📝 Translate English Phrases to Algebraic Expressions (14 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 14 questions available

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What is Translate English Phrases to Algebraic Expressions?

Definition:
Translating English phrases to algebraic expressions is the process of converting verbal statements into mathematical notation using variables, operation symbols, and numbers, by recognizing keywords that indicate specific operations and the order in which they should be written.

Working:
Identify the unknown quantity and assign a variable, then look for operation words: 'sum' ( $+$ ), 'difference' ( $-$ ), 'product' ( $\times$ ), 'quotient' ( $\div$ ), and phrases like 'less than' (reversal), and write the expression accordingly, paying attention to the correct order.

Example:
Translate 'The quotient of a number and 6, decreased by 4' into an expression.
Solution: Let number be $n$ ; 'quotient of a number and 6' is $\frac{n}{6}$ , 'decreased by 4' means subtract 4, so expression is $\frac{n}{6} - 4$ .

Reason:
This skill is crucial for solving word problems, as it transforms real-world scenarios into algebraic equations that can be manipulated and solved, making math applicable to daily life and professional fields.

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📝 All Translate English Phrases to Algebraic Expressions MCQs

Q1. A teacher says, “Seven less than twice a number $x$ .” Which algebraic expression correctly models the phrase, and why is the order important?

7-2x

, because “less than” means subtraction from seven

2x-7

, because seven is subtracted from twice the number ✅

2(x-7)

, because both quantities are affected by subtraction

7(2x)

, because “twice” means multiplication by seven

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The phrase “twice a number” gives $2x$ , while “seven less than” means subtract 7 from that quantity. Therefore $2x-7$ is correct; reversing the order changes the meaning and produces a different expression.

Q2. A student translates “the sum of a number $n$ and four times its opposite” as $n+4n$ . Which expression should be used instead, and what misconception caused the error?

n-4n

; the student ignored that the opposite of

n

-n

4n-n

; the student confused sum with difference

n+4(-n)

; the student correctly represents four times the opposite ✅

4(-n)-n

; the student reversed the order of the sum

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The opposite of $n$ is $-n$ , so four times its opposite is $4(-n)$ . The phrase “sum of” requires addition, giving $n+4(-n)$ . The error came from treating the opposite as though it were still $n$ .

Q3. A delivery company charges a fixed fee of $' in math mode at position 33: …s the distance \̲(̲d$ in miles. I…" style="color:#cc0000">8 plus three times the distance $d$ in miles. If a customer also receives a5 discount, which expression represents the final charge?

8+3d+5

8-3d-5

3(8+d)-5

8+3d-5

✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The variable charge is three times the distance, represented by $3d$ . Adding the fixed $' in math mode at position 9: 8 gives \̲(̲8+3d$ , and the…" style="color:#cc0000">8 gives $8+3d$ , and then subtracting the5 discount gives $8+3d-5$ . This models each part in the stated order.

Q4. A student claims that “five more than the quotient of $x$ and 3” can be written as $x+5\div3$ . What is the correct expression?

x+\frac{5}{3}

\frac{x}{3}+5

✅

\frac{x+5}{3}

3x+5

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The phrase “quotient of $x$ and 3” means $\frac{x}{3}$ . The phrase “five more than” then requires adding 5 to that entire quantity, producing $\frac{x}{3}+5$ . The incorrect expression changes which quantities are divided.

Q5. A graph shows a linear cost relationship passing through $(0,12)$ and $(4,28)$ , where $x$ represents items purchased. Which English phrase best matches the expression represented by the graph?

A.Twelve less than four times the number of items

B.Twelve more than four times the number of items ✅

C.Four more than twelve times the number of items

D.The product of twelve and four more than the number of items

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The graph has an initial value of 12 when $x=0$ . From $(0,12)$ to $(4,28)$ , the cost increases by 16, or 4 per item. Thus the expression is $12+4x$ , meaning twelve more than four times the number of items.

Q6. A fundraiser earns $6$ dollars for each ticket sold and starts with a donation of $20$ dollars. Afterward, the organizer pays a fee of $x$ dollars. Which expression correctly represents the remaining money?

6x+20-x

✅

6(x+20)-x

20-6x+x

6x-(20-x)

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Each ticket contributes 6 dollars, so $6x$ represents earnings from $x$ tickets. Adding the starting donation gives $6x+20$ , and subtracting the fee $x$ gives $6x+20-x$ . Each operation corresponds directly to a modeled quantity.

Q7. Two students translate the phrase “three less than the product of a number $y$ and the sum of $y$ and 2.” Student A writes $3-y(y+2)$ , while Student B writes $y(y+2)-3$ . Which student is correct?

A.Student A, because “less than” always places the smaller number first

B.Student B, because the product is formed before subtracting three ✅

C.Both are correct because subtraction can be reversed

D.Neither is correct; the expression should be

3y+2-3

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The phrase “product of $y$ and the sum of $y$ and 2” is $y(y+2)$ . Then “three less than” that entire product means subtract 3 from it, giving $y(y+2)-3$ . Student A reverses the subtraction and changes the meaning.

Q8. Which expression correctly represents “the product of 7 and a number $x$ , divided by 5”?

7x-5

\frac{7x}{5}

✅

7(x-5)

\frac{x}{35}

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The phrase “product of 7 and $x$ ” gives $7x$ . The words “divided by 5” mean the entire product is divided by 5, producing $\frac{7x}{5}$ . The distractors incorrectly change the operation or grouping.

Q9. A school buys $x$ notebooks at $4 each and divides the total cost equally among 2 departments. Which expression models the amount assigned to each department?

4x+2

\frac{x}{4}+2

\frac{4x}{2}

✅

4(x+2)

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The total cost of $x$ notebooks at $4 each is $4x$ . Since this total is divided equally between 2 departments, the correct model is $\frac{4x}{2}$ . The other choices confuse multiplication, addition, or division.

Q10. A student translates “the quotient of $3x$ and $x+4$ ” as $3+\frac{x}{x+4}$ . What is the correct expression?

3x(x+4)

\frac{3x}{x+4}

✅

\frac{x+4}{3x}

\frac{3}{x}+4

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: “Quotient of $3x$ and $x+4$ ” means the first quantity, $3x$ , is divided by the second quantity, $x+4$ . Therefore the correct expression is $\frac{3x}{x+4}$ . The incorrect choice breaks the numerator into separate terms.

Q11. A phone plan charges a base fee of $12 and then charges $x$ dollars for each of 3 additional services. The total is shared equally by 2 people. Which expression represents each person's share?

12+\frac{3x}{2}

\frac{12+3x}{2}

✅

2(12+3x)

\frac{12}{2}+3x

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Three services costing $x$ each produce $3x$ , which is added to the $12 base fee. Because the entire total is shared equally by 2 people, both terms must be divided by 2, giving $\frac{12+3x}{2}$ .

Q12. A graph of a cost function is a straight line passing through $(0,10)$ and $(5,30)$ . Which expression best represents the relationship between the number of items $x$ and total cost $C$ ?

C=10x+4

C=\frac{x+10}{4}

C=4x+10

✅

C=\frac{10x}{4}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The graph starts at 10 when $x=0$ , so 10 is the fixed amount. The increase is $30-10=20$ over 5 items, giving a rate of 4 per item. Thus the relationship is $C=4x+10$ .

Q13. A recipe uses $x$ grams of flour for each batch. A baker makes 6 batches and then divides the flour equally among 3 containers. Which expression represents the amount of flour in each container?

6x+3

\frac{6x}{3}

✅

\frac{x}{6}+3

3(6x)

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Six batches require $6x$ grams because the amount $x$ is multiplied by 6. Dividing the total equally among 3 containers gives $\frac{6x}{3}$ . This requires recognizing both the product and the subsequent quotient.

Q14. A student says that “half the quotient of $x$ and $y$ ” is $\frac{x}{2y}$ , while another says it is $\frac{1}{2}\left(\frac{x}{y}\right)$ . Which statement is correct?

A.Only the first student is correct

B.Only the second student is correct

C.Both expressions are equivalent ✅

D.Neither expression represents the phrase

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The quotient of $x$ and $y$ is $\frac{x}{y}$ . Taking half of that quotient gives $\frac{1}{2}\left(\frac{x}{y}\right)=\frac{x}{2y}$ . Therefore both students wrote algebraically equivalent expressions, even though their grouping looks different.

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