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📝 Checking if solution is reasonable (12 MCQs)

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

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What is Checking if solution is reasonable?

Definition:
Checking if a solution is reasonable means verifying that the answer obtained from solving an equation makes sense in the context of the original problem. This involves substituting the value back into the original wording and checking if it satisfies all conditions and is logical (e.g., not negative if counting items).

Working:
For example, if a problem says 'a number plus 5 is 12', solution $x = 7$ . Check: $7 + 5 = 12$ , which matches. Also, ensure that if the problem said 'positive number', the answer is positive.

Example:
Problem: 'The sum of two consecutive integers is 45.' Solution: $x=22$ , so integers are 22 and 23. Check: $22+23=45$ , correct and reasonable (both integers).

Reason:
This step catches arithmetic errors and ensures the answer is meaningful, which is critical for real-world applications.

Easy

Medium

Hard

📝 All Checking if solution is reasonable MCQs

Q1. A student solves $3x+7=25$ and obtains $x=6$ . Before accepting the answer, which is the strongest reasonableness check?

A.Check whether 6 is a whole number

B.Substitute

x=6

into the original equation and verify both sides are equal ✅

C.Compare 6 with the largest number in the equation

D.Multiply 6 by 3 and ignore the constant term

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Substitution directly tests whether the proposed value satisfies the original equation. Here, $3(6)+7=25$ , so both sides agree. This is stronger than checking whether the answer merely looks like a reasonable type of number.

Q2. A learner estimates $48.7\div5.1$ as approximately $10$ , then calculates $95.5$ . Which conclusion is most reasonable?

A.The calculator answer should be accepted because decimals are difficult to estimate

B.The estimate is irrelevant when a calculator gives an exact-looking answer

C.The result is suspicious because

48.7\div5.1

should be close to

10

, not nearly

100

✅

D.Both answers are equally reasonable because division can greatly increase numbers

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Since $5.1$ is close to $5$ , dividing $48.7$ by $5.1$ should produce a result near $9.5$ or $10$ . A result near $95.5$ is about ten times too large, indicating a likely calculation or decimal-placement error.

Q3. A shop calculates the total cost of 8 notebooks priced at Rs. $145$ each as Rs. $1,160$ . Which reasoning best confirms that the answer is reasonable?

A.Since

145

is close to

150

8\times150=1,200

, which is close to

1,160

✅

B.Since

145

is less than

1,160

, the total must be correct

C.Because 8 is even, the total must be even

D.The answer contains a zero, so it is probably correct

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Rounding $145$ to $150$ gives $8\times150=1,200$ , which provides a useful benchmark. The calculated Rs. $1,160$ is close to that estimate, supporting its reasonableness while still allowing for the exact price.

Q4. A rectangular garden has a perimeter of $54$ meters. A student claims its length is $20$ meters and width is $17$ meters. What is the best evaluation of the claim?

A.It is reasonable because

20+17=37

B.It is reasonable because both dimensions are positive

C.It is unreasonable because

2(20+17)=74

, not

54

✅

D.It is unreasonable because the length must be exactly twice the width

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: For a rectangle, the perimeter is determined by twice the sum of length and width. Substituting the proposed dimensions gives $2(20+17)=74$ , which differs substantially from $54$ , so the solution fails the original condition.

Q5. A taxi fare is modeled by $F=120+35m$ , where $m$ is the number of kilometers traveled. A student obtains $F=155$ for a trip of $4$ kilometers. What should the student conclude?

A.The answer is reasonable because

155

is greater than the starting fee

B.The answer is unreasonable because the distance should be subtracted from the starting fee

C.The answer is unreasonable because

120+35(4)=260

✅

D.The answer is reasonable because

35+120+4=159

, which is close

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The model charges a fixed $120$ plus $35$ for every kilometer. For $4$ kilometers, the variable charge is $35(4)=140$ , making the total $260$ . Therefore, $155$ is not a reasonable solution for the stated distance.

Q6. A student solves $0.25x=18$ and reports $x=4.5$ . Which check most clearly identifies the error?

A.Because

0.25

is less than

1

, multiplying it by

4.5

should give more than

18

B.Substituting

4.5

gives

0.25(4.5)=1.125

, which is far from

18

✅

C.Since

18

is even,

x

must also be even

D.The answer is reasonable because

4.5

is a positive decimal

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Substitution exposes the error immediately: $0.25(4.5)=1.125$ , not $18$ . Also, because multiplying by a number less than $1$ reduces a positive quantity, the value of $x$ should actually be substantially larger than $18$ .

Q7. A school expects 240 students to attend an event. Tickets cost Rs. $75$ each. A calculation gives expected revenue of Rs. $180,000$ . Which assessment is most appropriate?

A.The result is reasonable because

240+75=315

B.The result is reasonable because

180,000

is greater than

240

C.The result is unreasonable because

240\times75=18,000

, so the calculation is ten times too large ✅

D.The result is unreasonable because ticket revenue cannot exceed Rs.

10,000

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Multiplying $240$ students by Rs. $75$ per ticket gives Rs. $18,000$ . The proposed Rs. $180,000$ is exactly ten times larger, strongly suggesting a place-value or multiplication error rather than a plausible estimate.

Q8. A line graph shows the estimated distance traveled by a cyclist: at 1 hour, $18$ km; at 2 hours, $36$ km; at 3 hours, $54$ km; and at 4 hours, $72$ km. A student claims the 5-hour distance should be $90$ km. Which evaluation is best?

A.The claim is unreasonable because the graph stops at 4 hours

B.The claim is reasonable because the pattern increases by

18

km each hour ✅

C.The claim is unreasonable because distance should decrease over time

D.The claim is reasonable only if the cyclist rests between hours

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The graph shows a consistent increase of $18$ kilometers for every additional hour. Extending that observed pattern gives $72+18=90$ kilometers at 5 hours, so the student's prediction is consistent with the displayed relationship.

Q9. A farmer estimates that a rectangular field has area $1,000\text{ m}^2$ . One proposed solution uses dimensions $10$ m by $100$ m, while another uses $25$ m by $40$ m. Which statement is most accurate?

A.Only the first solution is reasonable because 100 is a larger dimension

B.Only the second solution is reasonable because its dimensions are closer together

C.Both solutions are mathematically reasonable for the stated area, although practical context may favor one ✅

D.Neither solution is reasonable because rectangular areas cannot equal

1,000

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Both $10\times100=1,000$ and $25\times40=1,000$ , so both satisfy the mathematical area requirement. Reasonableness in a real setting may depend on additional information such as land shape, fencing costs, or available space.

Q10. A student solves $2(x+5)=26$ by writing $2x+5=26$ , obtaining $x=10.5$ . Which explanation best diagnoses the problem?

A.The student should divide

26

2

before subtracting

5

B.The student incorrectly distributed the

2

; it must multiply both

x

and

5

✅

C.The student should add

5

after dividing by

2

D.The student's answer is reasonable because

10.5

is positive

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The factor $2$ applies to every term inside the parentheses, so $2(x+5)=2x+10$ , not $2x+5$ . Substituting $10.5$ into the original equation gives $31$ , confirming that the reported solution is incorrect.

Q11. A delivery service charges a fixed fee of Rs. $200$ plus Rs. $50$ per package. For 7 packages, one method gives Rs. $550$ , while another gives Rs. $1,350$ . Which conclusion is justified?

A.Rs.

550

is reasonable because

200+50+7=257

B.Rs.

1,350

is reasonable because

200(50)+7

is approximately that amount

C.Rs.

550

is reasonable because it uses the fixed fee and one variable charge only

D.Neither answer is reasonable; the model gives

200+50(7)=

Rs.

550

✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The model requires the fixed Rs. $200$ fee plus Rs. $50$ for each of 7 packages. Thus $200+50(7)=550$ . The Rs. $1,350$ result likely comes from incorrectly combining or multiplying the fixed and variable charges.

Q12. A student estimates the solution to $7x-12=44$ as $x\approx8$ . Another student solves it exactly and obtains $x=8$ . Which statement best explains why the exact answer passes a reasonableness check?

A.Since

7(8)=56

, subtracting

12

gives

44

, matching the original equation ✅

B.Because

8

is close to

44

, it must be correct

C.Because

7

and

12

are both integers, every answer must be an integer

D.Because adding

12

44

gives

56

, the equation no longer needs to be checked

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Substitution verifies the complete relationship: $7(8)-12=56-12=44$ . The exact result also agrees with the rough estimate, providing two independent reasonableness signals: the value fits the expected magnitude and satisfies the original equation.

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