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Math  /  Algebra

Question3. Malik's salary is $25,500\$ 25,500 per year, which he expects will increase by a constant dollar amount annually. In 12 years, his salary will have doubled. Assuming salar increases take place only at the end of a full year, how many years must Malik wait until his salary is at least \$40,000 annually?

Studdy Solution

STEP 1

1. Malik's current salary is 25,500peryear.<br/>2.Hissalaryincreasesbyaconstantdollaramountannually.<br/>3.In12years,hissalarywillhavedoubled.<br/>4.Weneedtofindouthowmanyyearsitwilltakeforhissalarytoreachatleast25,500 per year.<br />2. His salary increases by a constant dollar amount annually.<br />3. In 12 years, his salary will have doubled.<br />4. We need to find out how many years it will take for his salary to reach at least 40,000.

STEP 2

1. Determine the annual increase in salary.
2. Set up an equation to find the number of years until the salary reaches at least $40,000.
3. Solve the equation to find the number of years.

STEP 3

Determine the annual increase in salary.
Since Malik's salary doubles in 12 years, his salary after 12 years will be $2 \times 25,500 = 51,000.
The increase in salary over 12 years is $51,000 - 25,500 = 25,500.
The annual increase is therefore:
Annual Increase=25,50012 \text{Annual Increase} = \frac{25,500}{12}

STEP 4

Set up an equation to find the number of years until the salary reaches at least $40,000.
Let n n be the number of years it takes for the salary to reach at least $40,000. The salary after \( n \) years is given by:
25,500+n×Annual Increase≥40,000 25,500 + n \times \text{Annual Increase} \geq 40,000
Substitute the annual increase from Step 1:
25,500+n×25,50012≥40,000 25,500 + n \times \frac{25,500}{12} \geq 40,000

STEP 5

Solve the equation to find the number of years.
First, simplify the inequality:
n×25,50012≥40,000−25,500 n \times \frac{25,500}{12} \geq 40,000 - 25,500
n×25,50012≥14,500 n \times \frac{25,500}{12} \geq 14,500
Multiply both sides by 12 to clear the fraction:
n×25,500≥14,500×12 n \times 25,500 \geq 14,500 \times 12
n×25,500≥174,000 n \times 25,500 \geq 174,000
Divide both sides by 25,500 to solve for n n :
n≥174,00025,500 n \geq \frac{174,000}{25,500}
n≥6.8235 n \geq 6.8235
Since n n must be a whole number (as salary increases occur annually), round up to the nearest whole number:
n=7 n = 7
Malik must wait at least 7 \boxed{7} years for his salary to reach at least $40,000 annually.

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