📝 Translate Phrases to Expressions with Fractions (14 MCQs)

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

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What is Translate Phrases to Expressions with Fractions?

Definition:
Translating phrases to expressions with fractions involves converting verbal statements that include fractional quantities or parts into algebraic expressions, using variables for unknowns and operation words to determine the correct fractional representation, such as 'half of' indicating multiplication by $\frac{1}{2}$ .

Working:
Identify the fraction-related keywords: 'half', 'third', 'quarter', 'percent of', etc., assign variable for the unknown, write the fractional multiplication (e.g., $\frac{1}{3}$ times the quantity), and combine with other operations as needed.

Example:
Translate 'Three-fourths of a number increased by 2' into an expression.
Solution: Let number be $x$ ; 'three-fourths of a number' is $\frac{3}{4}x$ , 'increased by 2' means add 2, so expression is $\frac{3}{4}x + 2$ .

Reason:
This translation skill is vital for solving fraction word problems, as it enables students to model real-life situations involving parts of a whole, proportions, and percentages into algebraic equations that can be solved systematically.

Easy

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📝 All Translate Phrases to Expressions with Fractions MCQs

Q1. A recipe uses $\frac{3}{4}$ cup of flour for each batch. If $x$ represents the number of batches, which expression represents 2 cups less than the total flour used?

\frac{3}{4}x-2

✅

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

2-\frac{3}{4}x

\frac{3}{4}+2x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The phrase 'total flour used' is $\frac{3}{4}x$ because each batch contributes $\frac{3}{4}$ cup. The phrase '2 cups less than' means subtract 2 from that total, giving $\frac{3}{4}x-2$ .

Q2. A student translates 'the quotient of $x$ and $\frac{2}{5}$ , increased by 3' as $\frac{2}{5}x+3$ . What is the correct expression, and why?

\frac{2}{5}x+3

, because quotient means multiplication

x\div\frac{2}{5}+3

, because quotient indicates division ✅

\frac{2}{5}\div x+3

, because the fraction comes first

x\div(\frac{2}{5}+3)

, because increased by 3 changes the divisor

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: A quotient of $x$ and $\frac{2}{5}$ means $x\div\frac{2}{5}$ . The phrase 'increased by 3' then adds 3 to the entire quotient. Therefore the correct expression is $x\div\frac{2}{5}+3$ .

Q3. A phone plan charges $\frac{5}{6}$ dollar per gigabyte plus a fixed fee of $x$ dollars. Which expression represents the cost of using $g$ gigabytes and then receiving a discount of $\frac{1}{4}$ dollar?

\frac{5}{6}g+x-\frac{1}{4}

✅

\frac{5}{6}(g+x)-\frac{1}{4}

\frac{5}{6}g+(x-\frac{1}{4})

\frac{5}{6}(g+x-\frac{1}{4})

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The usage charge is $\frac{5}{6}g$ , while the fixed fee is $x$ . These costs are added first, producing $\frac{5}{6}g+x$ . Since the discount is a separate quarter-dollar reduction, subtracting $\frac{1}{4}$ gives the correct model.

Q4. A student claims that 'half the difference between $x$ and $\frac{3}{7}$ ' can be written as $x-\frac{3}{14}$ . Which expression correctly represents the phrase?

x-\frac{3}{14}

\frac{1}{2}x-\frac{3}{7}

\frac{1}{2}(x-\frac{3}{7})

✅

\frac{1}{2}x+\frac{3}{7}

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The phrase 'the difference between $x$ and $\frac{3}{7}$ ' is $x-\frac{3}{7}$ . The word 'half' means multiply the entire difference by $\frac{1}{2}$ , giving $\frac{1}{2}(x-\frac{3}{7})$ . The incorrect response halves only the fraction.

Q5. A graph shows a quantity increasing linearly from 2 when $x=0$ , with a vertical increase of $\frac{3}{5}$ for every increase of 1 in $x$ . Which expression best models the quantity $y$ ?

y=\frac{3}{5}x+2

✅

y=\frac{3}{5}(x+2)

y=2x+\frac{3}{5}

y=\frac{5}{3}x+2

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The graph indicates an initial value of 2, so the constant term is 2. The quantity increases by $\frac{3}{5}$ for every unit increase in $x$ , making $\frac{3}{5}$ the coefficient of $x$ . Thus $y=\frac{3}{5}x+2$ .

Q6. A rectangular garden has length $x+\frac{1}{2}$ meters and width $\frac{3}{4}$ meter. Which expression represents the perimeter after 1 meter is added to the width?

2(x+\frac{1}{2})+2(\frac{3}{4}+1)

✅

2(x+\frac{1}{2}+\frac{3}{4})+1

x+\frac{1}{2}+\frac{3}{4}+1

2(x+\frac{1}{2})(\frac{3}{4}+1)

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: After the change, the width becomes $\frac{3}{4}+1$ . A rectangle's perimeter is twice the sum of its length and width, so the expression must be $2(x+\frac{1}{2})+2(\frac{3}{4}+1)$ .

Q7. Which expression represents 'three-fourths of a number decreased by the quotient of 2 and $x$ '?

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

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

✅

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

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

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: 'Three-fourths of a number' translates to $\frac{3}{4}x$ . The quotient of 2 and $x$ is $\frac{2}{x}$ . Since the second quantity is decreased from the first, subtraction gives $\frac{3}{4}x-\frac{2}{x}$ .

Q8. Which expression represents the quotient of $x$ and $\frac{3}{5}$ , decreased by 2?

x\div\frac{3}{5}-2

✅

\frac{3}{5}x-2

x\div(\frac{3}{5}-2)

2-\frac{x}{\frac{3}{5}}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The quotient of $x$ and $\frac{3}{5}$ means $x\div\frac{3}{5}$ . The phrase 'decreased by 2' means subtracting 2 from that quotient. Therefore, $x\div\frac{3}{5}-2$ correctly preserves the order and meaning of both phrases.

Q9. A school has $x$ science books and $\frac{2}{3}x$ mathematics books. Which expression represents the ratio of mathematics books to science books, and what does it simplify to?

\frac{2}{3}x\div x=\frac{2}{3}

✅

x\div\frac{2}{3}x=\frac{3}{2}

\frac{2}{3}x+x=\frac{5}{3}x

\frac{2}{3}\div x=\frac{2}{3x}

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A ratio of mathematics books to science books is mathematics divided by science, so it is $\frac{2}{3}x\div x$ . Assuming $x\neq0$ , the $x$ factors cancel, leaving $\frac{2}{3}$ . This models the relative quantity rather than the total.

Q10. A student says the phrase 'the ratio of $\frac{4}{5}x$ to $y$ ' should be written as $\frac{4}{5y}x$ . Which response best evaluates the student's reasoning?

A.Correct, because ratios always place the second quantity in the denominator

B.Incorrect; the ratio is

\frac{\frac{4}{5}x}{y}

, which can be written as

\frac{4x}{5y}

✅

C.Incorrect; the ratio should be

\frac{5y}{4x}

D.Correct, because

\frac{4}{5}x

must be divided only by 5

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The ratio of one quantity to another is the first quantity divided by the second. Thus the expression is $\frac{\frac{4}{5}x}{y}$ . Multiplying numerator and denominator by 5 gives $\frac{4x}{5y}$ , not $\frac{4x}{5y}$ for a different grouping.

Q11. A delivery service charges $\frac{3}{4}$ dollar per mile and a fixed fee of $f$ dollars. If $m$ miles are traveled, which expression represents the ratio of the total distance cost to the fixed fee?

\frac{3}{4}m+f

\frac{\frac{3}{4}m}{f}

✅

\frac{f}{\frac{3}{4}m}

\frac{3}{4}\left(\frac{m}{f}\right)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The distance-based cost is $\frac{3}{4}m$ . The ratio of that cost to the fixed fee means divide the distance cost by $f$ , producing $\frac{\frac{3}{4}m}{f}$ . Adding the quantities would represent a total, not a ratio.

Q12. A graph shows $y$ increasing proportionally with $x$ . At $x=4$ , $y=3$ . Which expression represents the ratio of $x$ to $y$ for any nonzero $x$ ?

\frac{x}{y}=\frac{4}{3}

✅

\frac{x}{y}=\frac{3}{4}

\frac{x}{y}=x-\frac{3}{4}

\frac{x}{y}=\frac{3x}{4}

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Because the graph shows proportional growth, the ratio $y/x$ remains constant. Using the point $(4,3)$ , $y/x=3/4$ , so reversing the order gives $x/y=4/3$ . The ratio must remain constant as long as $x$ and $y$ are nonzero.

Q13. The phrase 'the quotient of $\frac{2}{3}x$ and $\frac{1}{4}y$ , increased by $\frac{1}{2}$ ' appears in a model. Which expression correctly translates the entire phrase?

\frac{2}{3}x\div\frac{1}{4}y+\frac{1}{2}

\frac{\frac{2}{3}x}{\frac{1}{4}y}+\frac{1}{2}

✅

\frac{2}{3}x\div(\frac{1}{4}y+\frac{1}{2})

\frac{1}{2}\left(\frac{\frac{2}{3}x}{\frac{1}{4}y}\right)

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The quotient of $\frac{2}{3}x$ and $\frac{1}{4}y$ is $\frac{\frac{2}{3}x}{\frac{1}{4}y}$ . The phrase 'increased by $\frac{1}{2}$ ' adds that fraction after the quotient, so B preserves the intended grouping.

Q14. For positive $x$ , compare these two expressions: $A=\frac{x}{\frac{1}{2}}$ and $B=\frac{1}{2}x$ . Which statement correctly describes their relationship?

A=B

because both contain

\frac{1}{2}

A=2B

because dividing by

\frac{1}{2}

doubles

x

✅

B=2A

because multiplying by

\frac{1}{2}

doubles the quotient

A+B=x

because the two expressions cancel

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Dividing by $\frac{1}{2}$ is equivalent to multiplying by 2, so $A=2x$ . Meanwhile, $B=\frac{x}{2}$ . Therefore $A=4B$ , not $2B$ , making none of the listed statements correct; this exposes a flawed distractor set and demonstrates why exact translation and simplification must be checked.

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