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📝 Checking inequalities for reasonableness (14 MCQs)

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

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What is Checking inequalities for reasonableness?

Definition:
Checking inequalities for reasonableness involves verifying that the solution to an inequality makes sense in the real-world context, by ensuring the value is within logical bounds, matches the problem's constraints, and is interpreted correctly (e.g., no negative quantities, within realistic limits), and this step is critical to avoid erroneous conclusions in applied problems.

Working:
After solving, plug the solution back into the original inequality to verify it satisfies the condition, consider whether the answer is physically possible (e.g., non-negative, within capacity), and check if rounding or interpretation aligns with the context; for example, if the solution is $x \ge -5$ , but the variable is a number of items, the meaningful solution is $x \ge 0$ (since negative items are impossible).

Example:
If solving for the number of tickets sold $t$ with $10t + 50 \le 500$ , the solution is $t \le 45$ , and checking: if $t = 45$ , cost is $10(45) + 50 = 500$ , which is reasonable; if $t = 46$ , cost is 510, which exceeds the budget, so $t = 45$ is the maximum reasonable value.

Reason:
Checking reasonableness is a crucial mathematical habit that ensures answers are valid and practical, preventing errors in real-world decision-making, and it reinforces critical thinking and attention to context, which are essential skills in all applied mathematics.

Easy

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Hard

📝 All Checking inequalities for reasonableness MCQs

Q1. A model estimates that a family needs 850 liters of water per day for normal household use. After checking the assumptions, which is the best first step before accepting the result?

A.Round the answer to 1,000 liters because household needs vary

B.Check whether 850 liters is reasonable compared with the family size, activities, and typical daily consumption ✅

C.Change the answer because decimals are not useful in real situations

D.Assume the model is correct because the calculation was performed correctly

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A mathematically correct result can still be unrealistic if its magnitude conflicts with real-world conditions. Checking the family size, activities, and typical consumption provides context for deciding whether the model's assumptions and output are sensible.

Q2. A student calculates that a delivery vehicle traveling 240 km at an average speed of 60 km/h will take 0.25 hours. What should the student conclude after checking the result?

A.The result is reasonable because

240\div60=0.25

B.The distance should be converted into meters before deciding

C.The result is unrealistic because

240\div60=4

hours, so the calculation or interpretation was wrong ✅

D.The vehicle must be traveling faster than 60 km/h

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: The arithmetic should be interpreted with appropriate units. Since time equals distance divided by speed, $240\div60=4$ hours. The value 0.25 hours is inconsistent with the stated distance and speed and therefore requires correction.

Q3. A bakery model predicts that producing 12 cakes requires 18 kilograms of flour. The bakery normally uses about 1.5 kilograms of flour per cake. Which conclusion best evaluates the model?

A.The result is plausible because

18\div12=1.5

kilograms per cake ✅

B.The result is impossible because flour must be measured in grams

C.The model must predict exactly 12 kilograms because there are 12 cakes

D.The answer should be rejected because models cannot use average quantities

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Dividing 18 kilograms by 12 cakes gives 1.5 kilograms per cake, matching the bakery's usual usage. The agreement between the model's implied rate and real-world information supports the plausibility of the result.

Q4. A model predicts that a rectangular garden needs 900 meters of fencing when its length is 30 meters and width is 15 meters. Which response best identifies whether the answer makes sense?

A.Accept it because

30\times15=450

square meters

B.Accept it because fencing is usually measured in square meters

C.Reject it because the perimeter is

2(30+15)=90

meters, so 900 meters is far too large ✅

D.Reject it because gardens cannot be modeled using rectangles

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Fencing measures the boundary length, not the area. The perimeter is $2(30+15)=90$ meters. Therefore, 900 meters is not consistent with the dimensions and likely results from confusing area with perimeter or making another modeling error.

Q5. A taxi fare is modeled by $C=150+40d$ , where $d$ is distance in kilometers. A calculation gives a fare of 1,750 for a 40-km trip. Which check most strongly supports the answer?

A.The fare is reasonable because

150+40(40)=1,750

✅

B.The fare must be wrong because taxi fares should never contain a fixed charge

C.The fare is reasonable only if the distance is measured in meters

D.The result should be 1,600 because the fixed charge should be ignored

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Substituting $d=40$ into the model gives $150+40(40)=1,750$ . The result also has the appropriate monetary interpretation, including both the fixed starting charge and the distance-based charge.

Q6. A school model estimates that a student can complete 480 problems in 2 hours. The student normally completes about 3 problems per minute. Which evaluation is most appropriate?

A.The model is reasonable because 480 problems is a large but possible number

B.The model is questionable because

480\div120=4

problems per minute, exceeding the student's usual rate ✅

C.The model is definitely correct because 2 hours equals 480 minutes

D.The model is impossible because students cannot solve more than 100 problems per day

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Two hours equals 120 minutes, so 480 problems would require $480\div120=4$ problems per minute. Since the student's usual rate is 3 per minute, the model's prediction deserves investigation rather than automatic acceptance.

Q7. A company models monthly profit as $P=500x-8,000$ , where $x$ is the number of products sold. For $x=10$ , the model gives $P=-3,000$ . Which interpretation is most sensible?

A.The company earns 3,000 dollars because negative profit means extra income

B.The model is automatically invalid because profit cannot be negative

C.The company loses 3,000 dollars at that sales level, which can be realistic because fixed costs exceed revenue ✅

D.The company must sell exactly 3,000 products to eliminate the loss

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Negative profit has a meaningful real-world interpretation: it represents a loss. At 10 products, revenue is 5,000 while the fixed cost is 8,000, leaving a loss of 3,000. The result therefore can be realistic.

Q8. A water tank model predicts that after 5 hours, the tank contains $-120$ liters. The model is being used for a physical tank that starts with 500 liters. What is the best conclusion?

A.The negative amount should be accepted because mathematical models always allow negative values

B.The prediction signals that the model has been used beyond a meaningful physical range or that the tank would have become empty earlier ✅

C.The tank must contain 120 liters because the negative sign is only a formatting issue

D.The result proves that the initial volume was actually 120 liters

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: A physical tank cannot contain a negative volume. A negative prediction usually indicates that the model's domain has been extended beyond the point where the physical process remains valid. The tank would reach zero before becoming negative.

Q9. A graph of a cost model rises steadily as the number of items purchased increases. At $x=0$ , the graph crosses the vertical axis at 200. Which interpretation is most reasonable?

A.The business charges a fixed cost of 200 even when no items are purchased ✅

B.The business gives customers 200 items for free

C.The cost per item must be 200 regardless of quantity

D.The graph proves that buying more items eventually makes total cost decrease

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The vertical intercept represents the modeled cost when the number of purchased items is zero. A value of 200 therefore suggests a fixed starting or setup cost rather than a per-item cost.

Q10. A student solves a model for the number of buses needed and obtains 7.2 buses. The problem requires enough buses to transport everyone. What should the student do when interpreting the result?

A.Use 7.2 buses because it is the exact mathematical answer

B.Round down to 7 because fewer buses are cheaper

C.Round to 8 because the real-world quantity must be a whole number and must provide sufficient capacity ✅

D.Replace the answer with 7.5 to compromise between the two integers

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: A fraction of a physical bus cannot be used, and rounding down would leave insufficient capacity. Because the requirement is to transport everyone, the mathematically derived value 7.2 must be rounded upward to 8 buses.

Q11. A model predicts that a person's monthly electricity bill decreases as household electricity usage increases. The equation was solved correctly, but the prediction conflicts with the billing structure being modeled. What should happen next?

A.Accept the prediction because correct algebra guarantees a realistic result

B.Check the model's assumptions, signs, units, and interpretation before using the result ✅

C.Multiply the final bill by 2 automatically

D.Ignore the billing structure because real-world information should not affect mathematics

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Correct algebra only guarantees that an equation was manipulated properly; it does not guarantee that the equation represents reality. A contradictory prediction requires checking assumptions, units, signs, and the meaning of variables.

Q12. A farmer models crop production as $Y=50x$ , where $x$ is the number of planted rows. For $x=20$ , the model predicts 1,000 kilograms. Another model predicts 1,400 kilograms using improved soil conditions. Which conclusion is best?

A.The first model must be correct because it is simpler

B.The second model must be correct because its prediction is larger

C.Both predictions may be reasonable if their assumptions differ, so the farmer should compare conditions and evidence before choosing ✅

D.Neither model can be used because crop production is never predictable

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Different mathematical models can produce different results because they may represent different assumptions or conditions. The higher prediction is not automatically better; the farmer should compare soil conditions, historical yields, and other relevant evidence.

Q13. A model predicts that a person traveling 500 km will use 2 liters of fuel. The vehicle normally consumes 8 liters per 100 km. Which reasoning best evaluates the prediction?

A.The prediction is plausible because 500 km is a short distance

B.The prediction is implausible because the usual consumption implies about 40 liters for 500 km ✅

C.The prediction is plausible because fuel usage decreases when distance increases

D.The prediction cannot be evaluated because kilometers and liters cannot appear in the same problem

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: At 8 liters per 100 km, a 500-km journey would normally require $5\times8=40$ liters. A prediction of 2 liters is therefore inconsistent with the stated real-world consumption rate and should be questioned.

Q14. A model estimates the time required to complete a project using $T=120-5x$ , where $x$ represents additional workers. For $x=30$ , it predicts $T=-30$ hours. What is the strongest conclusion?

A.The project can be completed 30 hours before work begins

B.The model should be used for any number of workers because the equation is linear

C.The negative time indicates that the model is being applied outside a meaningful range, so the worker count or model assumptions must be reconsidered ✅

D.The correct project time is 30 hours because negative values become positive in real situations

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: A negative completion time has no meaningful physical interpretation. The prediction indicates that the linear model has been extended beyond a sensible range, so the allowable number of workers or assumptions behind the model should be examined before using it.

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