QuestionDoes a $6.5\%$ salary cut followed by a $6.5\%$ raise return your salary to its original amount? Explain your answer.

Studdy Solution

STEP 1

Assumptions1. The original salary is denoted as. . The salary is first reduced by6.5% and then increased by6.5%.
3. The percentage changes are applied consecutively, not simultaneously.

STEP 2

First, we need to calculate the salary after the reduction of6.5%. We can do this by multiplying the original salary by (1 - the reduction rate).
$Reduced\, salary = \\times (1 - Reduction\, rate)$

STEP 3

Now, plug in the given value for the reduction rate to calculate the reduced salary.
$Reduced\, salary = \\times (1 -6.5\%)$

STEP 4

Convert the percentage to a decimal value.
$6.\% =0.065$ $Reduced\, salary = \\times (1 -0.065)$

STEP 5

Calculate the reduced salary.
$Reduced\, salary = \\times0.935$

STEP 6

Now that we have the reduced salary, we can find the new salary after the increase of6.5%. This can be done by multiplying the reduced salary by (1 + the increase rate).
$New\, salary = Reduced\, salary \\times (1 + Increase\, rate)$

STEP 7

Plug in the values for the reduced salary and the increase rate to calculate the new salary.
$New\, salary = \\times0.935 \\times (1 +6.5\%)$

STEP 8

Convert the percentage to a decimal value.
$6.5\% =0.065$ $New\, salary = \\times0.935 \\times (1 +0.065)$

STEP 9

Calculate the new salary.
$New\, salary = \\times.935 \\times.065$

STEP 10

implify the expression to compare the new salary with the original salary.
$New\, salary = \\times0.995455$ The new salary is approximately99.5455% of the original salary, which means it is less than the original salary. Therefore, the employer's plan to reduce the salary by6.5% and then increase it by6.5% does not make the new monthly salary equal to the original monthly salary. Instead, it results in a slight decrease in the salary.

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