Order of Operations: Whole Numbers - Accuplacer Arithmetic

1. What is the value of $10 + (24 \div 4 - 6) \times 8$?
   Answer: 10 First, evaluate the operations inside the parentheses. Inside the parentheses, division comes before subtraction: $24 \div 4 = 6$. The expression becomes $10 + (6 - 6) \times 8$. Next, complete the parentheses: $6 - 6 = 0$. The expression is now $10 + 0 \times 8$. Perform multiplication: $0 \times 8 = 0$. Finally, perform addition: $10 + 0 = 10$.
2. What is the final cost for the team's purchase?
   Answer: $120 First, calculate the cost of the jerseys: $5 \times 20 = 100$. Next, calculate the cost of the shorts: $2 \times 15 = 30$. The total cost before the discount is $100 + 30 = 130$. Finally, apply the $10 discount: $130 - 10 = 120$. The final cost is $120.
3. Find the value of $2 \times (8 - 3)^2 + 6 \times 3 - 4$
   Answer: 64 When you encounter expressions with multiple operations like this one, success depends on following the order of operations (PEMDAS/BODMAS) carefully: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right). Let's work through $2 \times (8 - 3)^2 + 6 \times 3 - 4$ step by step: First, handle the parentheses: $(8 - 3) = 5$ Next, apply the exponent: $(5)^2 = 25$ Now perform multiplication from left to right: $2 \times 25 = 50$ and $6 \times 3 = 18$ Finally, work left to right with addition and subtraction: $50 + 18 - 4 = 68 - 4 = 64$ The answer is C) 64. Let's examine why the other choices are wrong. Choice A) 46 likely results from incorrectly calculating $(8-3)^2$ as $5$ instead of $25$, giving $2 \times 5 + 18 - 4 = 28 - 4 = 24$. Wait, that's not 46 either—this suggests multiple errors compounding. Choice B) 58 might come from forgetting to square the 5, calculating $2 \times 5 + 6 \times 3 - 4 = 10 + 18 + 30 = 58$ (though this involves additional mistakes). Choice D) 72 could result from computational errors in the final steps, perhaps adding instead of subtracting the 4. Remember: Write out each step when applying order of operations. The most common mistakes happen when students rush through multiple operations or apply them in the wrong sequence. Double-check your exponent calculations—they're frequent error sources.
4. What is the result of subtracting 10 from the sum of 8 and the product of 4 and 5?
   Answer: 18 This problem translates to the expression $(8 + (4 \times 5)) - 10$. Following the order of operations, calculate the product inside the inner parentheses first: $4 \times 5 = 20$. The expression becomes $(8 + 20) - 10$. Next, calculate the sum in the parentheses: $8 + 20 = 28$. Finally, perform the subtraction: $28 - 10 = 18$.
5. What is the value of $100 - 60 \div 5 \times 2$?
   Answer: 76 According to the order of operations, multiplication and division are performed from left to right before subtraction. First, divide: $60 \div 5 = 12$. The expression becomes $100 - 12 \times 2$. Next, multiply: $12 \times 2 = 24$. Finally, subtract: $100 - 24 = 76$.
6. Find: $6 + 2 \times (5^2 - 20) \div 2 - 3$
   Answer: 8 When you encounter a complex expression like this, you're being tested on the order of operations (PEMDAS/BODMAS). You must work through operations in the correct sequence: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Let's solve $6 + 2 \times (5^2 - 20) \div 2 - 3$ step by step: First, handle what's inside the parentheses: $5^2 - 20$. Calculate the exponent: $5^2 = 25$, so $25 - 20 = 5$. Now you have: $6 + 2 \times 5 \div 2 - 3$ Next, work left to right with multiplication and division: $2 \times 5 = 10$, then $10 \div 2 = 5$. This gives you: $6 + 5 - 3$ Finally, work left to right with addition and subtraction: $6 + 5 = 11$, then $11 - 3 = 8$. Choice A (14) likely comes from adding all numbers without following proper order: $6 + 2 + 5 + 20 + 2 - 3$. Choice B (5) might result from incorrectly calculating the parentheses or mixing up the final arithmetic. Choice C (11) occurs if you forget the final subtraction of 3, stopping at $6 + 5$. The correct answer is D (8). Study tip: Write out each step when practicing order of operations. Don't try to do multiple steps mentally at once—this is where most errors occur. Always complete operations inside parentheses first, then work systematically through PEMDAS.
7. What is the total cost to paint a rectangular wall that is 10 feet long and 8 feet high?
   Answer: $210 First, calculate the area of the wall in square feet: $10 \times 8 = 80$ square feet. Next, calculate the cost based on the area: $2 \times 80 = 160$. Finally, add the flat fee to this cost: $50 + 160 = 210$. The total cost is $210.
8. What is the value of $60 \div (10 + 2) + 4 \times 3$?
   Answer: 17 First, evaluate the expression in the parentheses: $10 + 2 = 12$. The expression becomes $60 \div 12 + 4 \times 3$. Next, perform division and multiplication from left to right. $60 \div 12 = 5$ and $4 \times 3 = 12$. The expression is now $5 + 12$. Finally, perform the addition: $5 + 12 = 17$.
9. Calculate: $4^2 - 3 \times 2 + 20 \div (2 + 3)$
   Answer: 14 Following order of operations: First, evaluate parentheses: $(2 + 3) = 5$. Then evaluate the exponent: $4^2 = 16$. Next, perform multiplication and division from left to right: $3 \times 2 = 6$ and $20 \div 5 = 4$. Finally, perform addition and subtraction from left to right: $16 - 6 + 4 = 10 + 4 = 14$. Choice A (12) results from incorrectly calculating $16 - 6 + 2 = 12$. Choice B (8) results from incorrectly calculating $16 - 6 - 2 = 8$. Choice D (16) results from only calculating the exponent and ignoring the rest of the expression.
10. What is the value of $2 \times 3^2 - 15 \div 3 + 8$?
    Answer: 21 When you encounter an expression with multiple operations like this one, you need to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (left to right), then Addition and Subtraction (left to right). Let's work through $2 \times 3^2 - 15 \div 3 + 8$ step by step: First, handle the exponent: $3^2 = 9$ The expression becomes: $2 \times 9 - 15 \div 3 + 8$ Next, perform multiplication and division from left to right: - $2 \times 9 = 18$ - $15 \div 3 = 5$ The expression becomes: $18 - 5 + 8$ Finally, perform addition and subtraction from left to right: - $18 - 5 = 13$ - $13 + 8 = 21$ The answer is 21, which is choice D. Choice A (31) likely results from adding all numbers without following order of operations: $2 + 9 + 15 + 3 + 8 = 37$, or from similar calculation errors. Choice B (25) might come from incorrectly calculating $2 \times 3^2$ as $6^2 = 36$, then $36 - 15 + 3 + 8 = 32$, or other order mistakes. Choice C (19) could result from computing $18 - 15 + 3 + 8 = 14$ by forgetting to divide 15 by 3. Remember: always follow PEMDAS strictly. Write out each step to avoid rushing through the operations, especially when exponents and multiple operations are involved. This systematic approach prevents the calculation errors that create these wrong answer choices.
11. What is the total cost to ship 5 packages?
    Answer: $22 The shipment consists of 1 first package and $5 - 1 = 4$ additional packages. The cost of the additional packages is $4 \times 3 = 12$. The total cost is the sum of the first package fee and the additional package cost: $10 + 12 = 22$. The total cost is $22.
12. What is the value of $7 + (5-2) \times 4^2 \div 8$?
    Answer: 13 First, evaluate the parentheses: $5-2 = 3$. The expression becomes $7 + 3 \times 4^2 \div 8$. Next, evaluate the exponent: $4^2 = 16$. The expression is now $7 + 3 \times 16 \div 8$. Perform multiplication and division from left to right: $3 \times 16 = 48$, then $48 \div 8 = 6$. Finally, perform the addition: $7 + 6 = 13$.
13. What is the value of $3^2 \times 4 + 18 \div (2 + 1) - 7$?
    Answer: 35 Following order of operations: First, evaluate parentheses: $(2 + 1) = 3$. Then evaluate the exponent: $3^2 = 9$. Next, perform multiplication and division from left to right: $9 \times 4 = 36$ and $18 \div 3 = 6$. Finally, perform addition and subtraction from left to right: $36 + 6 - 7 = 42 - 7 = 35$. Choice B (41) results from calculation errors. Choice C (29) results from incorrectly subtracting instead of adding. Choice D (47) results from adding all terms without proper operations.
14. What is the result when 120 is divided by the product of 3 and 5?
    Answer: 8 The phrase 'the product of 3 and 5' implies that this calculation should be done first, as if it were in parentheses. The expression is $120 \div (3 \times 5)$. First, calculate the product: $3 \times 5 = 15$. Then, perform the division: $120 \div 15 = 8$.
15. What is the value of the expression $50 - 4 \times 2 + 6$?
    Answer: 48 Following the order of operations (PEMDAS/BODMAS), multiplication is performed before addition and subtraction. First, calculate $4 \times 2 = 8$. The expression becomes $50 - 8 + 6$. Then, perform subtraction and addition from left to right: $50 - 8 = 42$, and $42 + 6 = 48$.
16. What is the value of $2 \times (5 + 3^2) - 4$?
    Answer: 24 First, evaluate inside the parentheses. Within the parentheses, the exponent is calculated first: $3^2 = 9$. The expression becomes $2 \times (5 + 9) - 4$. Next, finish the parentheses: $5 + 9 = 14$. The expression is now $2 \times 14 - 4$. Perform multiplication: $2 \times 14 = 28$. Finally, perform subtraction: $28 - 4 = 24$.
17. What is the value of $3 + 2 \times (10 - (4 + 3))$?
    Answer: 9 The order of operations requires starting with the innermost parentheses. First, calculate $4 + 3 = 7$. The expression becomes $3 + 2 \times (10 - 7)$. Next, calculate the remaining parentheses: $10 - 7 = 3$. The expression is now $3 + 2 \times 3$. Perform multiplication: $2 \times 3 = 6$. Finally, add: $3 + 6 = 9$.
18. How many more cups of flour does the baker need?
    Answer: 13 First, calculate the total amount of flour needed for all batches. This is $3 \text{ cups/batch} \times 5 \text{ batches} = 15$ cups. The baker already has 2 cups, so the amount she still needs is found by subtraction: $15 - 2 = 13$ cups. The expression is $3 \times 5 - 2$.
19. What is $36 \div 4 \times 3 + 2^3 - 6$?
    Answer: 29 Following order of operations: First, evaluate the exponent: $2^3 = 8$. Then perform multiplication and division from left to right: $36 \div 4 = 9$, then $9 \times 3 = 27$. Finally, perform addition and subtraction from left to right: $27 + 8 - 6 = 35 - 6 = 29$. Choice A (23) results from incorrectly calculating $27 - 8 + 4 = 23$. Choice C (35) results from forgetting to subtract the final 6: $27 + 8 = 35$. Choice D (19) results from incorrectly calculating $27 - 8 = 19$.
20. What is the value of $3 \times (1 + 4)^2$?
    Answer: 75 First, evaluate the expression inside the parentheses: $1 + 4 = 5$. The expression becomes $3 \times 5^2$. Next, evaluate the exponent: $5^2 = 25$. Finally, perform the multiplication: $3 \times 25 = 75$.

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