At an architectural firm, the ratio of senior architects to junior architects to structura… — GMAT

At an architectural firm, the ratio of senior architects to junior architects to structural engineers was initially $3 : 7 : 4$ , respectively. After the firm hired $6$ additional senior architects and $4$ structural engineers left the firm, the ratio of senior architects to structural engineers became $3 : 2$ , while the number of junior architects remained unchanged. What was the total number of senior architects, junior architects, and structural engineers combined initially at the firm?

Answer: 56 employees

Answer

The total number of senior architects, junior architects, and structural engineers combined initially at the firm was 56.

Let the initial numbers of senior architects, junior architects, and structural engineers be

3x

7x

, and

4x

, respectively. After adding

6

senior architects and subtracting

4

structural engineers, the updated counts are

3x + 6

and

4x - 4

. Setting their ratio to

3 : 2

yields

\frac{3x + 6}{4x - 4} = \frac{3}{2}

. Cross-multiplying gives

6x + 12 = 12x - 12

, which simplifies to

6x = 24

, so

x = 4

. The total number of initial employees is

3x + 7x + 4x = 14x = 14(4) = 56

Step-by-Step Solution

Define initial quantities using ratio multiplier x

Senior Architects

= 3x

, Junior Architects

= 7x

, Structural Engineers

= 4x

Ratios specify relative proportions, so multiplying each term by

x

gives actual counts.

Formulate new quantities after staff changes

Senior Architects

= 3x + 6

, Structural Engineers

= 4x - 4

6 senior architects joined and 4 structural engineers left.

Set up ratio equation and solve for x

\frac{3x + 6}{4x - 4} = \frac{3}{2} \implies 2(3x + 6) = 3(4x - 4) \implies 6x + 12 = 12x - 12 \implies 6x = 24 \implies x = 4

Cross-multiplication converts the proportion into a linear equation.

Compute total initial staff count

Initial Total

= 3x + 7x + 4x = 14x = 14 \times 4 = 56

Summing all 3 initial ratio parts and substituting

x = 4

yields the total initial count.

Key Concept

Multi-part ratio formulation and linear equation setup based on additions and subtractions

Estimated Time:2m 0s

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