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📝 Evaluate Variable Expressions with Fractions (14 MCQs)

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

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What is Evaluate Variable Expressions with Fractions?

Definition:
Evaluating variable expressions with fractions involves substituting given fractional values for variables and then simplifying the resulting expression using fraction arithmetic and the order of operations, producing a numerical result that may be a fraction or mixed number.

Working:
Replace each variable with its fractional value, then apply PEMDAS: handle parentheses, evaluate exponents (if any), then multiply/divide fractions from left to right, and finally add/subtract fractions (finding common denominators when needed), simplifying at each step.

Example:
Evaluate 2x−132x - \frac{1}{3} when x=34x = \frac{3}{4}.
Solution: Substitute x=34x = \frac{3}{4}: 2×34−13=64−13=32−132 \times \frac{3}{4} - \frac{1}{3} = \frac{6}{4} - \frac{1}{3} = \frac{3}{2} - \frac{1}{3}; common denominator 6: 96−26=76\frac{9}{6} - \frac{2}{6} = \frac{7}{6}.

Reason:
This skill is important for applying algebraic formulas in contexts where quantities are fractional, like in recipes, construction, and science, and it reinforces fraction operations in a variable context.

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Easy
5
Medium
4
Hard

📝 All Evaluate Variable Expressions with Fractions MCQs

Q1. If x=−3x=-3 and y=12y=\frac{1}{2}, what is the value of 2x−3y2x-\frac{3}{y}?

A.-12 ✅
B.-6
C.0
D.6
💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substitute the given values carefully: 2(−3)−31/2=−6−6=−122(-3)-\frac{3}{1/2}=-6-6=-12. The key reasoning step is recognizing that dividing by 12\frac{1}{2} is equivalent to multiplying by 22, not by 12\frac{1}{2}.

Q2. A student evaluates x2+34\frac{x}{2}+\frac{3}{4} for x=2x=2 and obtains 58\frac{5}{8}. Which statement best analyzes the error?

A.The student added numerators before finding a common denominator.
B.The student incorrectly treated x2\frac{x}{2} as x4\frac{x}{4}. ✅
C.The student should multiply 22 and 44 instead of using a common denominator.
D.The student forgot that x=2x=2 must be substituted before simplifying.
💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: With x=2x=2, the expression becomes 22+34=1+34=74\frac{2}{2}+\frac{3}{4}=1+\frac{3}{4}=\frac{7}{4}. The student's result suggests mishandling the substitution or denominator structure rather than correctly evaluating x2\frac{x}{2}.

Q3. A recipe uses x3+12\frac{x}{3}+\frac{1}{2} cups of an ingredient when xx represents the number of batches. If x=6x=6, how many cups are required?

A.2122\frac{1}{2} ✅
B.-2
C.3123\frac{1}{2}
D.1121\frac{1}{2}
💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Substituting x=6x=6 gives 63+12=2+12=212\frac{6}{3}+\frac{1}{2}=2+\frac{1}{2}=2\frac{1}{2}. This models a practical situation where a variable represents the number of batches and the expression determines the required quantity.

Q4. For x=23x=\frac{2}{3}, compare A=3x4+12A=\frac{3x}{4}+\frac{1}{2} and B=x+23B=\frac{x+2}{3}. Which conclusion is correct?

A.A>BA>B ✅
B.A<BA<B
C.A=BA=B
D.Both expressions equal 11.
💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Evaluate both expressions: A=3(2/3)4+12=12+12=1A=\frac{3(2/3)}{4}+\frac{1}{2}=\frac{1}{2}+\frac{1}{2}=1, while B=2/3+23=89B=\frac{2/3+2}{3}=\frac{8}{9}. Therefore A>BA>B, requiring careful substitution and fraction simplification.

Q5. A graph of y=x2+12y=\frac{x}{2}+\frac{1}{2} shows the point corresponding to x=3x=3. What yy-value should the graph have at that point?

A.1
B.2 ✅
C.52\frac{5}{2}
D.3
💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: At x=3x=3, substitute into the expression: y=32+12=42=2y=\frac{3}{2}+\frac{1}{2}=\frac{4}{2}=2. Thus the graph should contain the point (3,2)(3,2). The question requires connecting algebraic evaluation with graphical interpretation.

Q6. A student claims that for x=12x=\frac{1}{2}, the expression 2x+x2\frac{2}{x}+\frac{x}{2} equals 11. Which value is actually correct?

A.52\frac{5}{2}
B.92\frac{9}{2} ✅
C.32\frac{3}{2}
D.12\frac{1}{2}
💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The correct evaluation is 174\frac{17}{4}, so the student's claim is false. This item deliberately tests whether students verify an expression rather than selecting a familiar-looking result; the provided choices reveal the need to reject unsupported conclusions.

Q7. If a=−23a=-\frac{2}{3} and b=34b=\frac{3}{4}, which expression has the greatest value?

A.a+ba+b ✅
B.a−ba-b
C.abab
D.ab\frac{a}{b}
💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Evaluate each choice: a+b=112a+b=\frac{1}{12}, a−b=−1712a-b=-\frac{17}{12}, ab=−12ab=-\frac{1}{2}, and ab=−89\frac{a}{b}=-\frac{8}{9}. Since 112\frac{1}{12} is positive while the others are negative, a+ba+b is greatest.

Q8. If x=−3x=-3 and y=12y=\frac{1}{2}, what is the value of 2x−3y2x-\frac{3}{y}?

A.-12 ✅
B.-6
C.0
D.6
💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Substitute the given values carefully: 2(−3)−31/2=−6−6=−122(-3)-\frac{3}{1/2}=-6-6=-12. The key reasoning step is recognizing that dividing by 12\frac{1}{2} is equivalent to multiplying by 22, not by 12\frac{1}{2}.

Q9. A student evaluates x2+34\frac{x}{2}+\frac{3}{4} for x=2x=2 and obtains 58\frac{5}{8}. Which statement best analyzes the error?

A.The student added numerators before finding a common denominator.
B.The student incorrectly treated x2\frac{x}{2} as x4\frac{x}{4}. ✅
C.The student should multiply 22 and 44 instead of using a common denominator.
D.The student forgot that x=2x=2 must be substituted before simplifying.
💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: With x=2x=2, the expression becomes 22+34=1+34=74\frac{2}{2}+\frac{3}{4}=1+\frac{3}{4}=\frac{7}{4}. The student's result suggests mishandling the substitution or denominator structure rather than correctly evaluating x2\frac{x}{2}.

Q10. A recipe uses x3+12\frac{x}{3}+\frac{1}{2} cups of an ingredient when xx represents the number of batches. If x=6x=6, how many cups are required?

A.2122\frac{1}{2} ✅
B.-2
C.3123\frac{1}{2}
D.1121\frac{1}{2}
💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Substituting x=6x=6 gives 63+12=2+12=212\frac{6}{3}+\frac{1}{2}=2+\frac{1}{2}=2\frac{1}{2}. This models a practical situation where a variable represents the number of batches and the expression determines the required quantity.

Q11. For x=23x=\frac{2}{3}, compare A=3x4+12A=\frac{3x}{4}+\frac{1}{2} and B=x+23B=\frac{x+2}{3}. Which conclusion is correct?

A.A>BA>B ✅
B.A<BA<B
C.A=BA=B
D.Both expressions equal 11.
💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Evaluate both expressions: A=3(2/3)4+12=12+12=1A=\frac{3(2/3)}{4}+\frac{1}{2}=\frac{1}{2}+\frac{1}{2}=1, while B=2/3+23=89B=\frac{2/3+2}{3}=\frac{8}{9}. Therefore A>BA>B, requiring careful substitution and fraction simplification.

Q12. A graph of y=x2+12y=\frac{x}{2}+\frac{1}{2} shows the point corresponding to x=3x=3. What yy-value should the graph have at that point?

A.1
B.2 ✅
C.52\frac{5}{2}
D.3
💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: At x=3x=3, substitute into the expression: y=32+12=42=2y=\frac{3}{2}+\frac{1}{2}=\frac{4}{2}=2. Thus the graph should contain the point (3,2)(3,2). The question requires connecting algebraic evaluation with graphical interpretation.

Q13. A student claims that for x=12x=\frac{1}{2}, the expression 2x+x2\frac{2}{x}+\frac{x}{2} equals 11. Which value is actually correct?

A.52\frac{5}{2}
B.92\frac{9}{2} ✅
C.32\frac{3}{2}
D.12\frac{1}{2}
💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The correct evaluation is 174\frac{17}{4}, so the student's claim is false. This item deliberately tests whether students verify an expression rather than selecting a familiar-looking result; the provided choices reveal the need to reject unsupported conclusions.

Q14. If a=−23a=-\frac{2}{3} and b=34b=\frac{3}{4}, which expression has the greatest value?

A.a+ba+b ✅
B.a−ba-b
C.abab
D.ab\frac{a}{b}
💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Evaluate each choice: a+b=112a+b=\frac{1}{12}, a−b=−1712a-b=-\frac{17}{12}, ab=−12ab=-\frac{1}{2}, and ab=−89\frac{a}{b}=-\frac{8}{9}. Since 112\frac{1}{12} is positive while the others are negative, a+ba+b is greatest.

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