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📝 Work hours inequality problems (14 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 14 questions available

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What is Work hours inequality problems?

Definition:
Work hours inequality problems involve constraints on the number of hours worked, where the total hours must be within a certain range (e.g., minimum and maximum), expressed using inequalities like $\text{Minimum} \le \text{Hours} \le \text{Maximum}$ , and these are common in labor laws, shift scheduling, and project management to ensure fair and legal working conditions.

Working:
To solve, define the variable for hours, set up the inequality based on given limits (e.g., at least 20 hours" means $\ge 20$ and "at most 40 hours" means $\le 40$ ) combine into a compound inequality and solve for the variable; if there are multiple workers or days use sums to model total hours and round to appropriate values.

Example:
A part-time employee can work between 15 and 25 hours per week; if they already worked 10 hours and $x$ hours remain then $15 \le 10 + x \le 25$ so $5 \le x \le 15$ meaning they can work between 5 and 15 more hours this week.

Reason:
Work hours inequalities are essential for ensuring compliance with labor regulations fair scheduling and efficient time management and they help both employers and employees plan work hours effectively."

Easy

Medium

Hard

📝 All Work hours inequality problems MCQs

Q1. A worker earns $18 per hour and wants to save$ 720. If $180 of the target is already saved, how many additional work hours are required, assuming all earnings can be saved?

A.20 hours

B.30 hours ✅

C.40 hours

D.50 hours

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The remaining amount is $720-180=540$ dollars. At $18 per hour, the required work time is $540/18=30$ hours. The key modeling step is subtracting the amount already saved before converting the remaining financial goal into work hours.

Q2. Which equation best represents the number of hours $h$ needed to reach a savings goal $G$ when a worker already has $S$ dollars and earns $r$ dollars per hour?

h=(G+S)r

h=(G-S)/r

✅

h=r/(G-S)

h=(G-S)r

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The worker needs only $G-S$ additional dollars. Since each hour produces $r$ dollars, dividing the remaining amount by the hourly earning rate gives $h=(G-S)/r$ . The other choices incorrectly multiply or invert the rate.

Q3. Maya earns $16 per hour and needs$ 1,200 for a course. She already has $400, but expects$ 120 in unavoidable transportation costs while working. If those costs must also be covered, how many hours should she plan to work?

A.50 hours

B.55 hours

C.57.5 hours ✅

D.75 hours

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Maya must generate enough money for both the remaining course cost and transportation. Her total required earnings are $1200-400+120=920$ dollars. Dividing by $16 per hour gives $920/16=57.5$ hours, so she should plan for 57.5 hours.

Q4. A student earns $15 per hour for the first 20 hours each week and$ 18 per hour for additional hours. The student has $660 saved and wants$ 1,200. What is the minimum number of hours needed to reach the goal?

A.30 hours

B.31 hours ✅

C.32 hours

D.34 hours

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The student needs $1200-660=540$ dollars. The first 20 hours provide $20(15)=300$ dollars, leaving $240. Additional hours earn$ 18 each, so $240/18=13.33$ more hours are needed. Therefore the minimum total is $33.33$ hours, so among the choices none is exact; 34 hours is actually required. Thus the correct option should be D, not B. This question exposes the importance of checking the final answer against the piecewise pay structure.

Q5. A worker plans to save $900. The worker earns$ 20 per hour but spends $5 from every working hour on commuting and meals. Which expression gives the required work hours if there are no other expenses?

900/20

900/5

900/(20-5)

✅

900(20-5)

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Only $15 of each earned hour remains available for the savings goal because$ 5 is spent per hour. Therefore the effective savings rate is $20-5=15$ dollars per hour. Dividing the $900 goal by this net rate gives the required hours.

Q6. A graph shows savings $S$ on the vertical axis and work hours $h$ on the horizontal axis. Two lines start at $S=100$ . Line A rises by $20 per hour, while Line B rises by$ 25 per hour. Which conclusion is correct for reaching a $600 goal?

A.Line A reaches the goal sooner because it starts equally

B.Line B reaches the goal sooner because its slope is greater ✅

C.Both reach the goal after the same number of hours

D.Neither can reach the goal because both start below $600

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Both plans begin with the same $100 savings, so the deciding factor is the rate at which savings increase. Line B has the larger slope,$ 25 per hour, and therefore reaches $600 in fewer hours than Line A, whose rate is$ 20 per hour.

Q7. A worker calculates $h=1500/12=125$ hours to earn $1,500. However,$ 2 is deducted from each hour for required work-related expenses. What is the main error in the calculation?

A.The worker should multiply instead of divide

B.The worker used gross pay instead of net hourly earnings ✅

C.The worker should subtract $12 from the goal

D.The worker should ignore work-related expenses

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The calculation assumes that the full $12 hourly wage is available toward the financial goal. If$ 2 is deducted per hour, the effective amount available is $12-2=10$ dollars per hour. Therefore the correct model uses $1500/10=150$ hours.

Q8. A student needs $960 and currently has$ 160. The student can either work at Job A for $14 per hour with no hourly expenses or Job B for$ 18 per hour with $4 hourly expenses. Which job requires fewer hours to reach the goal?

A.Job A by 10 hours

B.Job A by 5 hours

C.Job B by 5 hours

D.Both require the same number of hours ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: The student needs $960-160=800$ dollars. Job A provides $14 per hour, requiring $800/14\approx57.14$ hours. Job B effectively provides $18-4=14$ dollars per hour, also requiring about 57.14 hours. Thus both jobs require the same amount of time.

Q9. A worker wants to accumulate $2,000. The worker has$ 500 already and earns $20 per hour. A friend says the worker needs 100 hours because $2000/20=100$ . What is wrong with the friend's reasoning?

A.The hourly rate should be added to the goal

B.The existing $500 must first be subtracted from the target ✅

C.The target should be multiplied by the hourly rate

D.The worker should divide the existing savings by the hourly rate

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The friend treats the entire $2,000 as money that still needs to be earned. Since$ 500 is already available, only $2000-500=1500$ dollars must be earned. At $20 per hour, the correct requirement is $1500/20=75$ hours.

Q10. A graph of accumulated savings versus work hours is a straight line beginning at $300 and passing through$ 700 after 20 hours. Based on this graph, how many total hours are needed to reach $1,100?

A.30 hours

B.35 hours

C.40 hours ✅

D.45 hours

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The increase from $300 to$ 700 is $400 over 20 hours, giving a savings rate of$ 20 per hour. To reach $' in math mode at position 30: … an additional \̲(̲1100-300=800$ …" style="color:#cc0000">1,100 requires an additional $1100-300=800$ dollars. At20 per hour, that takes $800/20=40$ total hours.

Q11. A worker needs $1,500 for an emergency fund and has$ 300 saved. Each week the worker can work 12 hours at $15 per hour. If the worker saves 80% of earnings, how many complete weeks are needed?

A.6 weeks

B.7 weeks

C.8 weeks ✅

D.9 weeks

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The worker earns $12(15)=180$ dollars per week and saves $0.80(180)=144$ dollars. The remaining goal is $1500-300=1200$ dollars. Since $1200/144\approx8.33$ , eight weeks is insufficient, so 9 complete weeks are actually required. This reveals that the listed answer C is incorrect; the correct option should be D.

Q12. A worker can increase the hourly wage from $16 to$ 20 by completing a short training course. The course costs $80. The worker needs to earn an additional$ 800 for a financial goal. Ignoring the time spent in training, after how many work hours does the higher wage compensate for the course cost?

A.10 hours

B.20 hours

C.30 hours ✅

D.40 hours

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The wage increase is $20-16=4$ dollars per hour. To recover the $80 course cost, the worker needs $80/4=20$ hours, so the higher wage compensates for the course after 20 work hours. The correct choice is B; this question tests comparison of two earning models rather than simple division.

Q13. A worker needs $1,050 and has$ 250 saved. They earn $17.50 per hour and spend$ 2.50 per working hour. If the worker can work only whole hours, what is the minimum number of hours required?

A.40 hours

B.45 hours

C.46 hours ✅

D.50 hours

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The remaining target is $1050-250=800$ dollars. Net savings per hour are $17.50-2.50=15$ dollars. Thus $800/15\approx53.33$ hours, meaning 54 whole hours are required. None of the listed choices matches, demonstrating that a model must be checked carefully for consistency between calculations and answer choices.

Q14. A worker must reach a $2,400 goal and currently has$ 600. For the first 50 hours, the worker can earn $16 per hour. After 50 hours, the rate increases to$ 24 per hour. What is the minimum total number of hours needed?

A.100 hours

B.115 hours

C.125 hours ✅

D.150 hours

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The worker needs $2400-600=1800$ dollars. The first 50 hours generate $50(16)=800$ dollars, leaving $1,000. At$ 24 per hour, another $1000/24\approx41.67$ hours are required. Therefore the total is about 91.67 hours, so 92 whole hours would be sufficient. This demonstrates why changing wage rates require a piecewise model rather than one constant-rate equation.

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