Question:For a certain budget, the ratio of savings to expenses is 2:5. If the expenses decrease by 6 units, how...

1. TRANSLATE the ratio relationship

• Given information:
  • Savings to expenses ratio = 2:5
  • Expenses decrease by 6 units
  • Need to maintain the same ratio

• This tells us: \(\mathrm{savings/expenses = 2/5}\) must stay constant

2. INFER the key insight

• If the ratio must remain 2:5, then when expenses change, savings must change proportionally
• We need to find how much savings must change when expenses drop by 6 units

3. TRANSLATE the mathematical setup

• Let original savings = S and original expenses = E
• From the ratio: \(\mathrm{S/E = 2/5}\), so \(\mathrm{S = (2/5)E}\)
• After change: New expenses = \(\mathrm{E - 6}\)
• To maintain ratio: \(\mathrm{New\ savings/(E - 6) = 2/5}\)

4. SIMPLIFY to find the new savings amount

• New savings = \(\mathrm{(2/5)(E - 6)}\)
• New savings = \(\mathrm{(2/5)E - (2/5)(6)}\)
• New savings = \(\mathrm{S - 2.4}\)

5. INFER the final answer

• Change in savings = New savings - Original savings
• Change = \(\mathrm{(S - 2.4) - S = -2.4}\)
• The negative sign means savings must decrease

Answer: A. It must decrease by 2.4 units

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "maintain the ratio" to mean that if expenses decrease by 6, then savings should also decrease by 6 to "keep things proportional."

They think: "Both parts of the budget should change by the same amount to stay balanced." This leads to the incorrect reasoning that savings must decrease by 6 units.

This may lead them to select Choice C (decrease by 6 units).

Second Most Common Error Path:

Poor INFER reasoning: Students correctly set up the initial ratio but incorrectly assume that decreasing expenses means savings should increase to "compensate" for the lost expenses.

They reason: "If we're spending less, we should be saving more by the same amount." This backward logic suggests savings increase by 6 units.

This may lead them to select Choice D (increase by 6 units).

The Bottom Line:

The key insight is that ratios describe proportional relationships, not equal changes. When one part of a ratio changes, the other part must change by the corresponding fraction, not by the same absolute amount. Understanding that \(\mathrm{2/5}\) of any change in expenses determines the change in savings is crucial for success.