Was Store K's profit last month at least 10% greater than its profit the previous month? Store K's expenses last...

Understanding the Question

We need to determine whether Store K's profit last month was at least 10% greater than its profit the previous month. This is a yes/no question.

Let's restate this clearly:

• Previous month profit = Revenue - Expenses
• Last month profit = Revenue - Expenses
• Question: Is (Last month profit) \(\geq 1.1 \times\) (Previous month profit)?

Remember: For sufficiency, we need to definitively answer YES or NO - not calculate exact profit amounts.

Key Insight

This is a "competing percentages" problem. Revenue and expenses both changed, pulling profit in opposite directions. The critical question is: which effect dominates?

Analyzing Statement 1

Statement 1: Expenses last month were 5% greater than the previous month.

This tells us expenses increased, which reduces profit. But without knowing what happened to revenues, we can't determine the net effect on profit.

Testing Different Scenarios

Let's explore what could happen:

Scenario 1 - Revenues stayed the same:

• Previous: Revenue \(\$1000\), Expenses \(\$500\), Profit \(\$500\)
• Last month: Revenue \(\$1000\), Expenses \(\$525\) (+5%), Profit \(\$475\)
• Profit change: \(\$475/\$500 = 0.95 = 5\%\) decrease → Answer is NO

Scenario 2 - Revenues increased significantly:

• Previous: Revenue \(\$1000\), Expenses \(\$500\), Profit \(\$500\)
• Last month: Revenue \(\$1200\) (+20%), Expenses \(\$525\) (+5%), Profit \(\$675\)
• Profit change: \(\$675/\$500 = 1.35 = 35\%\) increase → Answer is YES

Since we can get both YES and NO answers, Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now we forget Statement 1 and analyze Statement 2 independently.

Statement 2: Revenues last month were 10% greater than the previous month.

This tells us revenues increased, which increases profit. But without knowing what happened to expenses, we can't determine if profit increased by at least 10%.

Testing Different Scenarios

Scenario 1 - Expenses also increased by 10%:

• Previous: Revenue \(\$1000\), Expenses \(\$600\), Profit \(\$400\)
• Last month: Revenue \(\$1100\) (+10%), Expenses \(\$660\) (+10%), Profit \(\$440\)
• Profit change: \(\$440/\$400 = 1.10 = 10\%\) increase → Answer is YES

Scenario 2 - Expenses increased by more than 10%:

• Previous: Revenue \(\$1000\), Expenses \(\$600\), Profit \(\$400\)
• Last month: Revenue \(\$1100\) (+10%), Expenses \(\$720\) (+20%), Profit \(\$380\)
• Profit change: \(\$380/\$400 = 0.95 = 5\%\) decrease → Answer is NO

Different expense changes lead to different answers, so Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Both Statements

Now we know from both statements:

• Revenues increased by 10%
• Expenses increased by 5%

The Critical Insight

Here's the key: revenues are growing faster than expenses (10% vs 5%). This creates a powerful mathematical relationship that guarantees our answer.

Let's test extreme cases to verify this always works:

Test Case 1 - High profit margin store:

• Previous: Revenue \(\$1000\), Expenses \(\$100\), Profit \(\$900\)
• Last month: Revenue \(\$1100\) (+10%), Expenses \(\$105\) (+5%), Profit \(\$995\)
• Profit growth: \((\$995 - \$900)/\$900 = \$95/\$900 = 10.6\%\) → Answer is YES ✓

Test Case 2 - Low profit margin store:

• Previous: Revenue \(\$1000\), Expenses \(\$900\), Profit \(\$100\)
• Last month: Revenue \(\$1100\) (+10%), Expenses \(\$945\) (+5%), Profit \(\$155\)
• Profit growth: \((\$155 - \$100)/\$100 = \$55/\$100 = 55\%\) → Answer is YES ✓

Why does this always work?

When revenues grow by 10% and expenses grow by only 5%:

• The extra revenue (10% of a larger number) always exceeds
• The extra expenses (5% of a smaller number)
• This guarantees profit growth exceeds 10%, regardless of the starting profit margin

With both statements together, we can definitively answer YES.

[STOP - Sufficient!] This eliminates choice E.

The Answer: C

Both statements together allow us to definitively answer that YES, Store K's profit last month was at least 10% greater than the previous month. Neither statement alone is sufficient.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."