Question

The given table shows the approximate enrollment at the University of Chicago every fifty years.

$\begin{array}{cc}\text{Year}& \text{Number of students}\\ & \text{enrolled}\\ 1900& x\\ 1950& 2.7\times {10}^4\\ 2000& 7.4\times {10}^4\end{array}$

If the total number of students enrolled in the year $2000$ is $20$ times the number of students enrolled in the year $1900,$ then how many more students were enrolled at the University of Chicago in $1950$ than in $1900?$

• A. $2.33\times {10}^4$
• B. Both B and C
• C. $1.1\times {10}^4$
• D. $23,300$

Answer 1

The correct option is B Both B and C

Detailed step-by-step solution:
Number of students enrolled in the year $1900=x$
Number of students student in the year $2000=20x$ (total number of students enrolled in the year $2000$ is $20$ times the number of students enrolled in the year $1900)$
$20x=7.4\times {10}^4$
$\Rightarrow 20x=7.4\times {10}^{3+1}$
$\Rightarrow 2\times 10x=7.4\times 10\times {10}^3$ (using product of powers property, i.e., $a^m\times a^n=a^{m+n}$ )
$\Rightarrow x=\frac{7.4\times 10}{2\times 10}\times {10}^3$
$\Rightarrow x=3.7\times {10}^3$ (evaluate $\frac{7.4}{2}=3.7)$
To get how many more students were enrolled at the University of Chicago in $1950$ than in $1900,$ we need to calculate the difference between the number of students enrolled in these two years,
Desired number of students (=\) (Number of students in $1950-$ Number of students in $1900)$
$\Rightarrow 2.7\times {10}^4-3.7\times {10}^3=2.7\times {10}^4-0.37\times 10\times {10}^3$ (the exponent of $10$ should be the same while subtracting, and making it $4,$ as options are given in the form of ${10}^4)$
$=2.7\times {10}^4-0.37\times {10}^{3+1}$ (using product of powers property, i.e., $a^m\times a^n=a^{m+n}$ )
$=2.7\times {10}^4-0.37\times {10}^4$
$=(2.7-0.37)\times {10}^4$ (using distributive property, i.e., $a\times (b-c)=a\times b-a\times c)$
$=2.33\times {10}^4$
$=23,300$ (writing in standard form)
$\Rightarrow$ Option D is correct.