Question 33·Hard·Linear Equations in One Variable
A landscaper rents a wood chipper. The rental company charges a delivery fee of $60 plus $45 for each hour the wood chipper is used. The landscaper has a coupon that reduces the entire bill (delivery fee and hourly charges together) by 15%. If the landscaper’s total charge after the coupon was applied was $1,071, which of the following equations could be used to determine , the number of hours the landscaper used the wood chipper?
For word problems about discounts, first write an expression for the full original amount in terms of the variable, then convert the percent discount into a multiplier (for example, 15% off means multiply by 0.85 or by 1 − 0.15). Multiply the entire original expression by this discount factor, set it equal to the given final amount, and then match this equation to the answer choices; always check whether the discount is applied to the whole amount or only part of it.
Hints
Find the total before the coupon
First write an expression for the total cost before any discount, using 60 dollars for delivery plus 45 dollars times the number of hours h.
Think about what 15% off means
If a bill is reduced by 15%, what fraction of the original bill do you still pay? How can you write that as a decimal or an expression like 1 minus something?
Connect the discount to the final total
Once you have the expression for the original total and the fraction you pay, multiply them together. That product should equal the final charge of 1,071 dollars.
Desmos Guide
Define the original total cost
In Desmos, type B(h) = 60 + 45h to represent the total cost before the coupon, where h is the number of hours.
Apply the 15% discount
Next, type C(h) = B(h) * (1 - 0.15) to represent the cost after the 15% discount is applied to the entire original bill.
Set the discounted cost equal to 1071
Now type C(h) = 1071. This equation in Desmos shows the relationship between h and the final charge of 1,071 dollars after the coupon. Compare this equation to the answer choices and select the one whose left-hand side matches the expression you used for C(h) when it is set equal to 1,071.
Step-by-step Explanation
Write the total cost before the coupon
The landscaper pays a 60-dollar delivery fee plus 45 dollars for each hour the chipper is used.
So the total cost before the coupon is applied is
Translate the 15% coupon into a multiplier
A 15% discount means the customer pays 100% − 15% = 85% of the original cost.
As a decimal, 85% is , and we can write this as
So to get the discounted price, multiply the original total by (or ).
Relate the discounted total to 1,071
Let the total before the coupon be . After applying a 15% discount, the landscaper pays
- discounted total = .
We are told that this discounted total is 1,071 dollars, so the key relationship is
- .
Substitute the expression for the original total
From Step 1, the total before the coupon is . Substitute this into the equation from Step 3:
This matches answer choice A, so the correct equation is .