SimaQ wrote:
A part-time employee whose hourly wage was increased by 25 percent decided to reduce the number of hours worked per week so that the employee's total weekly income would remain unchanged. By what percent should the number of hours worked be reduced?
(A) 12.5%
(B) 20%
(C) 25%
(D) 50%
(E) 75%
30-sec approach: By what percent must hours worked decrease so that total weekly income remains unchanged?
By 1 - (fractional inverse of the percent increase)When the original quantity is the start and end value, percent increase and percent decrease are inversely proportional. Use a fraction for the percent increase of one factor. Flip that fraction. Subtract from 1. The result is the percent by which the other factor must decrease. Thus:
\(xy = 1\)
\(x\) increases by 25% = 1.25* = \(\frac{5}{4}\)
Flip to \(\frac{4}{5}\) and subtract from 1 to yield percent by which \(y\) must decrease:
\(1 - \frac{4}{5} = \frac{1}{5}\) = 20 percent
(Or assess the flipped fraction: \(\frac{4}{5}\) = .8 = 80 percent = 20 percent decrease)
Note: if \(xy = 1\), then
\(\frac{5}{4}x *
\frac{4}{5}y =
1\)
Longer method, w/ numbers (no need!):
x = wage per hour = $20
y = # of hours worked = 10
(Wage per hour)(# of hours) = total weekly income
Total wages = $20 * 10 = $200
x, $20, increases by 25 percent:
x = \(\frac{5}{4}(20)\) = 25. We want $200. What is y?
\((25)(y)= 200\), so \(y = 8\)
By what percent did y decrease?
\(\frac{new-old}{old} =\frac{10-8}{10} =\frac{1}{5} = .20 * 100 =\) 20%
In other words, when you want the original quantity to remain the same and one factor increases by a percent, to calculate the corresponding percent decrease, finding and flipping the fraction works.
Answer B
*An increase of 25 percent =
\(1.25 =1\frac{25}{100} = 1\frac{1}{4} = \frac{5}{4}\)
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