At the beginning of a year, 40% of the members of a volunteer group were trail guides. During the year, 25% of the trail guides and 10% of the members who were not trail guides left the group. Then 32 new members joined the group, and none of the new members were trail guides.
After the new members joined, trail guides made up 30% of the group. How many members were in the group at the beginning of the year?
For percentage problems in which groups lose different percentages of members and then new members join, define the original total as a variable and track each group separately. Convert each departure into a remaining percentage, build the final total carefully, and use the stated final percentage to connect one group count to that total.
Hints
Use a variable for the starting total
Let represent the initial number of members. Express the initial numbers of trail guides and non-guides as percentages of .
Apply the departures separately
If 25% of a group leaves, 75% remains. If 10% leaves, 90% remains.
Focus on the final total
The 32 new members are all non-guides, so they change the final total but do not change the number of trail guides. Set the remaining trail-guide count equal to 30% of the final total.
Desmos Guide
Set up the two quantities
Algebra is generally faster here, but Desmos can verify the equation. Enter for the number of trail guides remaining.
Graph the final-percentage expression
Enter . This represents 30% of the final group size.
Find the intersection
Click the intersection of the two graphs. Its -coordinate is the initial number of members.
Step-by-step Explanation
Represent the original groups
Let be the number of members at the beginning of the year.
The number of trail guides was , and the number of members who were not trail guides was .
Find the number who remained
Because 25% of the trail guides left, 75% remained:
Because 10% of the non-guides left, 90% remained:
After 32 non-guides joined, the total number of members was , or .
Use the final percentage
Trail guides were 30% of the final group, so