\(\mathrm{C(d) = 35 + 15(d - 4)}\)The function C gives the total cost, in dollars, for a mobile phone plan...
GMAT Algebra : (Alg) Questions
The function C gives the total cost, in dollars, for a mobile phone plan that uses d gigabytes of data in a month, for \(\mathrm{d \geq 4}\). If a customer's data usage, already over 4 gigabytes, increases by an additional 1.5 gigabytes, by how much will their cost increase, in dollars?
- 15.00
- 22.50
- 37.50
- 57.50
1. TRANSLATE the problem information
- Given information:
- Function: \(\mathrm{C(d) = 35 + 15(d - 4)}\)
- Data usage increases by an additional 1.5 gigabytes
- Need to find: increase in cost
- What this tells us: We need to determine how cost changes when data usage changes by 1.5 GB
2. INFER the key insight about linear functions
- This function is in the form \(\mathrm{C(d) = 35 + 15(d - 4)}\)
- In a linear function, the coefficient of the variable term represents the rate of change
- Here, 15 is the coefficient, meaning cost increases by $15 for every 1 gigabyte increase
3. SIMPLIFY the calculation
- Since cost increases by $15 per gigabyte
- For 1.5 gigabyte increase: \(\mathrm{15 \times 1.5 = 22.50}\)
Answer: B ($22.50)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students don't recognize that the coefficient 15 represents the rate of change. Instead, they try to substitute specific values into the function, picking arbitrary starting points like d = 4 or d = 5, then calculating \(\mathrm{C(d + 1.5) - C(d)}\). While this approach can work, it's unnecessarily complex and prone to arithmetic errors.
This leads to confusion about which values to substitute and may result in calculation mistakes that point to wrong answer choices.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret "increases by an additional 1.5 gigabytes" as meaning the new total usage is 1.5 GB (ignoring that it's already over 4 GB). This fundamental misunderstanding leads them to calculate \(\mathrm{C(1.5)}\) or focus on the wrong part of the function.
This may lead them to select Choice A ($15.00), thinking 1 gigabyte costs $15 so the answer must be $15.
The Bottom Line:
This problem tests whether students understand that in linear functions, the coefficient tells you the rate of change directly. Students who recognize this pattern can solve it in seconds, while those who don't may get bogged down in unnecessary substitution work.