At the start of a community festival, of the visitors had purchased their tickets online. During the festival, 18 visitors who had purchased tickets online left, and 12 visitors who had purchased tickets at the gate entered. At that point, of the visitors at the festival had purchased their tickets online.
By what percent did the number of visitors who had purchased tickets online decrease from the start of the festival?
When a percentage describes a subgroup before and after people enter or leave, define the original total first. Write one expression for the subgroup after the changes and another expression using its new percentage of the changed total. Once the original subgroup count is known, use the standard percent-decrease formula: decrease divided by original amount, multiplied by .
Hints
Choose a variable for the initial total
Let the initial total number of visitors be . Express the initial number of online-ticket visitors as a percentage of .
Track both changes
The online-ticket group loses 18 visitors, while the overall total decreases by 6 visitors.
Use the correct reference amount
After finding the initial total, calculate the decrease as a fraction of the initial number of online-ticket visitors, not the later number.
Desmos Guide
Use algebra first
Algebra is generally faster here because the percentage relationship produces one linear equation. Desmos can verify the initial total.
Graph the two later online-ticket counts
Enter and . Their intersection represents the initial total for which both expressions give the same later number of online-ticket visitors.
Calculate the decrease
Use the intersection's -coordinate in the expression . This calculates the decrease divided by the initial online-ticket count.
Step-by-step Explanation
Represent the initial and later counts
Let be the initial number of visitors. Initially, the number who purchased tickets online was .
After 18 of those visitors leave, the number of online-ticket visitors is . The total number of visitors changes by , so the later total is .
Use the later percentage
The later number of online-ticket visitors is of the later total:
Solving gives , so .
Find the percentage decrease
Initially, there were online-ticket visitors. The decrease was 18 visitors, so the percent decrease is