An account shows a final balance of $22 after two transactions. First, the balance increases by $6 from a deposit,...

1. TRANSLATE the problem information

• Given information:
  • Final balance after two transactions: \(\$22\)
  • First transaction: deposit of \(\$6\) (increases balance)
  • Second transaction: withdrawal of \(\$14\) (decreases balance)
  • Need to find: initial balance before these transactions

2. INFER the approach

• Since we know the final result and the changes, we need to work backwards
• We can represent the initial balance as a variable and track how it changes
• Let's call the initial balance x

3. TRANSLATE each transaction step

• Starting balance: \(\mathrm{x}\)
• After deposit: \(\mathrm{x + 6}\)
• After withdrawal: \(\mathrm{(x + 6) - 14 = x - 8}\)
• This final amount equals \(\$22\)

4. SIMPLIFY by setting up and solving the equation

• Set up the equation: \(\mathrm{x - 8 = 22}\)
• Solve for x: \(\mathrm{x = 22 + 8 = 30}\)

5. Check our answer

• Start with \(\$30\)
• After +\(\$6\) deposit → \(\$36\)
• After -\(\$14\) withdrawal → \(\$22\) ✓

Answer: C (30)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students might misinterpret which direction to work in the problem. Instead of working backwards from the final balance, they might think: "If the balance increased by \(\$6\) and decreased by \(\$14\), then I need to add both amounts to \(\$22\) to get the starting balance."

This leads them to calculate \(\mathrm{22 + 6 + 14 = 42}\), causing them to select Choice D (42).

Second Most Common Error:

Poor INFER strategy: Students might correctly set up that they need to work backwards, but get confused about the signs. They might think the initial balance needs both transactions added to it: \(\mathrm{x + 6 + 14 = 22}\), which gives \(\mathrm{x = 22 - 20 = 2}\). Since 2 isn't an option, this leads to confusion and guessing.

The Bottom Line:

The key challenge is recognizing that when working backwards from a final balance, you need to "undo" the transactions in reverse. A deposit that was added needs to be subtracted from the final balance, and a withdrawal that was subtracted needs to be added back.