Question 63·Hard·Linear Equations in One Variable
Birchwood Outfitters Daily Jacket Sales
On one day, Birchwood Outfitters sold 18 Premium Down jackets and 22 Lightweight Canvas jackets, taking in a total of $3,440 in revenue. The price of a Lightweight Canvas jacket is $40 less than the price of a Premium Down jacket.
What is the price, in dollars, of one Lightweight Canvas jacket?
For word problems about sales with a fixed price difference, start by defining clear variables for each price and immediately translate the “ dollars less than” or “more than” phrase into an equation relating the two. Next, write a total revenue equation using (number sold)(price) for each item and set it equal to the given total. Use substitution to reduce the system to one variable, solve carefully (watching your arithmetic with the difference amount), then plug back in to find the requested price and match it to the answer choices.
Hints
Set up the variables
Decide what each variable will represent. Let one variable be the price of a Premium Down jacket and the other be the price of a Lightweight Canvas jacket.
Use the price difference
Translate the phrase "the price of a Lightweight Canvas jacket is $40 less than the price of a Premium Down jacket" into an equation connecting your two variables. Which price is larger?
Write the total revenue equation
Think about how to express the total money made: it is (number of Premium Down)(their price) plus (number of Canvas)(their price). Set this equal to $3,440.
Solve using substitution
Use the equation that relates the two prices to substitute into the revenue equation so that you only have one variable, then solve step by step.
Desmos Guide
Represent the prices with variables
In Desmos, let be the price of a Premium Down jacket and be the price of a Lightweight Canvas jacket. Type the equation for the price relationship: y = x - 40.
Enter the revenue equation
Type the total revenue equation: 18x + 22y = 3440. Desmos will display this as a line on the same coordinate plane as the first equation.
Find the intersection point
Look for the point where the two lines intersect. The -coordinate of this point is the Premium Down price, and the -coordinate is the Lightweight Canvas price.
Read off the canvas price
Click on the intersection point and note the -value shown by Desmos. That -value is the price of one Lightweight Canvas jacket; compare it to the answer choices.
Step-by-step Explanation
Define the variables
Let:
- be the price of one Premium Down jacket (in dollars).
- be the price of one Lightweight Canvas jacket (in dollars).
We are asked to find .
Translate the price relationship into an equation
The problem says: "The price of a Lightweight Canvas jacket is $40 less than the price of a Premium Down jacket."
That means the canvas jacket is cheaper, so:
Write the revenue equation for total sales
Birchwood sold 18 Premium Down and 22 Lightweight Canvas jackets for a total of $3,440.
So the total revenue equation is:
This represents: (number of Premium Down)(price of Premium Down) plus (number of Canvas)(price of Canvas) equals total dollars collected.
Substitute to get an equation in one variable
Use in the revenue equation so everything is in terms of :
Distribute the 22:
Combine like terms:
Add 880 to both sides:
Divide both sides by 40:
So one Premium Down jacket costs $108.
Find the Lightweight Canvas price and match the answer
Use the relationship with :
So, the price of one Lightweight Canvas jacket is $68, which corresponds to choice B.