Question

The number of employees who work at Company A increased by 18% during the year 2003. During 2004, the number of employees increased by 26%. Calculate the approximate percent of the increase, to the nearest percent, in employment over the two-year period.
(a) 31%
(b) 44%
(c) 49%
(d) 53%
(e) 65%

Answer 1

Hint: To solve this problem, we should know the basic concepts of calculation of percentage increase in a quantity. In general, percentage increase = $\dfrac{\text{Change in the amount of quantity}}{\text{Original amount of quantity}}\times 100$. We will use this formula to solve this problem.

Complete step-by-step answer:
We will first convert the word problem into mathematical problem. Thus, by the first condition, we have that the number of employees who work at Company A increased by 18% during the year 2003. Let’s assume that originally there were x employees and x’ employees in the year 2003. Now, in 2003, there was 18% increase, thus, by using the required formula of percentage increase, we have,
Percentage increase = $\dfrac{\text{Change in the amount of quantity}}{\text{Original amount of quantity}}\times 100$
18 = $\dfrac{\text{Change in the amount of quantity}}{\text{Original amount of quantity}}\times 100$
0.18 = $\dfrac{\text{Change in the amount of quantity}}{\text{Original amount of quantity}}$
0.18 = $\dfrac{x'-x}{x}$
0.18x = x’ – x
1.18x = x’ -- (1)
Now, we do the same for the year 2004. Let’s assume there are y employees in the year 2004. Now, there is a 26% increase from the year 2003. Thus, we have,
Percentage increase = $\dfrac{\text{Change in the amount of quantity}}{\text{Original amount of quantity}}\times 100$
26 = $\dfrac{\text{Change in the amount of quantity}}{\text{Original amount of quantity}}\times 100$
0.26 = $\dfrac{\text{Change in the amount of quantity}}{\text{Original amount of quantity}}$
0.26 = $\dfrac{y-x'}{x'}$
0.26x’ = y – x’
1.26x’ = y
Now, from (1), x’ = 1.18x, thus, we have,
1.26(1.18x) = y
y = 1.4868x
Now, calculating percentage increase from the original number of employees (x) to those in the year 2004 (y), we have,
Percentage increase = $\dfrac{\text{Change in the amount of quantity}}{\text{Original amount of quantity}}\times 100$
Percentage increase = $\dfrac{\text{y-x}}{\text{x}}\times 100$
Percentage increase = $\dfrac{\text{1}\text{.4868x-x}}{x}\times 100$
Percentage increase = 48.68 %
The nearest value is 49%. Hence, the correct answer is (c) 49%.

Note: The problems related to percentage increase/decrease generally require only the use of one formula given by Percentage increase/decrease = $\dfrac{\text{Change in the amount of quantity}}{\text{Original amount of quantity}}\times 100$. However, it is important to convert the necessary word problem into an appropriate mathematical question to apply this. One other way to proceed the problem is to assume an arbitrary value of the original number of employees (say 100). One can then proceed by repeating similar steps to solve the problem to obtain the same answer. This is because the formula involves the use of ratios and thus any arbitrary value gets cancelled due to this fact.