Maya is selling concert tickets. She earns a commission of $2 for each of the first 100 tickets she sells and $3.50 for each additional ticket she sells. Maya pledges to donate 15% of her total commission to a local charity.
What is the least number of tickets beyond the first 100 that Maya must sell so that her donation is at least $120?
For percent-and-inequality word problems like this, start by defining a variable for the unknown quantity (here, the number of extra tickets) and write an expression for the total amount (total commission) using the rates given. Then multiply by the percent (converted to a decimal) to get the donation and set up an inequality using phrases like “at least” or “no more than” to decide the inequality direction. Solve the inequality carefully—keeping decimals exact or using a calculator—and finally adjust to a whole number if the context (like tickets or people) cannot be fractional, rounding up when you need at least a certain amount.
Hints
Account for the first 100 tickets
First find how much commission Maya earns from the first 100 tickets before thinking about any extra tickets.
Define a variable for the additional tickets
Let a variable (for example, ) represent the number of tickets she sells beyond the first 100. How can you write her total commission in terms of ?
Write an inequality for the donation
Her donation is 15% of her total commission. Turn 15% into a decimal and multiply it by your expression for total commission. Then set this expression greater than or equal to 120.
Solve and think about whole numbers
After solving the inequality for your variable, remember she cannot sell part of a ticket. How do you use this to decide the smallest possible number of extra tickets?
Desmos Guide
Enter the donation function
In Desmos, type the equation y = 0.15(200 + 3.5x) to represent Maya's donation as a function of the number of extra tickets .
Graph the target donation level
On a new line, type y = 120 to draw a horizontal line showing the required minimum donation.
Find the boundary x-value
Look at the point where the graph of y = 0.15(200 + 3.5x) intersects the line y = 120. The x-coordinate of this intersection is the cutoff value from your inequality.
Choose the least whole number
Since Maya can only sell whole tickets, take the smallest integer greater than or equal to that intersection x-value. That integer is the number of tickets beyond the first 100 that she must sell.
Step-by-step Explanation
Define the variable and write total commission
Let be the number of tickets that Maya sells beyond the first 100.
- From the first 100 tickets, she earns $2 each: dollars.
- From the additional tickets, she earns $3.50 each: dollars.
So her total commission (money she earns before donating) is:
Express the donation and set up the inequality
Maya donates 15% of her total commission. As a decimal, 15% is .
Her donation is:
We are told this donation must be at least 120 dollars, so we write the inequality:
Solve the inequality for x
First distribute the 0.15:
So the inequality becomes:
Subtract 30 from both sides:
Now divide both sides by 0.525:
Compute the division:
So must be at least about 171.43.
Interpret the result and choose the least whole number
Maya cannot sell a fraction of a ticket, so must be a whole number of tickets that is at least 171.43.
The smallest integer greater than or equal to 171.43 is 172.
So the least number of tickets beyond the first 100 that Maya must sell is 172, which corresponds to choice C.