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📝 Maximum capacity inequality problems (12 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 12 questions available

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What is Maximum capacity inequality problems?

Definition:
Maximum capacity inequality problems involve constraints where the total quantity of items, people, or weight must not exceed a given capacity, expressed as $\text{Total Load} \le \text{Maximum Capacity}$ , and these problems are common in logistics, engineering, and event planning to ensure safety and efficiency by respecting upper limits.

Working:
To solve, define the variable for the number of items or people, multiply by the weight or space per unit, add any fixed load, and set the total less than or equal to the capacity, then solve the inequality; for example, if a truck can carry at most 10 tons, and each box weighs 0.5 tons, then $0.5x \le 10$ , so $x \le 20$ , meaning at most 20 boxes can be carried.

Example:
A concert hall has a maximum capacity of 500 people; if 320 tickets are already sold, and $x$ more can be sold, then $320 + x \le 500$ , so $x \le 180$ , meaning at most 180 additional tickets can be sold.

Reason:
Maximum capacity problems are essential for safety compliance, resource management, and operational planning, as they help prevent overloading and ensure that systems function within safe and efficient limits.

Easy

Medium

Hard

📝 All Maximum capacity inequality problems MCQs

Q1. An elevator has a posted maximum capacity of 12 passengers. Which mathematical inequality correctly represents the number $x$ of passengers allowed inside?

x<12

x\le12

✅

x>12

x\ge12

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A maximum capacity means the stated limit may be reached but cannot be exceeded. Therefore, 12 passengers are permitted, while any number greater than 12 is prohibited. The correct mathematical model is $x\le12$ , not a strict inequality.

Q2. A boat is rated for a maximum of 18 people. If 7 people are already aboard, which expression represents the greatest number of additional people who can safely board?

18+7

18-7

✅

7-18

18\div7

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The total number of people cannot exceed 18. Since 7 people are already aboard, the remaining capacity is found by subtracting the current number from the maximum: $18-7=11$ . Thus, 11 additional people can board.

Q3. A stage can hold at most 800 kg. Four performers each weigh 65 kg, and equipment weighs 420 kg. If $x$ is the number of additional 50-kg speakers, which inequality models the situation?

260+420+50x\le800

✅

260+420+50x\ge800

65+420+50x\le800

260+50+420x\le800

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The four performers contribute $4(65)=260$ kg, while equipment contributes 420 kg. Each additional speaker contributes 50 kg. Because the total must not exceed 800 kg, the correct model is $260+420+50x\le800$ .

Q4. An elevator allows a maximum combined load of 900 kg. Six passengers have a combined weight of 510 kg. A delivery worker weighs 75 kg and carries boxes weighing 120 kg. How many more 60-kg passengers can enter after the worker and boxes are included?

A.2

B.3 ✅

C.4

D.5

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: After including the worker and boxes, the load is $510+75+120=705$ kg. The remaining capacity is $900-705=195$ kg. Since each passenger weighs 60 kg, at most 3 additional passengers can enter because $3(60)=180$ , while four would add 240 kg.

Q5. A boat has a maximum capacity of 24 people. A group of 31 tourists is divided among two identical boats. Which conclusion is necessarily true if every boat must satisfy its capacity limit?

A.One boat must contain exactly 24 people.

B.At least one boat must contain 16 or more people. ✅

C.Both boats must contain 15 or more people.

D.Each boat can contain at most 12 people.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Together the two boats can hold up to $24+24=48$ people, so the group can fit. However, dividing 31 people between two boats means one boat must have at least $\lceil31/2\rceil=16$ people. The exact distribution is not determined.

Q6. A theater manager models seating with $s\le450$ , where $s$ is the number of guests. Another manager writes $s<450$ . What is the key difference between the models?

A.The first allows exactly 450 guests; the second does not. ✅

B.The second allows exactly 450 guests; the first does not.

C.Both models allow exactly 450 guests.

D.Neither model represents a capacity restriction.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The phrase 'maximum capacity' means the boundary value is permitted unless additional safety rules say otherwise. Therefore, $s\le450$ includes 450, whereas $s<450$ incorrectly excludes exactly 450 guests.

Q7. A stage manager says, 'The stage supports 1,200 kg, and we already have 900 kg of performers and equipment. Since $1,200-900=300$ , we can add six 50-kg items.' What is wrong with this reasoning?

A.The subtraction should be reversed.

B.Six items weigh 250 kg.

C.Six items weigh 300 kg, which reaches the limit exactly. ✅

D.The manager forgot that capacity is a minimum, not a maximum.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The arithmetic is correct: $1,200-900=300$ kg of capacity remains. Six items at 50 kg each weigh $6(50)=300$ kg, so the total becomes exactly 1,200 kg. Therefore, six items are allowed if the stated capacity may be reached.

Q8. An elevator's maximum load is 1,000 kg. Three adults weigh 82, 76, and 91 kg, and a maintenance cart weighs 310 kg. A technician incorrectly calculates the remaining capacity as $1,000-(82+76+91)+310$ . What error did the technician make?

A.The cart weight should also be subtracted from the capacity. ✅

B.The adult weights should be multiplied instead of added.

C.The maximum load should be added to the passenger weights.

D.The cart weight should be divided by the number of adults.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The three adults weigh $82+76+91=249$ kg, and the cart adds another 310 kg. Both loads consume capacity, so the correct remaining capacity is $1,000-(249+310)=441$ kg. Adding 310 instead incorrectly treats the cart as available capacity.

Q9. A graph of allowable passengers $x$ against total elevator load $y$ is a straight increasing line until it reaches the horizontal capacity $y=1,000$ . If each passenger adds 75 kg and the empty elevator load is 250 kg, which point represents the capacity boundary?

(8,850)

(9,925)

(10,1,000)

✅

(11,1,075)

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The load is modeled by $y=250+75x$ . Setting the load equal to the maximum gives $250+75x=1,000$ , so $75x=750$ and $x=10$ . Thus the boundary occurs at $(10,1,000)$ ; the next passenger would exceed capacity.

Q10. A boat has a maximum passenger capacity of 20. Children count as 0.5 passenger for a particular safety calculation, while adults count as 1 passenger. If 12 adults and $c$ children are aboard, which inequality models the allowable number of children?

12+c\le20

12+0.5c\le20

✅

12+2c\le20

0.5(12)+c\le20

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The model must account for the different safety weights assigned to adults and children. Twelve adults contribute 12 capacity units, while each child contributes 0.5 unit. Therefore, the total is $12+0.5c$ , which must be at most 20.

Q11. A ferry has capacity 60 people. At departure, 42 passengers are aboard. At the next stop, $x$ passengers want to board, but 8 passengers will leave first. Which is the greatest possible value of $x$ ?

A.18

B.20

C.26 ✅

D.28

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: After 8 passengers leave, the ferry has $42-8=34$ passengers. The remaining capacity is $60-34=26$ . Therefore, at most 26 additional passengers may board. Choosing 28 would produce 62 passengers, exceeding the stated maximum.

Q12. A stage has a rectangular safe standing area that can accommodate at most 3 people per square meter. If its dimensions are 10 m by 8 m, and a 20-square-meter area is reserved for equipment, what is the greatest number of people allowed?

A.180

B.200

C.220 ✅

D.240

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The usable area is $10(8)-20=60$ square meters. At most 3 people per square meter gives $60(3)=180$ people, so the correct answer is A. This problem requires recognizing that reserved space must be removed before applying the density limit.

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