← Back to 3. Mathematical Models in Algebra

📝 Define variables and write expressions (14 MCQs)

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

←Previous Topic:Consecutive odd integers📝 Define variables and write expressionsChecking if solution is reasonable:Next Topic→

What is Define variables and write expressions?

Definition:
Defining variables and writing expressions involves choosing a letter (e.g., $x, y, n$ ) to represent an unknown quantity and then creating algebraic expressions that describe relationships involving that unknown. This is the first step in solving any word problem.

Working:
For example, if a problem says 'a number increased by 8', define $x$ as the number and write $x + 8$ . If it says 'twice the number decreased by 3', write $2x - 3$ .

Example:
Problem: 'The product of a number and 5, plus 7.' Let $x$ be the number. Expression: $5x + 7$ .

Reason:
Clear variable definition is essential because it sets up the foundation for the equation and ensures that all parts of the problem are correctly represented.

Easy

Medium

Hard

📝 All Define variables and write expressions MCQs

Q1. A theater sells tickets for \ $8 each and charges a one-time booking fee of \$ 12. If $t$ represents the number of tickets, which expression represents the total amount paid?

8+12t

8t+12

✅

20t

8(t+12)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The ticket cost depends on the number of tickets, so it is represented by $8t$ . The booking fee is paid only once, so it remains $12$ . Therefore, the total is $8t+12$ .

Q2. A student chooses $x$ to represent the number of notebooks purchased. Each notebook costs \ $4, and a pen costs \$ 3. Which expression represents the total cost when one pen is purchased?

4+3x

7x

4x+3

✅

4(x+3)

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Since $x$ represents notebooks, their total cost is $4x$ . The single pen contributes a fixed \$3 regardless of the number of notebooks. Combining these quantities gives $4x+3$ , not an expression where the pen cost varies with $x$ .

Q3. A gym charges a membership fee of \ $25 plus \$ 6 for every training session. Which choice correctly defines a variable and models the total monthly cost?

A.Let

C

be sessions, so

C=25+6

B.Let

s

be sessions, so

6+25s

represents cost.

C.Let

s

be sessions, so

25+6s

represents cost. ✅

D.Let

s

be cost, so

25+6s

represents sessions.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The variable should represent the quantity that changes, which is the number of training sessions. The fixed membership fee is \ $25, while each session adds \$ 6. Thus $25+6s$ correctly models the total monthly cost.

Q4. A delivery company charges \ $10 for the first package and \$ 4 for each additional package. If $n$ represents the total number of packages, which expression models the cost for $n\ge1$ ?

10+4n

10n+4

10+4(n-1)

✅

14n

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The first package already costs \ $' in math mode at position 30: … the remaining \̲(̲n-1$ packages …" style="color:#cc0000">10, while only the remaining $n-1$ packages cost \4 each. Therefore, the correct model is $10+4(n-1)$ . The common error is charging \$4 for the first package as well.

Q5. A rectangular garden has a length that is 5 meters greater than its width. If $w$ represents the width, which expression represents the perimeter?

2w+5

4w+10

✅

2(w+5)

w(w+5)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: If the width is $w$ , then the length is $w+5$ . Perimeter requires adding all four sides, giving $2w+2(w+5)$ , which simplifies to $4w+10$ . The product $w(w+5)$ would represent area instead.

Q6. A school club has $m$ members. Each member contributes \ $7, but the club receives a fixed donation of \$ 50. Which expression gives the amount available after the contributions and donation are combined?

7(m+50)

50m+7

7m+50

✅

57m

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The contribution depends on the number of members, producing $7m$ . The donation is fixed and does not depend on membership, so it contributes $50$ . Adding these parts gives $7m+50$ .

Q7. A taxi charges a \ $5 starting fee and \$ 2.50 per kilometer. A passenger travels $d$ kilometers. After identifying the variable and fixed quantity, which expression represents the fare?

2.50+5d

5+2.50d

✅

7.50d

5(2.50+d)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Distance is the changing quantity, so $d$ is the appropriate variable. The \ $5 starting fee is fixed, while each kilometer adds \$ 2.50. Therefore, the total fare is modeled by $5+2.50d$ .

Q8. A student says, "If $x$ is the number of hours worked and the wage is \$15 per hour, the earnings are $15+x$ because both numbers are involved." What is the best analysis of the student's reasoning?

A.Correct, because addition always combines quantities.

B.Incorrect, because hourly wage must be multiplied by hours worked, giving

15x

. ✅

C.Correct, because

x

represents money earned.

D.Incorrect, because earnings should always be

x-15

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The student's error is treating a rate as though it were a fixed amount. A rate of \$15 applies to every hour, so the number of hours must multiply the rate. Thus $15x$ represents earnings.

Q9. A parking lot charges \ $3 for the first hour and \$ 2 for every hour after the first. A student models the cost for $h$ hours as $3+2h$ . For $h=4$ , why does this model overestimate the intended cost?

A.It charges \$3 for every hour. ✅

B.It subtracts the first hour instead of adding it.

C.It charges \

2 for the first hour even though the first hour costs \

D.It ignores the number of hours completely.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The model $3+2h$ adds the \ $' in math mode at position 78: …d structure is \̲(̲3+2(h-1)$ , bec…" style="color:#cc0000">2 charge four times, including for the first hour. The intended structure is $3+2(h-1)$ , because only the three hours after the first receive the \2 additional-hour charge.

Q10. A learner defines $x$ as the total number of books in a library and writes $x+12$ for "12 more books than the number of books borrowed." What is the key modeling problem?

A.The variable should represent books borrowed, not the total library collection. ✅

B.Addition cannot be used with book quantities.

C.The number 12 must be multiplied by

x

D.The variable must always represent a monetary quantity.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The expression must represent the relationship described, so the variable should refer to the quantity being increased. If $x$ represents books borrowed, then $x+12$ means 12 more than borrowed. Defining $x$ as total library books changes the meaning.

Q11. A graph shows a straight line passing through $(0,20)$ and $(5,45)$ , where the horizontal axis represents hours and the vertical axis represents dollars earned. Which expression best models the amount earned after $h$ hours?

20+5h

✅

20+25h

25+20h

45h

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The graph begins at \ $' in math mode at position 9: 20 when \̲(̲h=0$ , so \" style="color:#cc0000">20 when $h=0$ , so \20 is the fixed starting amount. From $(0,20)$ to $(5,45)$ , earnings increase by \ $25 over 5 hours, giving a rate of \$ 5 per hour. Therefore, $20+5h$ is correct.

Q12. A restaurant bill includes a fixed service charge of \ $6, a meal cost of \$ 14 per person, and a coupon reducing the bill by \$10. If $p$ is the number of people, which expression models the final bill?

6+14p+10

14(p+6)-10

6+14p-10

✅

20p-10

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The service charge contributes a fixed \ $' in math mode at position 27: …als contribute \̲(̲14p$ . The coup…" style="color:#cc0000">6, while meals contribute $14p$ . The coupon decreases the entire bill by \10, so it must be subtracted. Combining these components gives $6+14p-10$ , which simplifies to $14p-4$ .

Q13. A square has side length $x+3$ . A student claims its perimeter is $x^2+9$ , arguing that squaring the side expression gives the total boundary. Which correction is most appropriate?

A.The perimeter is

4x+12

, because four equal sides are added. ✅

B.The perimeter is

x^2+6x+9

, because perimeter requires squaring.

C.The perimeter is

x+12

, because only one side changes.

D.The perimeter is

4x+9

, because 9 is added once.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Perimeter measures total boundary length, so the four equal sides must be added rather than multiplied or squared. Each side is $x+3$ , giving $4(x+3)=4x+12$ . Squaring the side would instead relate to area.

Q14. A number is represented by $x$ . Another quantity is 3 less than twice that number, and a third quantity is 5 more than the first number. Which expression represents the sum of all three quantities?

4x+2

✅

3x+8

2x-3+5x

4x-8

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The three quantities are $x$ , $2x-3$ , and $x+5$ . Adding them gives $x+(2x-3)+(x+5)=4x+2$ . The problem requires translating several relationships before combining like terms, making careful variable interpretation essential.

🔗 Related Topics (MCQs)

📝 How Use a Problem Solving Strategy MCQs 📝 How to Approach word problems MCQs 📝 How to Use a problem solving strategy for word problems MCQs 📝 How to Solve number problems MCQs 📝 Translate English sentences into algebraic equations MCQs

←Previous Topic:Consecutive odd integers📝 Define variables and write expressionsChecking if solution is reasonable:Next Topic→