Question: A home repair service uses a pricing structure with a flat fee plus an hourly rate. The service charges a flat fee of b dollars and a constant hourly rate of m dollars per hour. For any job, the hourly rate is twice the flat fee, and a 5-hour job costs 44 dollars. What is the total cost, in dollars, of a 10-hour job?

Question:A home repair service uses a pricing structure with a flat fee plus an hourly rate. The service charges a...

GMAT Algebra : (Alg) Questions

Source: Prism

Algebra

Linear functions

MEDIUM

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Question:

A home repair service uses a pricing structure with a flat fee plus an hourly rate. The service charges a flat fee of $\mathrm{b}$ dollars and a constant hourly rate of $\mathrm{m}$ dollars per hour. For any job, the hourly rate is twice the flat fee, and a 5-hour job costs 44 dollars. What is the total cost, in dollars, of a 10-hour job?

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Solution

1. TRANSLATE the problem information

Given information:
- Flat fee: $\mathrm{b}$ dollars
- Hourly rate: $\mathrm{m}$ dollars per hour
- Hourly rate is twice the flat fee: $\mathrm{m = 2b}$
- A 5-hour job costs $44
- Need: cost of 10-hour job

What this tells us: We have a linear cost function $\mathrm{C(h) = mh + b}$ with a constraint relating $\mathrm{m}$ and $\mathrm{b}$

2. INFER the approach

Since we have one equation with two unknowns ($\mathrm{m}$ and $\mathrm{b}$), we need to use the constraint $\mathrm{m = 2b}$ to eliminate one variable
We'll substitute this constraint into our cost equation for the 5-hour job to solve for $\mathrm{b}$ first
Once we find $\mathrm{b}$, we can find $\mathrm{m}$, then calculate the 10-hour cost

3. Set up the equation for the 5-hour job

$\mathrm{C(5) = m(5) + b = 44}$
This gives us: $\mathrm{5m + b = 44}$

4. SIMPLIFY using substitution

Substitute $\mathrm{m = 2b}$ into $\mathrm{5m + b = 44}$:
$\mathrm{5(2b) + b = 44}$
$\mathrm{10b + b = 44}$
$\mathrm{11b = 44}$
$\mathrm{b = 4}$

5. Find the hourly rate

Since $\mathrm{m = 2b}$ and $\mathrm{b = 4}$:
$\mathrm{m = 2(4) = 8}$

6. Calculate the 10-hour job cost

$\mathrm{C(10) = m(10) + b}$
$\mathrm{C(10) = 8(10) + 4}$
$\mathrm{C(10) = 80 + 4}$
$\mathrm{C(10) = 84}$

Answer: 84

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle to correctly identify what "twice the flat fee" means mathematically, writing $\mathrm{b = 2m}$ instead of $\mathrm{m = 2b}$.

When they use $\mathrm{b = 2m}$, they get:

$\mathrm{5m + b = 44}$

$\mathrm{5m + 2m = 44}$

$\mathrm{7m = 44}$

$\mathrm{m = \frac{44}{7} \approx 6.29}$

Then $\mathrm{b = 2(\frac{44}{7}) = \frac{88}{7} \approx 12.57}$

For 10 hours:

$\mathrm{C(10) = (\frac{44}{7})(10) + \frac{88}{7}}$

$\mathrm{C(10) = \frac{440}{7} + \frac{88}{7}}$

$\mathrm{C(10) = \frac{528}{7} \approx 75.4}$

This leads to confusion since 75.4 isn't a clean integer answer, causing them to guess or round incorrectly.

Second Most Common Error:

Poor INFER reasoning: Students set up the equations correctly but don't recognize they need to use substitution to solve the system. Instead, they try to solve $\mathrm{5m + b = 44}$ and $\mathrm{m = 2b}$ separately or get confused about which equation to use first.

This causes them to get stuck early in the problem and resort to guessing.

The Bottom Line:

This problem tests whether students can correctly translate verbal constraints into mathematical relationships and then systematically use substitution to solve a constrained system. The key insight is recognizing that "hourly rate is twice the flat fee" means $\mathrm{m = 2b}$, not the reverse.

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