Selling the same item two ways with a total item count and a total revenue calls for a one-variable revenue equation. Write the count not sold individually using the total, then turn that count into boxes before assigning the box price. Enter the finished equation as a Desmos regression to find the individual count. The tempting slip is charging the box price for every muffin in a box.
Hints
- Hint 1
If counts individual muffins, the muffins in boxes are whatever remains after taking from the total. What expression gives that remaining count? You don't need a second variable.
- Hint 2
The box price is per box, not per muffin. Four muffins make one box, so divide the remaining muffin count by to find how many boxes were sold.
- Hint 3
A revenue equation adds dollars from each kind of sale. Multiply the individual count by its price and the box count by its price. What total must those two amounts reach?
Step-by-step
Approach 1: Write a revenue equation
Step 1Find the muffins sold in boxes
The problem says muffins were sold individually out of sold in all. Subtract those from the total to get the muffins sold in boxes:
- Step 2
Turn boxed muffins into boxes
A box's price applies to boxes, not muffins. Each box holds muffins, so divide the boxed-muffin count by to get the number of boxes:
- Step 3
Add the revenue from both kinds of sale
Individual muffins bring in dollars. Boxes bring in dollars. Their combined revenue is $134.40, so:
- Step 4
Solve for the individual count
Type in Desmos. Use instead of and instead of so Desmos finds the unknown rather than graphing an equation. Under PARAMETERS, it shows . Since counts muffins sold individually, muffins were sold individually. Grid in 16.
Approach 2: Compare with selling everything in boxes
Step 1Find an all-boxes starting amount
Imagine all muffins were sold in boxes. Type in Desmos: counts the boxes, and multiplying by gives $132 in revenue. Desmos shows .
- Step 2
Find the boxed price per muffin
To compare the two ways of selling one muffin, divide a box's price among its muffins. The boxed price per muffin is:
- Step 3
Find the extra revenue per individual muffin
Each individually sold muffin brings in more than a boxed muffin. Subtract the boxed price per muffin from the individual price:
- Step 4
Count the muffins that earned the extra revenue
The actual revenue exceeds the all-boxes amount. Divide that difference by the extra $0.15 per individual muffin. Add in Desmos; it shows . So muffins were sold individually. Grid in 16.