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Difficulty: MediumOrder of Operations and Number Properties

Evaluate each of the following mathematical expressions using the standard order of operations. Arrange the expressions in order from least to greatest value.

  1. 12−3×4−(6+5)2 - 3 \times 4 - (6 + 5)
  2. 2(2−3)×4−(6+5)(2 - 3) \times 4 - (6 + 5)
  3. 32−(3×4−6)+52 - (3 \times 4 - 6) + 5
  4. 42−3×(4−6)+52 - 3 \times (4 - 6) + 5

Answer

The correct order of the expressions from least to greatest value is: 2−3×4−(6+5)2 - 3 \times 4 - (6 + 5) (evaluates to −21-21), (2−3)×4−(6+5)(2 - 3) \times 4 - (6 + 5) (evaluates to −15-15), 2−(3×4−6)+52 - (3 \times 4 - 6) + 5 (evaluates to 11), and 2−3×(4−6)+52 - 3 \times (4 - 6) + 5 (evaluates to 1313).
Evaluating the four expressions using the correct order of operations (parentheses first, then multiplication, and finally addition/subtraction from left to right) gives the values −21-21, −15-15, 11, and 1313. Arranged from least to greatest, the correct order is 2−3×4−(6+5)2 - 3 \times 4 - (6 + 5) followed by (2−3)×4−(6+5)(2 - 3) \times 4 - (6 + 5), then 2−(3×4−6)+52 - (3 \times 4 - 6) + 5, and finally 2−3×(4−6)+52 - 3 \times (4 - 6) + 5.

Step-by-Step Solution

1
Evaluate the expression 2−3×4−(6+5)2 - 3 \times 4 - (6 + 5).
−21-21
First evaluate the operations in parentheses: 6+5=116 + 5 = 11. Then perform multiplication: 3×4=123 \times 4 = 12. The expression is now 2−12−112 - 12 - 11. Perform subtraction from left to right: 2−12=−102 - 12 = -10, and −10−11=−21-10 - 11 = -21.
2
Evaluate the expression (2−3)×4−(6+5)(2 - 3) \times 4 - (6 + 5).
−15-15
First evaluate the operations in parentheses: 2−3=−12 - 3 = -1 and 6+5=116 + 5 = 11. The expression is now −1×4−11-1 \times 4 - 11. Perform multiplication: −1×4=−4-1 \times 4 = -4. Finally, perform subtraction: −4−11=−15-4 - 11 = -15.
3
Evaluate the expression 2−(3×4−6)+52 - (3 \times 4 - 6) + 5.
11
First evaluate inside the parentheses, performing multiplication before subtraction: 3×4=123 \times 4 = 12, and 12−6=612 - 6 = 6. The expression is now 2−6+52 - 6 + 5. Perform subtraction and addition from left to right: 2−6=−42 - 6 = -4, and −4+5=1-4 + 5 = 1.
4
Evaluate the expression 2−3×(4−6)+52 - 3 \times (4 - 6) + 5.
1313
First evaluate the subtraction in parentheses: 4−6=−24 - 6 = -2. The expression is now 2−3×(−2)+52 - 3 \times (-2) + 5. Next, perform multiplication: −3×(−2)=6-3 \times (-2) = 6. Finally, perform addition from left to right: 2+6+5=132 + 6 + 5 = 13.
5
Order the evaluated results from least to greatest.
−21<−15<1<13-21 < -15 < 1 < 13
Comparing the values: −21-21 is the smallest value, followed by −15-15, then 11, and 1313 is the largest value.

Key Concept

Evaluating arithmetic expressions containing multiple operations using PEMDAS order of operations.
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Topics

Order of Operations and Number Properties

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