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📝 How to Use a problem solving strategy for word problems (15 MCQs)

📖 From Digital SAT Algebra • 3. Mathematical Models in Algebra • 15 questions available

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What is How to Use a problem solving strategy for word problems?

Definition:
Using a problem-solving strategy for word problems means applying a consistent framework: define the unknown with a variable, translate the English sentences into an algebraic equation, solve the equation using properties of equality, and then interpret the solution back in the context of the problem to ensure it makes sense.

Working:
For example, 'The sum of twice a number and 6 is 22.' Let $x$ be the number. Translate to $2x + 6 = 22$ . Solve: subtract 6: $2x = 16$ , divide by 2: $x = 8$ . Interpret: the number is 8. Check: twice 8 plus 6 is $16 + 6 = 22$ , correct.

Example:
Problem: 'A number decreased by 15 is 30.' Let $x$ be the number. Equation: $x - 15 = 30$ . Add 15: $x = 45$ . The number is 45.

Reason:
This strategy provides a repeatable process that turns confusing word problems into manageable algebra, which is crucial for success in math courses.

Easy

Medium

Hard

📝 All How to Use a problem solving strategy for word problems MCQs

Q1. A student reads a word problem and immediately starts calculating without identifying what is known and what must be found. Which change would most improve the reliability of the solution?

A.Perform all calculations mentally before writing anything

B.Identify the unknown, organize known information, choose a model, and then verify the result ✅

C.Choose the largest number in the problem as the answer

D.Convert every number into a fraction before solving

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: A reliable problem-solving strategy begins by identifying the unknown and organizing the given information before selecting an appropriate mathematical model. This prevents irrelevant details or misleading numbers from controlling the calculation.

Q2. A rectangular garden has a perimeter of $54$ meters. Its length is $3$ meters more than twice its width. Which equation correctly represents the situation if $w$ is the width?

2w+3=54

2(2w+3)+2w=54

✅

2(2w)+3=54

w+(2w+3)=54

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The length is $2w+3$ , while the width is $w$ . A rectangle has two lengths and two widths, so the perimeter model must be $2(2w+3)+2w=54$ .

Q3. A taxi charges a fixed fee plus a constant amount per kilometer. A $6$ -km trip costs \ $' in math mode at position 13: 14, while a \̲(̲10$ -km trip co…" style="color:#cc0000">14, while a $10$ -km trip costs \22. Which strategy best determines the cost of a $15$ -km trip?

A.Assume the cost is proportional to distance and calculate

15(14/6)

B.Find the rate from the change in cost and distance, then determine the fixed fee before evaluating

15

km ✅

C.Subtract

6

from

14

, then multiply the result by

15

D.Average the two given costs and multiply by

15/8

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The situation contains both a fixed charge and a distance-dependent charge, so direct proportionality is inappropriate. Comparing changes gives the per-kilometer rate, after which the fixed fee can be identified and the new cost calculated.

Q4. A school is buying notebooks. Supplier A charges \ $3 per notebook with no delivery fee. Supplier B charges \$ 2 per notebook plus a \$40 delivery fee. For $30$ notebooks, which model and conclusion are correct?

A.A:

3(30)=90

; B:

2(30)+40=100

, so A is cheaper ✅

B.A:

3+30=33

; B:

2+40=42

, so B is cheaper

C.A:

3(30)+40=130

; B:

2(30)=60

, so B is cheaper

D.Both cost \$90 because the per-notebook prices are close

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Modeling each supplier separately reveals the total cost rather than relying on unit price alone. Supplier A costs $90$ , while Supplier B costs $100$ , so the apparently cheaper unit price does not produce the lower total.

Q5. A water tank initially contains $120$ liters and is being drained at $8$ liters per minute. Another tank contains $40$ liters and is filled at $4$ liters per minute. After how many minutes will the amounts be equal?

A.5 minutes

B.10 minutes ✅

13\frac{1}{3}

minutes

D.20 minutes

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The first amount can be modeled by $120-8t$ , while the second is $40+4t$ . Setting them equal gives $120-8t=40+4t$ , so $80=12t$ , yielding $t=10$ minutes.

Q6. A learner models a situation with $5x+20=100$ , obtaining $x=16$ . The original problem states that $x$ represents the number of identical items and that each item costs \$5. Which verification is most important?

A.Check whether

16

is positive and satisfies the equation and whether

16

items make sense in the stated context ✅

B.Check only whether

5+20=100

C.Replace

x

with

20

because it is the fixed amount

D.Ignore the context because solving the equation is sufficient

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Verification should occur at both the mathematical and contextual levels. The value $16$ satisfies the equation, but a complete solution also requires checking whether a whole-number quantity of $16$ items is meaningful in the original situation.

Q7. A rectangular field has an area of $240$ square meters. Its length is $4$ meters greater than its width. Which equation should be solved to determine the dimensions?

w+4=240

2w+4=240

w(w+4)=240

✅

2w(w+4)=240

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: If the width is $w$ , the length is $w+4$ . Area is obtained by multiplying length and width, so the correct model is $w(w+4)=240$ . The perimeter formula would represent a different quantity.

Q8. A student solves a problem about a $20\%$ discount on a \ $' in math mode at position 24: …by calculating \̲(̲75(0.20)=15$ a…" style="color:#cc0000">75 item by calculating $75(0.20)=15$ and reports \15 as the final price. What is the error?

A.The student used multiplication instead of division

B.The student calculated the discount amount but failed to subtract it from the original price ✅

C.The student should have added the discount to the original price

D.The student should have used

20

instead of

0.20

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Multiplying $75$ by $0.20$ correctly finds the discount amount of \ $15, not the final price. The final price requires subtracting the discount from the original price, giving \$ 60.

Q9. A worker claims that if $4$ workers complete a job in $18$ hours, then $8$ workers will need $36$ hours because $8$ is twice $4$ . Which reasoning best identifies the mistake?

A.More workers generally require more time because there are more people to manage

B.For a fixed amount of work, doubling the workers should halve the time under the stated constant-rate assumption ✅

C.The worker should always add the number of workers to the hours

D.The correct time must always remain

18

hours regardless of worker count

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The worker incorrectly treated the relationship as direct when it is inverse under the constant-product assumption. Doubling the number of workers changes the time from $18$ hours to $9$ hours, assuming equal productivity.

Q10. A graph shows two cost lines. Plan A starts at \ $30 and increases by \$ 6 per unit. Plan B starts at \ $60 and increases by \$ 3 per unit. At approximately how many units will the two plans cost the same?

A.5 units

B.10 units ✅

C.15 units

D.20 units

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The two models are $30+6x$ and $60+3x$ . Setting them equal gives $30+6x=60+3x$ , so $3x=30$ and $x=10$ . This comparison identifies the break-even point.

Q11. A delivery company charges \ $12 for the first kilogram and \$ 4 for every additional kilogram. A package weighs $7$ kilograms. Which reasoning correctly models the charge?

12(7)=84

, because every kilogram costs \$12

12+4(7)=40

, because seven additional kilograms are added

12+4(6)=36

, because only the six kilograms after the first incur the additional charge ✅

4(7)-12=16

, because the first kilogram is deducted

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The first kilogram costs \ $12, while only the remaining six kilograms incur the \$ 4 additional charge. Therefore the model is $12+4(7-1)=36$ . This carefully distinguishes the initial fee from the marginal cost.

Q12. A farmer wants to fence a rectangular enclosure using $80$ meters of fencing while maximizing area. One proposed solution chooses dimensions $10$ m by $30$ m. Which evaluation is strongest?

A.It is optimal because

10+30=40

B.It is invalid because its perimeter is

80

, so it cannot be used

C.It is feasible because its perimeter is

80

, but comparing other dimensions with the same perimeter is necessary before claiming maximum area ✅

D.It is automatically optimal because the length is three times the width

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The dimensions $10$ by $30$ satisfy the perimeter requirement because $2(10+30)=80$ , so the solution is feasible. However, feasibility does not establish optimality; other dimension pairs must be compared or analyzed to determine which gives the greatest area.

Q13. A graph represents the distance traveled by a cyclist over time. From $0$ to $2$ hours, the graph rises from $0$ to $30$ km. From $2$ to $3$ hours, it remains horizontal at $30$ km. From $3$ to $5$ hours, it rises to $70$ km. Which conclusion follows?

A.The cyclist traveled fastest during the first interval

B.The cyclist stopped for one hour and then traveled at

20

km/h during the final interval ✅

C.The cyclist traveled

30

km during every interval

D.The cyclist's average speed was

30

km/h for the entire trip

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A horizontal distance-time segment indicates no additional distance was traveled, so the cyclist stopped between hours $2$ and $3$ . During the final interval, $40$ km were covered in $2$ hours, giving $20$ km/h.

Q14. A store offers two consecutive discounts of $20\%$ and $10\%$ on an item originally priced at \ $' in math mode at position 49: …al discount as \̲(̲30\%$ and conc…" style="color:#cc0000">500. A student calculates the total discount as $30\%$ and concludes the final price is \350. Which is correct?

A.The final price is \$350 because percentages can always be added

B.The final price is \$360 because the second discount applies to the already reduced price ✅

C.The final price is \$375 because only the first discount applies

D.The final price is \$400 because the discounts cancel each other

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The first discount reduces \ $500 to \$ 400. The second $10\%$ discount is applied to \ $400, giving \$ 40 off and a final price of \$360. Consecutive percentage changes generally cannot be combined by simple addition.

Q15. A positive integer is increased by $5$ , and the result is multiplied by $3$ . This produces the same value as multiplying the original integer by $4$ and subtracting $7$ . Which integer satisfies the situation?

A.7

B.9 ✅

C.11

D.13

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Let the integer be $x$ . The statement gives $3(x+5)=4x-7$ . Expanding gives $3x+15=4x-7$ , so $x=22$ , which is not among the options. Therefore the listed choices are inconsistent with the stated problem, and no option is correct.

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