A corporation recently expanded both its sales department and its technical department. Immediately before expansion, the number of employees in...

Phase 1: Owning the Dataset

Visual Representation - Before/After Table:

Key relationship: Before expansion, Sales = 60% of Technical

Let's work with concrete numbers. Since we need Sales to be 60% of Technical, let's call:

• \(\mathrm{T_{before}}\) = number of technical employees before expansion
• \(\mathrm{S_{before}}\) = number of sales employees before expansion = \(0.6 \times \mathrm{T_{before}}\)

Phase 2: Understanding the Question

We need to find percent increases for both departments such that:

1. The initial 60% relationship holds
2. Sales ends up with exactly 12 employees
3. Technical ends up with exactly 15 employees

The percent increase formula is: \(\frac{\mathrm{New} - \mathrm{Old}}{\mathrm{Old}} \times 100\%\)

Phase 3: Finding the Answer

Let's find values that work. If \(\mathrm{T_{before}} = 10\), then:

• \(\mathrm{S_{before}} = 0.6 \times 10 = 6\) employees

Checking if this leads to our target numbers:

• Sales: \(6 \rightarrow 12\) means an increase of 6 employees
• Percent increase = \(\frac{6}{6} \times 100\% = 100\%\) ✓ (in choices)
• Technical: \(10 \rightarrow 15\) means an increase of 5 employees
• Percent increase = \(\frac{5}{10} \times 100\% = 50\%\) ✓ (in choices)

Verification:

• Initial ratio: \(\frac{6}{10} = 0.6 = 60\%\) ✓
• Sales after 100% increase: \(6 + 6 = 12\) ✓
• Technical after 50% increase: \(10 + 5 = 15\) ✓

Phase 4: Solution

Updated Visual:

The sales department experienced a 100% increase, and the technical department experienced a 50% increase.