}$

After 5^th year :

The number of students again increased by 2.5%, i.e.,

$\Rightarrow n = 1086.152 + \left (1086.152\times\frac{2.5}{100} \right )$

$\Rightarrow n = 1086.152 + (27.154) = \mathbf{1113.306 \simeq \underline{1113}}$

Therefore, the number of students attending summer school in 5 years is  $\mathbf{\underline{"1113"}}$.