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?
A survey split into groups calls for part-whole percentages: the original shares account for all respondents. If one group later grows, apply its percent increase to its original count, not to a new survey total. Then compare the groups as shares of the same original whole and use a Desmos equation to find that whole. The trap is treating the increase as the entire new count.
Hints
- Hint 1
The whole is everyone who answered the original survey, so its commuting shares account for every respondent. Subtract the three stated shares to find the original bike share.
- Hint 2
A increase adds of the original bike count to the count already there. What percent of the original bike count is there afterward?
- Hint 3
The drive-alone count stays the same. Express both that count and the new bike count as shares of the original survey total. Their difference represents employees.
Step-by-step
Compare shares of the original survey
Step 1Find the original bike share
The whole is all original respondents, or . The remaining share belongs to bikers. Type in Desmos; it shows , so bikers were of the original survey.
- Step 2
Turn the increase into a new count
An increase of is added to the original bike count; it isn't the entire new count. Keep the original of bikers and add of that count. Add the two shares:
- Step 3
Find the new bike share
The new bike count is of the original bike count, which was of the original survey. Type in Desmos; it shows . So the new bike count equals of the original survey total, not of a new total.
- Step 4
Write the gap as an equation
Let be the number of original survey respondents. Because the drive-alone count stays the same, it is still of . The new bike count is of . Driving minus biking is employees, so write:
- Step 5
Find the original total
Type in Desmos. Use for the unknown original total and to tell Desmos to solve the equation. Under PARAMETERS, it shows . So 100 employees responded to the original survey. Grid in 100.