Question 54·200 Super-Hard SAT Math Questions·Algebra

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A rehearsal studio charges a monthly membership fee of $120 that includes up to 6 hours of rehearsal time. Each additional hour beyond 6 hours costs $22. In any month when a member uses more than 6 hours, the studio also applies a one-time $18 credit to that month’s bill before tax is calculated.

After the credit is applied, the studio charges an 8% tax on the remaining amount. After the tax is added, the studio applies a 25% promotional discount to the taxed amount. Finally, the studio adds a $6 booking fee (this fee is added at the end and is not taxed or discounted).

Which equation gives the total amount $y$ , in dollars, that a member is charged for $x$ hours of rehearsal in a month, where $x>6$ ?

A fixed fee plus a charge for each extra hour gives a linear model: each extra hour raises the bill by the same amount. Count only hours beyond the included allowance, then apply the tax and discount in order. Use Desmos to find the adjusted hourly rate and the charge for one valid number of hours. The booking fee goes on last, untouched by either percentage.

Hints

Hint 1
The membership includes the first $6$ hours. If a member uses $x$ hours, the studio charges for $x-6$ extra hours, not $x$ hours. What does that make the hourly charge?
Hint 2
An $8\%$ tax multiplies the amount remaining after the credit by $1.08$ . A $25\%$ discount then multiplies the taxed amount by $0.75$ . Keep the booking fee outside both multiplications.
Hint 3
All four equations have the same number multiplying $x$ , called the slope. Find the bill for a valid input such as $x=7$ , then use that bill and the slope to find the number added to $x$ .

Step-by-step

Build the bill, then find its constant term
Step 1
Write the amount before tax
The membership covers the first $6$ hours, so $x$ hours means $x-6$ hours at $22 each. Add the $120 membership fee and subtract the one-time credit to get the pre-tax amount:
$120+22(x-6)-18$
Step 2
Apply the changes in order
Tax adds $8\%$ , so multiply the amount after the credit by $1.08$ . The discount leaves $75\%$ , so multiply the taxed amount by $0.75$ . Add the untaxed, undiscounted booking fee last:
$y=0.75\bigl(1.08(120+22(x-6)-18)\bigr)+6$
Step 3
Find the adjusted hourly rate
Each extra hour adds $22 before the percentages. The fixed fees and credit don't change with the hours, so multiply only $22 by the tax and discount factors. Type $22\cdot1.08\cdot0.75$ in Desmos; it shows $17.82$ . That is the slope, or increase per extra hour, in the final bill.
Step 4
Get one bill from the rule
Choose $x=7$ because it satisfies $x>6$ . In Desmos, type $x_{1}=7$ , then evaluate the bill expression with $x_{1}$ in place of $x$ . Desmos shows $106.44$ , so $7$ hours costs $106.44 after every adjustment.
Step 5
Find the constant and match the equation
A linear equation has the form $y=mx+b$ , where $m$ is the slope and $b$ is the constant term. Use the bill at $x=7$ :
$106.44=17.82(7)+b$
Subtract $17.82(7)$ :
$b=106.44-17.82(7)$
Type $106.44-17.82x_{1}$ in Desmos; it shows $-18.3$ . So the equation for the charge when $x>6$ is $y=17.82x-18.3$ . The negative constant isn't a bill for zero hours: this rule applies only when $x>6$ . Choice C.

Question 54·200 Super-Hard SAT Math Questions·Algebra

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