Order of Operations - ISEE Middle Level Mathematics Achievement

What is the value of the expression $15 - 3^2 + (5-2)^3 \div 9$ ?
Answer: 9 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 $6 + 3 = 9$ .
Evaluate: $( 8 \div 2 ) \times ( 3^2 - 5 ) - \frac{1}{2}$
Answer: $\frac{31}{2}$ 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 $(8 ÷ 2) × (3^2 - 5) - \frac{1}{2}$ , 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 $16 - \frac{1}{2} = \frac{32}{2} - \frac{1}{2} = \frac{31}{2}$ . Choice C is correct because it reflects the proper application of the order of operations, computing the expression correctly to arrive at $\frac{31}{2}$ . 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.
A game's score is calculated using the formula $S = 50 + 20c - 5p$ , where $c$ is the number of coins collected and $p$ is the number of penalties. If a player collects 8 coins and gets 3 penalties, what is their final score?
Answer: 195 Substitute $c=8$ and $p=3$ into the formula: $S = 50 + 20(8) - 5(3)$ . Following the order of operations, perform the multiplications first: $20 \times 8 = 160$ and $5 \times 3 = 15$ . The expression becomes $S = 50 + 160 - 15$ . Now, perform addition and subtraction from left to right. First, add: $50 + 160 = 210$ . Then, subtract: $210 - 15 = 195$ . The player's final score is 195.
Which placement of parentheses in the expression $6 \times 8 - 5 + 2$ results in a value of 20?
Answer: $6 \times (8 - 5) + 2$ 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.
If $x=4$ and $y=-2$ , what is the value of the expression $x^2 - (10 - y) \div 3$ ?
Answer: 12 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: $16 - 4 = 12$ .
What is the value of $48 \div 8 \times 3 - 2$ ?
Answer: 16 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: $18 - 2 = 16$ .
An expression is defined as $8 \times (-3)^2 - 10 \div 2$ . What is the value of this expression?
Answer: 67 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: $72 - 5 = 67$ .
What is the value of the expression $5 + 2[-4 + (9-3)^2 \div 12]$ ?
Answer: 3 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: $5 + (-2) = 3$ .
What is the value of the expression $36 - [18 \div (2+4) \times 5]$ ?
Answer: 21 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: $36 - 15 = 21$ .
What number must be subtracted from the result of $4 \times (5+3)^2$ to get 200?
Answer: 56 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 $n = 256 - 200 = 56$ .
Evaluate: $( 10 - 2^3 ) \times \frac{3}{2} + \frac{1}{4}$
Answer: $\frac{13}{4}$ 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 $(10 - 2^3) × \frac{3}{2} + \frac{1}{4}$ , we first evaluate the exponent $2^3 = 8$ , then the parentheses $(10 - 8) = 2$ , followed by multiplication $2 × \frac{3}{2} = 3$ , and finally addition $3 + \frac{1}{4} = \frac{12}{4} + \frac{1}{4} = \frac{13}{4}$ . Choice A is correct because it reflects the proper application of the order of operations, computing the expression correctly to arrive at $\frac{13}{4}$ . 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.
The temperature in a town was 12°F. Over the next three hours, the temperature dropped 5°F each hour. In the fourth hour, the temperature rose by 7°F. Which expression correctly represents the final temperature?
Answer: $12 - 5 \times 3 + 7$ When you encounter word problems involving sequential changes to a starting value, the key is to translate each step of the process into mathematical operations in the correct order. Let's trace through what happens to the temperature step by step. You start with 12°F. Over three hours, the temperature drops 5°F each hour, so the total drop is $5 \times 3 = 15°F$ . This gives you $12 - 15 = -3°F$ after three hours. Then in the fourth hour, the temperature rises 7°F, bringing you to $-3 + 7 = 4°F$ . The expression that captures this entire process is $12 - 5 \times 3 + 7$ , which is choice D. Choice A is wrong because $(12 - 5) \times 3 + 7$ calculates the temperature after one hour (7°F), then multiplies by 3, giving 21°F, then adds 7°F for a final answer of 28°F. This doesn't represent the actual temperature changes. Choice B is incorrect because $12 - 5 + 3 \times 7$ follows order of operations to give $12 - 5 + 21 = 28°F$ . This treats the "3" as a multiplier for the temperature rise rather than the number of hours. Choice C is wrong because $12 - (5+3) + 7$ gives $12 - 8 + 7 = 11°F$ . This incorrectly adds the hourly drop to the number of hours instead of multiplying them. Remember: when temperature changes happen repeatedly, multiply the change per period by the number of periods. Watch out for expressions that mix up the numbers or apply operations in the wrong sequence.
What is the value of the expression $\frac{5^2 - (3+4)}{6 \times 2 - 3}$ ?
Answer: 2 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.
What is the value of $0.5 \times (10.6 - 4.6) + 12 \div 0.3$ ?
Answer: 43 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: $3 + 40 = 43$ .
If $m = 6$ , what is the value of $3m - (m+2)^2 \div 4$ ?
Answer: 2 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: $18 - 16 = 2$ .
What is the result of evaluating the expression $20 - |-12| \div 3 + |5 - 9|$ ?
Answer: 20 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 $16 + 4 = 20$ .
To evaluate $5 + 3 \times (8-2) \div 3$ , a student wrote the following steps: Step 1: $5 + 3 \times 6 \div 3$ Step 2: $8 \times 6 \div 3$ Step 3: $48 \div 3$ Step 4: $16$ In which step did the student make the first mistake?
Answer: Step 2 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.
Which expression has the greatest value?
Answer: $10 + 20 \div (5 - 2)$ 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.
A positive integer $k$ is used in the expression $20 - 2 \times k$ . If the value of the expression is greater than 8 but less than 12, what is one possible value of $k$ ?
Answer: 5 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.
What is the value of $3 \times [20 - (2+3)^2 \div 5]$ ?
Answer: 45 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: $3 \times 15 = 45$ .

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