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Difficulty: HardRatios, Rates, and Proportions

At a marketing firm, the ratio of designers to writers is 3:43:4, and the ratio of writers to managers is 5:25:2. The firm hires 1010 new designers and 44 new managers, while the number of writers remains the same. If the new ratio of designers to managers is 2:12:1, how many writers are employed at the firm?

  1. A
    8
  2. B
    20
  3. C
    30
  4. 40Answer
  5. E
    120

Answer

There are 40 writers employed at the firm.
To find the number of writers, we combine the ratios of designers to writers (3:43:4) and writers to managers (5:25:2) by finding a common multiple for the number of writers (20). This gives a combined ratio of 15:20:815:20:8 for designers, writers, and managers. Representing these quantities as 15x15x, 20x20x, and 8x8x, we can write the equation for the new ratio of designers to managers: 15x+108x+4=2\frac{15x + 10}{8x + 4} = 2. Solving this equation gives 15x+10=16x+815x + 10 = 16x + 8, which simplifies to x=2x = 2. The number of writers is 20(2)=4020(2) = 40.

Step-by-Step Solution

1
Align the two given ratios to find a common ratio.
The ratio of designers to writers is 3:4=15:203:4 = 15:20. The ratio of writers to managers is 5:2=20:85:2 = 20:8. The combined ratio of designers to writers to managers is 15:20:815:20:8.
To analyze the changes across the different categories, we need a single ratio that shares the common term (writers).
2
Express the quantities of employees in terms of a variable xx.
Let the number of designers be 15x15x, writers be 20x20x, and managers be 8x8x.
This allows us to represent the actual number of employees algebraically.
3
Set up an equation based on the new ratio of designers to managers after the new hires.
15x+108x+4=21\frac{15x + 10}{8x + 4} = \frac{2}{1}
The firm hired 10 designers and 4 managers, and the ratio of these new quantities is 2:12:1.
4
Solve the equation for xx.
15x+10=2(8x+4)  ⟹  15x+10=16x+8  ⟹  x=215x + 10 = 2(8x + 4) \implies 15x + 10 = 16x + 8 \implies x = 2.
Solving for xx gives us the multiplier to find the actual employee counts.
5
Calculate the total number of writers.
W=20(2)=40W = 20(2) = 40.
The number of writers is represented by 20x20x.

Key Concept

Combining multiple ratios and setting up algebraic equations from changing rates/proportions.

Alternative Method

Express the number of designers and managers in terms of the number of writers, WW. The number of designers is 34W\frac{3}{4}W, and the number of managers is 25W\frac{2}{5}W. After hiring, the equation is 34W+1025W+4=2\frac{\frac{3}{4}W + 10}{\frac{2}{5}W + 4} = 2. Solving for WW yields 34W+10=45W+8  ⟹  2=120W  ⟹  W=40\frac{3}{4}W + 10 = \frac{4}{5}W + 8 \implies 2 = \frac{1}{20}W \implies W = 40.
Estimated Time:2m 0s
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Topics

Ratios, Rates, and Proportions

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