Hana deposited a fixed amount into her bank account each month. The function \(\mathrm{f(t) = 100 + 25t}\) gives the...

1. INFER the function structure

• Given: \(\mathrm{f(t) = 100 + 25t}\) where \(\mathrm{t}\) = number of monthly deposits
• This is a linear function in the form \(\mathrm{f(t) = b + mt}\)
• We need to understand what each part represents in context

2. INFER what each component means

• In \(\mathrm{f(t) = 100 + 25t}\):
  • 100 is the constant term (what happens when \(\mathrm{t = 0}\))
  • 25 is the coefficient of \(\mathrm{t}\) (how much \(\mathrm{f(t)}\) changes per unit of \(\mathrm{t}\))
• Since \(\mathrm{t}\) represents monthly deposits, 25 tells us how much the account changes with each deposit

3. TRANSLATE the coefficient to real-world meaning

• The coefficient 25 means: for every 1 additional monthly deposit, the account balance increases by $25
• This directly matches option A: "With each monthly deposit, the amount increased by $25"

4. Verify by checking other options

• Option B: Initial amount = \(\mathrm{f(0) = 100 + 25(0) = \$100}\) (not $25)
• Option C: After 1 deposit = \(\mathrm{f(1) = 100 + 25(1) = \$125}\) (not $25)
• Option D: 25 is the rate per deposit, not total number of deposits

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the constant term with the coefficient

Many students see both 100 and 25 in the function and don't clearly understand which represents what. They might think the question is asking about the initial amount (100) or get confused about which number corresponds to "increase per deposit." This leads to random guessing between the options.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what "with each monthly deposit" means

Some students understand that 25 is important but think it represents the total after 1 deposit rather than the increase per deposit. They calculate \(\mathrm{f(1) = 125}\) and incorrectly think this means the account "increased by $125" rather than "is now $125." This may lead them to select Choice C ($25) by incorrectly reasoning that if the total is $125 and started at $100, the deposit was $25 (confusing deposit amount with rate of increase).

The Bottom Line:

This problem tests whether students understand the difference between the constant term (initial value) and the coefficient (rate of change) in linear functions, and can correctly interpret these in real-world contexts.