ISEE Middle Level Mathematics Achievement Quiz: Order Of Operations

What this quiz covers

This quiz focuses on Order Of Operations, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Mathematics Achievement.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Step 1
2. Step 2 (correct answer)
3. Step 3
4. Step 4

Explanation: The original expression is (5 + 3 \times (8-2) \div 3). Step 1 correctly evaluates the parentheses: (8-2=6), resulting in (5 + 3 \times 6 \div 3). Step 2 shows (8 \times 6 \div 3), which means the student incorrectly added (5+3) before performing multiplication and division. According to the order of operations, multiplication and division must be done before addition. Therefore, the first mistake was made in Step 2.

Question 2

1. 12
2. 18
3. 20 (correct answer)
4. 28

Explanation: Following the order of operations, treat the absolute value bars as grouping symbols. First, evaluate the expressions inside them: (|-12| = 12) and (|5 - 9| = |-4| = 4). The expression becomes (20 - 12 \div 3 + 4). Next, perform the division: (12 \div 3 = 4). The expression is now (20 - 4 + 4). Finally, perform addition and subtraction from left to right: (20 - 4 = 16), and (16 + 4 = 20).

Question 3

1. $\frac{27}{2}$
2. $\frac{29}{2}$
3. $\frac{31}{2}$ (correct answer)
4. $\frac{33}{2}$

Explanation: This question tests the ISEE Middle Level skill of evaluating expressions using the order of operations. The order of operations ensures consistent results in mathematical expressions, following the hierarchy of parentheses, exponents, multiplication and division, and addition and subtraction (PEMDAS). In this expression (8 ÷ 2) × (3^2 - 5) - rac{1}{2}, we first evaluate within each set of parentheses: $8 ÷ 2 = 4$ and $3^2 - 5 = 9 - 5 = 4$, then multiply $4 × 4 = 16$, and finally subtract 16 - rac{1}{2} = rac{32}{2} - rac{1}{2} = rac{31}{2}. Choice C is correct because it reflects the proper application of the order of operations, computing the expression correctly to arrive at rac{31}{2}. The other choices represent common errors such as performing operations out of order or making arithmetic mistakes. Teaching strategies: Use practice problems that require identifying and correcting common errors, and have students verbalize each step as they work through the problem.

Question 4

1. -4
2. 3 (correct answer)
3. 7
4. 12

Explanation: Follow the order of operations, starting with the innermost parentheses: (9-3 = 6). The expression becomes (5 + 2[-4 + 6^2 \div 12]). Inside the brackets, evaluate the exponent: (6^2 = 36). The expression is now (5 + 2[-4 + 36 \div 12]). Still inside the brackets, perform the division: (36 \div 12 = 3). The expression becomes (5 + 2[-4 + 3]). Evaluate inside the brackets: (-4 + 3 = -1). The expression is now (5 + 2(-1)). Perform the multiplication: (2(-1) = -2). Finally, perform the addition: (5 + (-2) = 3).

Question 5

1. (6 \times (8 - 5 + 2))
2. ((6 \times 8) - 5 + 2)
3. (6 \times 8 - (5 + 2))
4. (6 \times (8 - 5) + 2) (correct answer)

Explanation: When you see a problem asking about parentheses placement, you're working with the order of operations. Parentheses change which calculations happen first, which can dramatically affect the final result. You need to evaluate each option systematically to find which gives you 20. Let's work through the correct answer first. In choice D, $6 \times (8 - 5) + 2$, you solve the parentheses first: $8 - 5 = 3$. Then multiply: $6 \times 3 = 18$. Finally add: $18 + 2 = 20$. This matches our target value. Now let's see why the other options don't work. Choice A gives us $6 \times (8 - 5 + 2) = 6 \times 5 = 30$, which is too large. Choice B, $(6 \times 8) - 5 + 2$, equals $48 - 5 + 2 = 45$, also too large. Choice C yields $6 \times 8 - (5 + 2) = 48 - 7 = 41$, again too large. Notice that choices A, B, and C all produce values much larger than 20. This happens because they either multiply 6 by a larger number (choice A) or start with the full product of 48 (choices B and C). Only choice D reduces the multiplication factor by subtracting within the parentheses first. When tackling order of operations problems, always work through each option completely rather than trying to shortcut. Small changes in parentheses placement create big differences in results, so careful calculation is essential.

Question 6

1. 0
2. 4
3. 16 (correct answer)
4. 22

Explanation: According to the order of operations, multiplication and division have the same level of priority and should be performed from left to right. First, perform the division: (48 \div 8 = 6). The expression becomes (6 \times 3 - 2). Next, perform the multiplication: (6 \times 3 = 18). The expression becomes (18 - 2). Finally, perform the subtraction: (18 - 2 = 16).

Question 7

1. 1
2. 7
3. 9 (correct answer)
4. 15

Explanation: Following the order of operations (PEMDAS): First, evaluate the expression in the parentheses: (5-2 = 3). The expression is now (15 - 3^2 + 3^3 \div 9). Next, evaluate the exponents from left to right: (3^2 = 9) and (3^3 = 27). The expression becomes (15 - 9 + 27 \div 9). Next, perform division: (27 \div 9 = 3). The expression is now (15 - 9 + 3). Finally, perform subtraction and addition from left to right: (15 - 9 = 6), and (6 + 3 = 9).

Question 8

1. 3
2. 5 (correct answer)
3. 6
4. 8

Explanation: We are given (8 < 20 - 2k < 12). Let's test the answer choices for (k). A) If (k=3), (20 - 2(3) = 20 - 6 = 14). 14 is not less than 12. B) If (k=5), (20 - 2(5) = 20 - 10 = 10). (8 < 10 < 12) is true. C) If (k=6), (20 - 2(6) = 20 - 12 = 8). 8 is not greater than 8. D) If (k=8), (20 - 2(8) = 20 - 16 = 4). 4 is not greater than 8.

Question 9

1. 56 (correct answer)
2. 64
3. 156
4. 256

Explanation: First, evaluate the expression (4 \times (5+3)^2). According to the order of operations, start with the parentheses: (5+3 = 8). The expression becomes (4 \times 8^2). Next, evaluate the exponent: (8^2 = 64). The expression becomes (4 \times 64). Finally, multiply: (4 \times 64 = 256). The question asks what number must be subtracted from this result to get 200. Let the number be (n). So, (256 - n = 200). Solving for (n), we get (n = 256 - 200 = 56).

Question 10

1. (10 + 20 \div 5 - 2)
2. ((10 + 20) \div 5 - 2)
3. ((10 + 20) \div (5 - 2))
4. (10 + 20 \div (5 - 2)) (correct answer)

Explanation: When you encounter expressions with multiple operations, the order of operations (PEMDAS/BODMAS) determines which calculations to perform first. You must work through parentheses, then multiplication and division (left to right), then addition and subtraction (left to right). Let's evaluate each expression systematically: For choice A: $10 + 20 \div 5 - 2$ First divide: $20 \div 5 = 4$ Then work left to right: $10 + 4 - 2 = 12$ For choice B: $(10 + 20) \div 5 - 2$ Parentheses first: $10 + 20 = 30$ Then divide: $30 \div 5 = 6$ Finally subtract: $6 - 2 = 4$ For choice C: $(10 + 20) \div (5 - 2)$ Both sets of parentheses: $30 \div 3 = 10$ For choice D: $10 + 20 \div (5 - 2)$ Parentheses first: $5 - 2 = 3$ Then divide: $20 \div 3 = 6\frac{2}{3}$ Finally add: $10 + 6\frac{2}{3} = 16\frac{2}{3}$ Choice D gives the greatest value at $16\frac{2}{3}$. Choice A incorrectly ignores that division comes before addition and subtraction. Choice B reduces the dividend by putting addition in parentheses with division. Choice C reduces the divisor, but dividing the same number by a smaller divisor (3 instead of 5) doesn't increase the result enough to exceed choice D. Remember: parentheses can dramatically change an expression's value by altering the order of operations. Always identify what's in parentheses first, then follow PEMDAS carefully.

Question 11

1. 15
2. 27
3. 45 (correct answer)
4. 57

Explanation: First, evaluate inside the innermost grouping symbols, the parentheses: (2+3=5). The expression becomes (3 \times [20 - 5^2 \div 5]). Next, inside the brackets, evaluate the exponent: (5^2=25). The expression is now (3 \times [20 - 25 \div 5]). Still inside the brackets, perform the division: (25 \div 5 = 5). The expression becomes (3 \times [20 - 5]). Evaluate inside the brackets: (20 - 5 = 15). Finally, perform the multiplication: (3 \times 15 = 45).

Question 12

1. -12
2. 20 (correct answer)
3. 24
4. 32

Explanation: First, evaluate the parentheses: (1+2 = 3). The expression becomes (60 \div 5(3) - 4^2). The notation 5(3) means (5 \times 3). Next, evaluate the exponent: (4^2=16). The expression is now (60 \div 5 \times 3 - 16). Perform division and multiplication from left to right. First, (60 \div 5 = 12). The expression becomes (12 \times 3 - 16). Then, (12 \times 3 = 36). The expression is now (36 - 16). Finally, subtract: (36 - 16 = 20).

Question 13

1. -2
2. 1 (correct answer)
3. 5
4. 13

Explanation: Substitute the values of (a) and (b) into the expression: (4^2 \div 4 - 6(1/2)). According to the order of operations, evaluate the exponent first: (4^2 = 16). The expression becomes (16 \div 4 - 6(1/2)). Next, perform division and multiplication from left to right. (16 \div 4 = 4) and (6 \times 1/2 = 3). The expression is now (4 - 3). Finally, perform the subtraction: (4 - 3 = 1).

Question 14

1. 2 (correct answer)
2. 8
3. 14
4. 17

Explanation: Substitute (m=6) into the expression: (3(6) - (6+2)^2 \div 4). Following the order of operations, start with the parentheses: (6+2=8). The expression becomes (3(6) - 8^2 \div 4). Next, evaluate the exponent: (8^2 = 64). The expression is now (3(6) - 64 \div 4). Perform multiplication and division from left to right: (3 \times 6 = 18) and (64 \div 4 = 16). The expression becomes (18 - 16). Finally, subtract: (18 - 16 = 2).

Question 15

1. 0
2. 8
3. 12 (correct answer)
4. 20

Explanation: First, substitute the given values for (x) and (y) into the expression: (4^2 - (10 - (-2)) \div 3). Following the order of operations, start with the innermost parentheses: (10 - (-2) = 10 + 2 = 12). The expression becomes (4^2 - 12 \div 3). Next, evaluate the exponent: (4^2 = 16). The expression is now (16 - 12 \div 3). Perform the division before subtraction: (12 \div 3 = 4). Finally, perform the subtraction: (16 - 4 = 12).

Question 16

1. $\frac{13}{4}$ (correct answer)
2. $\frac{7}{4}$
3. $\frac{9}{4}$
4. $\frac{11}{4}$

Explanation: This question tests the ISEE Middle Level skill of evaluating expressions using the order of operations. The order of operations ensures consistent results in mathematical expressions, following the hierarchy of parentheses, exponents, multiplication and division, and addition and subtraction (PEMDAS). In this expression (10 - 2^3) × rac{3}{2} + rac{1}{4}, we first evaluate the exponent $2^3 = 8$, then the parentheses $(10 - 8) = 2$, followed by multiplication 2 × rac{3}{2} = 3, and finally addition 3 + rac{1}{4} = rac{12}{4} + rac{1}{4} = rac{13}{4}. Choice A is correct because it reflects the proper application of the order of operations, computing the expression correctly to arrive at rac{13}{4}. Students often make errors by evaluating $2^3$ as 6 instead of 8, leading to incorrect answers. Teaching strategies: Use practice problems focusing specifically on exponents, and create memory devices to help students remember that exponents mean repeated multiplication.

Question 17

1. 2 (correct answer)
2. 3
3. 6
4. 18

Explanation: The fraction bar acts as a grouping symbol, so evaluate the numerator and the denominator separately before dividing. In the numerator: (5^2 - (3+4)). Parentheses first: (3+4=7). Then exponents: (5^2=25). Then subtraction: (25-7=18). So the numerator is 18. In the denominator: (6 \times 2 - 3). Multiplication first: (6 \times 2=12). Then subtraction: (12-3=9). So the denominator is 9. The expression simplifies to (\frac{18}{9}), which equals 2.

Question 18

1. 33
2. 43 (correct answer)
3. 50
4. 70

Explanation: Following the order of operations, first evaluate the expression in parentheses: (10.6 - 4.6 = 6.0). The expression becomes (0.5 \times 6 + 12 \div 0.3). Next, perform multiplication and division from left to right. (0.5 \times 6 = 3). To calculate (12 \div 0.3), we can multiply the numerator and denominator by 10 to get (120 \div 3 = 40). The expression is now (3 + 40). Finally, add: (3 + 40 = 43).

Question 19

1. 6
2. 21 (correct answer)
3. 33
4. 165

Explanation: To evaluate the expression, follow the order of operations (PEMDAS/BODMAS). First, solve the operation inside the parentheses: (2+4 = 6). The expression becomes (36 - [18 \div 6 \times 5]). Next, solve the operations inside the brackets. Since division and multiplication have equal priority, work from left to right. First, do the division: (18 \div 6 = 3). The expression becomes (36 - [3 \times 5]). Then, do the multiplication: (3 \times 5 = 15). The expression becomes (36 - 15). Finally, perform the subtraction: (36 - 15 = 21).

Question 20

1. -53
2. 31
3. 67 (correct answer)
4. 77

Explanation: According to the order of operations, evaluate exponents first. ((-3)^2 = (-3) \times (-3) = 9). The expression is now (8 \times 9 - 10 \div 2). Next, perform multiplication and division from left to right. (8 \times 9 = 72) and (10 \div 2 = 5). The expression becomes (72 - 5). Finally, perform the subtraction: (72 - 5 = 67).