Problem Solving

I'm having difficulty following the OG answer explanation and therefore would greatly appreciate alternative, easier to follow explanations, please. Thank you in advance, experts!

Q: A survey of employers found that during 1993 employment costs rose 3.5 percent, where employment costs consist of salary costs and fringe-benefit costs. If salary cost rose 3 percent and fringe-benefit costs rose 5.5 percent during 1993, then fringe-benefit costs represented what percent of employment costs at the beginning of 1993?

A) 16.5%
B) 20%
C) 35%
D) 55%
E) 65%

basso25@ wrote: A survey of employers found that during 1993 employment costs rose 3.5 percent, where employment costs consist of salary costs and fringe-benefit costs. If salary cost rose 3 percent and fringe-benefit costs rose 5.5 percent during 1993, then fringe-benefit costs represented what percent of employment costs at the beginning of 1993?

A) 16.5%
B) 20%
C) 35%
D) 55%
E) 65%

This is a MIXTURE problem:
Increase in salary costs = 3%.
Increase in fringe-benefit costs = 5.5%.
Increase in the cost of the MIXTURE (salaries + fringe-benefits) = 3.5%.

Use ALLIGATION -- a very good way to handle MIXTURE PROBLEMS.

Let S = the increase in salary costs and F = the increase in fringe-benefit costs.

Step 1: Plot the 3 percentages on a number line, with the percentages attributed to the two ingredients (3% and 5.5%) on the ends and the percentage attributed to the mixture (3.5%) in the middle.
S 3%----------3.5%-----------F 5.5%

Step 2: Calculate the distances between the percentages.
S 3%----.5----3.5%-----2----F 5.5%

Step 3: Determine the ratio in the mixture.
The ratio of S to F in the mixture is the RECIPROCAL of the distances in red.
S:F = 2 : .5 = 4:1.

Since S:F = 4:1, and 4+1=5, of every $5 spent, $4 was spent on salaries and $1 was spent on fringe benefits.
Thus:
F/total = 1/5 = 20%.

The correct answer is B.

For two other problems that I solved with alligation, check here:

https://www.beatthegmat.com/ratios-fract ... tml#484583

basso25@ wrote:I'm having difficulty following the OG answer explanation and therefore would greatly appreciate alternative, easier to follow explanations, please. Thank you in advance, experts!

Q: A survey of employers found that during 1993 employment costs rose 3.5 percent, where employment costs consist of salary costs and fringe-benefit costs. If salary cost rose 3 percent and fringe-benefit costs rose 5.5 percent during 1993, then fringe-benefit costs represented what percent of employment costs at the beginning of 1993?

A) 16.5%
B) 20%
C) 35%
D) 55%
E) 65%

Salary Costs = S; Fringe = F

S+F = 100 ..........[1]
1.03S+1.055F = 103.5 ...........[2][Salary went up by 3% and Fringe went up by 5.5% amounting to a 3.5 % rise in the total]

Multiplying [1] by 1.03 and solvong we get : 0.025F = 0.5 => F = 20.
Total = 100 => F = 20%-- [Ans]

Let total cost @ beginning of 1993 be 100
After 3.5% rise, total cost at end of 1993 = 103.5

Now Total Cost = EMployment cost (EC) + Fringe benefit cost (FBC)
beg. of 1993: Let EC be x and FBC be y such that x + y = 100
End of 1993: EC becomes 1.03X and FBC becomes 1.055Y such that 1.03X + 1.055Y = 103.5

Now solving the 2 equations: 1.03 (X + Y) + 0.025Y = 103.5
103 + 0.025Y = 103.5
0.025Y = 0.5
Y = 20%
Ans B