On January 1, 2024, the monthly price of a certain streaming plan is $6.99. The price increases by $1.20 on...
GMAT Algebra : (Alg) Questions
On January 1, 2024, the monthly price of a certain streaming plan is \(\$6.99\). The price increases by \(\$1.20\) on January 1 of each of the next 4 years. Let \(\mathrm{y}\) denote the calendar year, where \(\mathrm{y = 2025, 2026, 2027, or 2028}\). Which function \(\mathrm{P(y)}\) models the monthly price, in dollars, in year \(\mathrm{y}\)?
1. TRANSLATE the problem information
- Given information:
- Starting price January 1, 2024: $6.99
- Price increases by $1.20 each January 1st for the next 4 years
- Need function \(\mathrm{P(y)}\) for years 2025, 2026, 2027, 2028
- What this tells us: The price grows linearly, increasing by the same amount each year
2. INFER the mathematical relationship
- Key insight: By year y, the number of $1.20 increases that have occurred is \(\mathrm{(y - 2024)}\)
- This is because the first increase happens in 2025, second in 2026, etc.
- So the function structure is: Starting price + (Number of increases) × (Amount per increase)
3. Set up the initial function
\(\mathrm{P(y) = 6.99 + 1.20(y - 2024)}\)
4. SIMPLIFY to match the answer format
- Expand the expression:
\(\mathrm{P(y) = 6.99 + 1.20y - 1.20(2024)}\)
- Calculate \(\mathrm{1.20 \times 2024 = 2428.8}\) (use calculator)
- Combine the constants:
\(\mathrm{P(y) = 6.99 + 1.20y - 2428.8}\)
\(\mathrm{P(y) = 1.20y - 2421.81}\)
5. Verify with a test point
- For \(\mathrm{y = 2025}\): \(\mathrm{P(2025) = 1.20(2025) - 2421.81 = 2430 - 2421.81 = 8.19}\) ✓
- This matches our expected price of $6.99 + $1.20 = $8.19
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students incorrectly determine when increases occur, thinking the number of increases by year y is just y, leading to \(\mathrm{P(y) = 6.99 + 1.20y}\).
They don't realize that calendar year 2025 corresponds to exactly 1 increase (not 2025 increases), year 2026 to 2 increases, etc. This fundamental misunderstanding of the time relationship makes them gravitate toward the simpler but incorrect relationship.
This may lead them to select Choice A (6.99 + 1.20y).
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret which year serves as the reference point, using \(\mathrm{(y - 2025)}\) instead of \(\mathrm{(y - 2024)}\), thinking increases start being counted from 2025.
This creates \(\mathrm{P(y) = 6.99 + 1.20(y - 2025)}\), which gives $6.99 for year 2025 instead of the correct $8.19, missing the fact that one increase has already occurred by 2025.
This may lead them to select Choice D (1.20(y - 2025) + 6.99).
The Bottom Line:
This problem challenges students to correctly model the timing relationship between calendar years and cumulative price increases. Success requires understanding that "by year y" means counting all increases up to and including that year, making the reference point crucial for the correct function.