📝 Order of operations PEMDAS (21 MCQs)

📖 From Digital SAT Algebra • 1. Basics of Algebra • 21 questions available

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What is Order of operations PEMDAS?

Definition:
PEMDAS is an acronym that stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), which defines the correct sequence for evaluating mathematical expressions to ensure a single, unambiguous result.

Working:
First, simplify inside parentheses or grouping symbols, then evaluate exponents like $2^3$ ; next, perform all multiplication and division as they appear from left to right, and finally, perform all addition and subtraction from left to right, respecting the priority levels.

Example:
Evaluate $3 + 2 \times (5 - 3)^2$ .
Solution: Parentheses first: $5-3=2$ , exponent: $2^2=4$ , multiplication: $2 \times 4 = 8$ , addition: $3+8=11$ .

Reason:
PEMDAS ensures consistency in mathematical communication, so everyone interprets expressions the same way, which is critical in algebra, programming, and science to avoid errors in calculations.

Easy

Medium

Hard

📝 All Order of operations PEMDAS MCQs

Q1. A student evaluates $18-3(4+2)\div2$ as $45$ by subtracting first and then multiplying. What is the correct value, and which error best explains the student's result?

A.9; subtraction was performed before grouping ✅

B.9; multiplication and division were handled incorrectly

C.15; addition inside parentheses should be delayed

D.45; the student's method is valid

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: First evaluate the parentheses: $4+2=6$ . Then multiply and divide from left to right: $3(6)\div2=9$ . Finally, $18-9=9$ . The student's answer results from ignoring the priority of the grouped expression and applying subtraction too early.

Q2. A rectangular garden has a length represented by $2(5+3)$ meters and a width of $6-2$ meters. Which expression correctly calculates its area and gives the correct result?

2(5)+3(6)-2=26

2(5+3)(6-2)=64

✅

2(5+3)+6-2=20

2(5)+3(6-2)=22

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The length is $2(5+3)=16$ meters because the parentheses must be evaluated before multiplication. The width is $6-2=4$ meters. Therefore, the area is $16\times4=64$ square meters, making option B correct.

Q3. Two students simplify $24\div(3+1)\times2$ . Student A obtains $3$ , while Student B obtains $12$ . Which evaluation is correct and why?

A.Student A, because division must always come after multiplication

B.Student B, because parentheses are evaluated first and division and multiplication proceed left to right ✅

C.Both are correct because multiplication and division can be rearranged

D.Student A, because multiplication always has priority over division

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The parentheses give $3+1=4$ , leaving $24\div4\times2$ . Division and multiplication have equal precedence, so they are evaluated from left to right: $24\div4=6$ , then $6\times2=12$ . Student B correctly follows that sequence.

Q4. A calculator-free solution to $7+2[9-3(2+1)]$ is written as $7+2[9-9]=25$ . What should the final value be, and what mistake occurred?

A.7; multiplication by 2 was performed too early ✅

B.25; the solution is correct

C.21; the brackets were evaluated incorrectly

D.14; the subtraction inside the brackets should be done first

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Evaluate the innermost grouping first: $2+1=3$ . Then $3(3)=9$ , so the bracket becomes $9-9=0$ . Finally, $7+2(0)=7$ . The incorrect solution treats $2[0]$ as $2$ , effectively forgetting that multiplication by zero gives zero.

Q5. A graphing tool shows the values of two expressions for $x=1,2,3$ : $A(x)=2x+3$ and $B(x)=2(x+3)$ . At which value of $x$ do the two expressions produce the same result?

x=0

x=1

x=3

D.They never produce the same result ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: The expressions simplify to $A(x)=2x+3$ and $B(x)=2x+6$ . Their difference is always $3$ , so their graph lines remain parallel and never have the same value. Therefore, no value of $x$ makes them equal.

Q6. A machine calculates the expression $30-4^2\div2+3$ . One student says the answer is $17$ , while another says $25$ . Which result is correct?

A.17, because subtraction comes before division

B.25, because

4^2

must be evaluated before division and subtraction ✅

C.25, because addition is evaluated before subtraction

D.17, because division should be performed before the exponent

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Evaluate the exponent first: $4^2=16$ . Then divide: $16\div2=8$ . The remaining expression is $30-8+3$ . Addition and subtraction are handled from left to right, giving $22+3=25$ . Thus the second student's answer is correct.

Q7. Find the value of $48\div[2(3+1)]-5$ . A student claims the answer is $19$ because they calculate $48\div2$ before evaluating the bracket. What is the correct value?

A.1

B.7 ✅

C.19

D.43

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The grouping symbol contains $2(3+1)$ , so $3+1=4$ and then $2\times4=8$ . Therefore, $48\div8=6$ , followed by $6-5=1$ . The student's $19$ comes from incorrectly separating the division from the grouped denominator instead of evaluating the complete grouping first.

Q8. Which statement best describes what it means to simplify an algebraic expression?

A.Changing the expression into a form that is easier to interpret while keeping its value unchanged ✅

B.Finding only the numerical value of every variable

C.Removing all variables from the expression

D.Changing the expression into any shorter expression, even if its value changes

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Simplifying an expression means rewriting it into an equivalent form that is usually more compact, organized, or easier to evaluate. The essential requirement is that the new expression must have the same value for every permitted value of its variables.

Q9. A student rewrites $3x+5x-2$ as $8x-2$ . Another student says this is not simplification because the number of terms decreased. Which conclusion is most accurate?

A.The first student is correct because like terms were combined without changing the expression's value ✅

B.The second student is correct because simplification must remove every constant

C.Both students are incorrect because variables cannot be combined

D.The first student is incorrect because

3x+5x=15x

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The terms $3x$ and $5x$ are like terms because both contain $x$ to the first power. Combining their coefficients gives $8x$ , producing $8x-2$ . The result is equivalent but contains fewer terms, making it a simpler form.

Q10. A shop uses the expression $4n+2n+7$ to represent the cost of $n$ identical items plus a fixed fee. Which result best demonstrates successful simplification?

6n+7

, because equivalent variable terms are combined ✅

6n+7n

, because all numbers should be attached to

n

8n

, because the constant fee can be combined with the variable terms

24n+7

, because

4n

and

2n

should be multiplied

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The two variable terms represent the same quantity type, so $4n+2n=6n$ . The fixed fee $7$ remains separate because it is not a like term with $n$ . Thus $6n+7$ preserves the original meaning while using fewer terms.

Q11. A student claims that $2(x+4)$ and $2x+4$ are both simplified forms of the same expression because each contains fewer symbols than a longer expression. How should this reasoning be evaluated?

A.Correct, because any shorter expression is automatically equivalent

B.Incorrect, because

2(x+4)=2x+8

, not

2x+4

✅

C.Correct, because the

2

can be distributed to only one term

D.Incorrect, because expressions containing parentheses can never be simplified

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: Simplification requires equivalence, not merely fewer symbols. Distributing $2$ across both terms gives $2(x+4)=2x+8$ . Since $2x+4$ differs from $2x+8$ , replacing the original expression with it changes the value and is therefore not simplification.

Q12. A graph represents $y=2x+6$ . A student proposes $y=2(x+3)$ as a simplified version. What can be concluded from the graph and the algebraic forms?

A.The proposed form is equivalent because both produce the same

y

-value for every

x

✅

B.The proposed form is incorrect because parentheses always make an expression unsimplified

C.The forms are equivalent only when

x=0

D.The proposed form changes the slope of the graph

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Expanding $2(x+3)$ gives $2x+6$ , so the two expressions are equivalent. Their graphs therefore coincide completely, with the same slope and intercept. This illustrates that simplification can produce different-looking expressions without changing their mathematical relationship.

Q13. Which pair provides the strongest evidence that simplification preserves meaning rather than merely reducing the number of symbols?

5x+2x

and

7x

✅

x+4

and

x+5

3(x+2)

and

3x+2

8x-3

and

8x+3

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The expressions $5x+2x$ and $7x$ are equivalent because both represent seven groups of $x$ . The other pairs are not generally equivalent. Therefore, this pair demonstrates that simplification changes form while preserving mathematical meaning.

Q14. For $x=4$ , a student evaluates $3x+2x+5$ directly and obtains $25$ . Another student first simplifies it to $5x+5$ and then substitutes $x=4$ . What does this comparison demonstrate?

A.Simplification can create an equivalent form that makes later evaluation more efficient ✅

B.Simplification is valid only when variables have no assigned values

C.The first method must be wrong because substitution is required before combining terms

D.The two methods must produce different results because their forms are different

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Direct evaluation gives $3(4)+2(4)+5=12+8+5=25$ . Simplifying first gives $5(4)+5=20+5=25$ . Both methods preserve the same value, demonstrating that simplification creates an equivalent form that can make subsequent calculations easier.

Q15. A student evaluates $18-2(5+1)\div3$ as $10$ . Which sequence correctly applies the order of operations?

A.Evaluate parentheses, then multiplication and division from left to right, then subtraction ✅

B.Perform subtraction first, then multiplication, and finally division

C.Perform division first, then subtraction, and ignore the parentheses

D.Multiply

2

5

, add

1

, then divide the result by

3

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: First evaluate the parentheses: $5+1=6$ . Then multiplication and division proceed from left to right: $2(6)\div3=4$ . Finally, $18-4=14$ . The student's answer comes from applying operations in an incorrect sequence rather than following the required priority.

Q16. A calculator program processes $8+12\div3\times2-5$ . Which result should the program return if it follows the standard order of operations?

A.5

B.11 ✅

C.17

D.19

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Division and multiplication have equal priority and are evaluated from left to right. Thus $12\div3=4$ , followed by $4\times2=8$ . The expression becomes $8+8-5$ , giving $11$ .

Q17. A café uses $50-3(4+2)+8\div2$ to model a customer's remaining balance after several charges. What balance does the expression represent?

A.12

B.20

C.36 ✅

D.42

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Evaluate the grouping first: $4+2=6$ . Then $3\times6=18$ and $8\div2=4$ . The expression becomes $50-18+4=32+4=36$ . Therefore, the correct balance is $36$ .

Q18. Two students solve $24\div(2+4)\times3$ . Student A obtains $12$ , while Student B obtains $4$ . Which analysis is correct?

A.Student A is correct because multiplication always comes before division

B.Student B is correct because the parentheses are evaluated first, followed by division and multiplication from left to right ✅

C.Both are correct because multiplication and division may be performed in either order

D.Student A is correct because division must always be performed last

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: The parentheses give $2+4=6$ , leaving $24\div6\times3$ . Since division and multiplication have equal precedence, they are evaluated left to right: $24\div6=4$ , then $4\times3=12$ . Therefore, Student A is correct.

Q19. A graphing program compares $y=2x+6$ with $y=2(x+3)$ . What does the overlapping graph indicate about the two expressions?

A.They are equivalent for every value of

x

✅

B.They are equivalent only when

x=3

C.The first expression always has a larger value

D.The second expression has twice the slope of the first

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Distributing the $2$ in $2(x+3)$ gives $2x+6$ , exactly matching the first expression. Therefore, both produce the same $y$ -value for every $x$ , so their graphs completely overlap. The graph provides evidence of algebraic equivalence.

Q20. A student claims that $6+4\times3^2$ equals $30^2$ because addition, multiplication, and the exponent can be combined into one calculation. Which correction is most appropriate?

A.The value is

90

, because the exponent is evaluated first, then multiplication, then addition ✅

B.The value is

36

, because addition must occur before multiplication

C.The value is

54

, because multiplication should be performed before the exponent

D.The student's method is valid because all operations can be grouped together

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: The exponent is evaluated first: $3^2=9$ . Then multiplication gives $4\times9=36$ , followed by addition: $6+36=42$ . Therefore, the correct value is $42$ , and treating all operations as one combined step is the student's mistake.

Q21. A puzzle asks for the value of $36\div[3(2+1)] + 2^3$ . A solver evaluates the bracket as $3(2)+1$ . What is the correct result when the expression is evaluated properly?

A.10

B.11

C.12 ✅

D.20

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: The innermost parentheses give $2+1=3$ . The bracket then becomes $3\times3=9$ , so $36\div9=4$ . Next, $2^3=8$ . Finally, $4+8=12$ . The error comes from failing to evaluate the entire grouped expression correctly before division.

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