📝 Translating words to algebraic expressions (12 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 12 questions available

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What is Translating words to algebraic expressions?

Definition:
Translating words to algebraic expressions involves converting verbal phrases and statements into mathematical symbols, using key words to identify operations, such as 'sum' for addition, 'difference' for subtraction, 'product' for multiplication, and 'quotient' for division, with variables representing unknown quantities.

Working:
Read the phrase carefully, assign variables to unknown numbers, replace operation words with symbols ( $+, -, \times, \div$ ), and pay attention to order (e.g., 'less than' reverses subtraction), then write the expression in symbolic form.

Example:
Translate 'Five more than twice a number' into an expression.
Solution: Let the number be $x$ ; 'twice a number' is $2x$ , and 'five more than' means add $5$ , so expression is $2x + 5$ .

Reason:
This skill bridges everyday language and mathematics, enabling us to model real-world situations into equations that can be solved, making it crucial for word problems in algebra and applied math.

Easy

Medium

Hard

📝 All Translating words to algebraic expressions MCQs

Q1. A school charges a fixed registration fee of 500 plus 75 for each workshop attended. If $w$ represents the number of workshops, which expression correctly models the total cost?

500w+75

500+75w

✅

575w

500(w+75)

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: The fixed registration fee is paid once, so 500 is the constant term. The workshop cost is 75 for each of $w$ workshops, giving $75w$ . Therefore, the total cost is $500+75w$ .

Q2. A student translates seven less than three times a number $x$ " as $7-3x$ . What is the best evaluation of the student's reasoning?"

A.Correct, because less than" always means subtraction from the first number mentioned"

B.Incorrect, because less than" reverses the order of subtraction" ✅

C.Correct, because multiplication must always be written before subtraction

D.Incorrect, because the expression must contain division

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The phrase seven less than three times a number" means subtract 7 from $3x$ , not subtract $3x$ from 7. Thus the correct expression is $3x-7$ . The student's main error is reversing the subtraction order."

Q3. A taxi fare is modeled by a starting charge of 120 and an additional 40 per kilometer. A passenger travels $d$ kilometers and then receives a 50 discount. Which expression represents the final amount paid?

120+40d+50

120(40d)-50

120+40d-50

✅

40(d+120)-50

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The starting charge contributes 120, while traveling $d$ kilometers contributes $40d$ . The 50 discount reduces the total, so it must be subtracted after combining the original charges. Hence the expression is $120+40d-50$ .

Q4. A number-line graph shows a quantity beginning at $x$ , moving 6 units to the right, and then being doubled. Which expression describes the resulting quantity?

2x+6

2(x+6)

✅

x+12

6(x+2)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Moving 6 units to the right changes $x$ to $x+6$ . The graph then indicates that this entire quantity is doubled, so the multiplication applies to both terms. Therefore, the correct expression is $2(x+6)$ .

Q5. Two students translate the difference between twice a number $n$ and 9." Student A writes $2n-9$ , while Student B writes $9-2n$ . Which conclusion is correct?"

A.Only Student A is correct because difference always starts with the smaller number

B.Only Student B is correct because 9 appears second in the phrase

C.Both are correct because difference is commutative

D.Student A is correct because the stated order is twice the number followed by 9 ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The phrase specifies the difference between $2n$ and 9 in that order, giving $2n-9$ . Subtraction is not commutative, so reversing the terms generally changes the value. Student A correctly preserves the phrase's intended order.

Q6. A rectangle has a length described as 5 more than twice its width". If the width is $w$ , which expression should be used for the length before finding the perimeter?"

2(w+5)

5w+2

2w+5

✅

w+10

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: Twice its width" means $2w$

Q7. A puzzle asks for three times the sum of a number $x$ and 4, decreased by 2." Which expression preserves the order of operations implied by the wording?"

3x+4-2

3(x+4)-2

✅

3(x+4-2)

3x+3(4-2)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The phrase first forms the sum $x+4$ , then takes three times that entire sum, and finally decreases the result by 2. Parentheses are essential because the multiplier applies to both $x$ and 4, giving $3(x+4)-2$ .

Q8. A delivery service charges 150 as a fixed fee plus 25 for every package. If $p$ is the number of packages, which expression represents the charge after receiving a 50 discount?

150+25(p-50)

150+25p-50

✅

150-25p+50

25(150+p)-50

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The fixed fee is 150, and $25p$ represents the cost of $p$ packages. Since the discount is a separate reduction of 50 from the total, it is subtracted afterward, giving $150+25p-50$ .

Q9. A number-line representation starts at $x$ , moves 7 units left, and then multiplies the resulting quantity by 3. Which algebraic expression matches the movement?

3x-7

3(x-7)

✅

3(x+7)

x-21

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Moving 7 units left changes $x$ to $x-7$ . The instruction then multiplies the entire resulting quantity by 3. Therefore, parentheses are necessary, producing $3(x-7)$ , rather than multiplying only $x$ .

Q10. A teacher asks for an expression representing the sum of twice $n$ and the quotient of $n$ by 5." Which student response is correct?"

2n+\frac{n}{5}

✅

2(n+\frac{n}{5})

2n+\frac{5}{n}

\frac{2n+n}{5}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The word sum" indicates addition between two quantities. "Twice $n$ " becomes $2n$ , while "the quotient of $n$ by 5" becomes $\frac{n}{5}$ . Combining them gives $2n+\frac{n}{5}$ ."

Q11. A graph shows a quantity represented by $x$ first being increased by 4 and then multiplied by 2. A student writes $2x+4$ . Why does the student's expression fail to match the graph?

A.The student should divide by 2 instead

B.The increase of 4 must occur before the multiplication ✅

C.The student should replace

x

with 4

D.The multiplication should apply only to 4

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The graph describes two sequential operations: first $x$ becomes $x+4$ , then that entire result is multiplied by 2. Therefore, the correct expression is $2(x+4)$ , not $2x+4$ , because the latter doubles only the original $x$ .

Q12. A contest problem describes a quantity as the difference between the product of 4 and $x+2$ , and the quotient of $x$ by 3." Which expression most accurately preserves every operation in the wording?"

4x+2-\frac{x}{3}

4(x+2)-\frac{x}{3}

✅

4(x+2-\frac{x}{3})

4x+8-\frac{3}{x}

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The phrase product of 4 and $x+2$ " requires $4(x+2)$ , while "quotient of $x$ by 3" gives $\frac{x}{3}$ . The word "difference" means subtract the second quantity from the first, so $4(x+2)-\frac{x}{3}$ is correct."

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