Question 27·Hard·Percentages
A technology firm surveyed its employees about their primary commuting method. The results showed that 48% of respondents drive alone, 30% carpool, 14% take public transit, and the remaining respondents bike to work.
Several months later, after the firm installed secure on-site bicycle storage, the number of employees who bike to work increased by 150%, while the counts in the other categories stayed the same. After this change, there are exactly 28 fewer employees who bike to work than employees who drive alone.
How many employees responded to the original survey?
(Express the answer as an integer)
For percent word problems like this, first make sure all categories add to 100%, and find any missing percentage by subtraction. When a group 'increases by k%', translate that to a multiplier of on the original amount. Then assign a variable for the total (often ), write each group's count as (percent as a decimal) × , and carefully translate comparison phrases like '28 fewer than' into an equation. Solve the resulting linear equation and quickly check that your answer makes the percentages and differences in the story work out cleanly.
Hints
Account for all commuting methods
First, add the given percentages (drive alone, carpool, public transit). What percent is left for biking?
Understand what 'increased by 150%' means
If some group has an increase of 150%, how many times the original size is the new size? Think of it as original + 1.5×original.
Translate the situation into an equation
Let be the total number of employees. Express the number biking after the increase and the number driving alone as percentages of . How can you write 'there are exactly 28 fewer employees who bike than who drive alone' as an equation relating these two expressions?
Desmos Guide
Enter expressions for bikers and drivers
In Desmos, let represent the total number of employees. Enter these two expressions as functions:
y1 = 0.20x(number who bike after the increase: 20% of )y2 = 0.48x - 28(drivers minus 28, since bikers are 28 fewer than drivers)
Find where the two counts are equal
Look at the graph and tap the intersection point of the lines and . The -coordinate of this intersection is the value of that makes the two expressions equal; that is the total number of employees in the original survey.
Step-by-step Explanation
Find the original biking percentage
Add the given percentages for the first three commuting methods:
.
The remaining percentage must be the biking group:
.
So, originally, of the employees biked to work.
Interpret the 150% increase in bikers
An 'increase by 150%' means:
- New amount = original amount + times the original
- So new amount = times the original
The biking percentage after the change becomes:
.
So after the storage is installed, of the employees bike to work.
Write an equation for the total number of employees
Let be the total number of employees who responded to the survey.
- Number who drive alone: of
- Number who bike after the increase: of
We are told that after the change, there are 28 fewer bikers than drivers. That means:
This equation compares the new number of bikers to the (unchanged) number of drivers.
Solve the equation for N
Solve
Subtract from both sides:
Now divide both sides by :
So, 100 employees responded to the original survey.