GED Math: Order of Operations
Word problems often ask you to choose an expression or to say which operation is done last. To write an expression, build it in pieces and use parentheses so that the operations happen in the intended order. For example, “a 10% discount on the total of 3 shirts at $18 and 2 pants at $35” needs parentheses around the total before the discount is applied: 0.9(3 × 18 + 2 × 35). To find the last operation, follow the order of operations: addition and subtraction happen last (left to right) unless a grouping symbol or fraction bar encloses them. The last operation is the final step that combines the whole expression.
1. The Order of Operations
When an expression has more than one operation, the answer can change depending on the order you work in, so mathematicians agree on one order. 1. Grouping symbols: do operations inside parentheses, brackets, and braces first. 2. Exponents and roots. 3. Multiplication and division, from left to right as they appear. 4. Addition and subtraction, from left to right as they appear. Multiplication and division are equal partners, as are addition and subtraction, so 18 ÷ 3 × 2 is (18 ÷ 3) × 2 = 12, not 18 ÷ 6 = 3. Many students remember the steps with a phrase, but what matters is the rule: multiply and divide before you add and subtract, and go left to right within each level.
Worked Example: Find 2 + 62 ÷ 4 × 3 − 5.
Exponent first: 62 = 36, so 2 + 36 ÷ 4 × 3 − 5. Multiply and divide left to right: 36 ÷ 4 = 9, then 9 × 3 = 27, so 2 + 27 − 5. Add and subtract left to right: 2 + 27 = 29, then 29 − 5 = 24.
1. Find 3 + 4 × 5.
Explanation: Multiply first: 4 × 5 = 20. Then 3 + 20 = 23. (35 results from adding first.)
2. Find 20 − 12 ÷ 4.
Explanation: Divide first: 12 ÷ 4 = 3. Then 20 − 3 = 17.
3. Find 18 ÷ 3 × 2.
Explanation: Work left to right: 18 ÷ 3 = 6, then 6 × 2 = 12.
4. Find 100 ÷ 10 ÷ 2.
Explanation: Left to right: 100 ÷ 10 = 10, then 10 ÷ 2 = 5.
5. Find 2 + 62 ÷ 4.
Explanation: Exponent: 36. Divide: 36 ÷ 4 = 9. Add: 2 + 9 = 11.
6. Find 5 × 3 − 2 × 4.
Explanation: Multiply both products first: 15 − 8 = 7.
2. Grouping Symbols and Nested Groups
Grouping symbols tell you what to do first. Parentheses ( ), brackets [ ], and braces { } all mean “simplify what is inside before anything else.” Writing a number right next to a group, as in 4(3), means multiply. When one set of symbols is nested inside another, start with the innermost set and work outward. Inside each group, still follow the order of operations (for example, multiply before you add). Notice that parentheses can be present without enclosing an operation: in (−4)(6) nothing needs to be calculated inside them, so only the multiplication matters.
Worked Example: Find 3[2(−5 + 3) + 10].
1. Find 4(3) − 2 + (6 + 4 × 2).
Explanation: Inside the parentheses, multiply first: 4 × 2 = 8, so 6 + 8 = 14. Then 12 − 2 + 14 = 24.
2. Find 16 ÷ (10 − 6)2.
Explanation: Parentheses: 10 − 6 = 4. Exponent: 42 = 16. Divide: 16 ÷ 16 = 1.
3. Find 52 − (5 − 7)(2).
Explanation: 25 − (−2)(2) = 25 − (−4) = 25 + 4 = 29.
4. Find 3[2(−5 + 3) + 10].
Explanation: (−5 + 3) = −2; 2(−2) = −4; −4 + 10 = 6; 3(6) = 18.
5. Find 50 − 2[(7 − 4)2 + 1].
Explanation: Parentheses: 3. Exponent: 9. Brackets: 9 + 1 = 10. Multiply: 2 × 10 = 20. Subtract: 50 − 20 = 30.
6. Find 2{[(9 − 3) × 2] + 4}.
Explanation: Innermost: 9 − 3 = 6. Then 6 × 2 = 12. Then 12 + 4 = 16. Then 2 × 16 = 32.
3. The Fraction Bar as a Grouping Symbol
The fraction bar is a grouping symbol too. Everything in the numerator is one group and everything in the denominator is another group. Before you divide, simplify the numerator and the denominator separately (using the order of operations in each), and then divide the two results. The same idea applies when a fraction bar appears inside a longer expression: finish the fraction first, then continue with the rest. On a calculator, put the numerator and denominator in parentheses, or use the fraction key, so the division happens after both parts are done.
Worked Example: Find (5 + 3)2 ÷ 4 + 3(−1), written with a fraction bar: the numerator is (5 + 3)2 and the denominator is 4.
Parentheses: 5 + 3 = 8. Exponent: 82 = 64. Divide by the bar: 64 ÷ 4 = 16. Multiply: 3(−1) = −3. Add: 16 + (−3) = 13.
1. Find (15 + 25) ÷ (2 × 5) + 6, where the bar separates the two groups.
Explanation: Numerator: 40. Denominator: 10. 40 ÷ 10 = 4. Then 4 + 6 = 10.
2. Find 33 over (5 − 2) minus (4 − 2)2 over 2.
Explanation: First fraction: 27 ÷ 3 = 9. Second fraction: 4 ÷ 2 = 2. Then 9 − 2 = 7.
3. Find 25 over (4 + 1), plus 3(6 − 1).
Explanation: 25 ÷ 5 = 5, and 3(5) = 15. Then 5 + 15 = 20.
4. Find (62 − 4) over (2 + 6).
Explanation: Numerator: 36 − 4 = 32. Denominator: 8. 32 ÷ 8 = 4.
5. Find 48 over (3 + 5), minus 22.
Explanation: 48 ÷ 8 = 6, and 22 = 4. Then 6 − 4 = 2.
6. Which is the correct first step in simplifying (7 + 5) over (2 × 3) + 4?
Explanation: Simplify the numerator (12) and denominator (6) before dividing. Then 12 ÷ 6 = 2, and 2 + 4 = 6.
4. Exponents, Roots, and Negative Numbers in Expressions
Many GED expressions combine several skills. Exponents and roots come right after grouping symbols. A square root symbol acts like a grouping symbol: simplify what is under the radical first, so √(16 + 9) = √25 = 5, which is different from √16 + √9 = 7. When negative numbers appear, apply the sign rules at each step and keep parentheses around negative bases: (−3)2 = 9, but −32 = −9. Subtracting a negative becomes adding. Work in small steps, rewriting the expression after each operation so that nothing is lost.
Worked Example: Find (4 − 12)(−6) + (10 − 3), then 23 + (8 − 5)2 − 3.
First: (4 − 12) = −8, so (−8)(−6) = 48; (10 − 3) = 7; 48 + 7 = 55. Second: 23 = 8 and (8 − 5)2 = 32 = 9, so 8 + 9 − 3 = 14.
1. Find (4 − 12)(−6) + (10 − 3).
Explanation: (−8)(−6) = 48, and 48 + 7 = 55.
2. Find 23 + (8 − 5)2 − 3.
Explanation: 8 + 9 − 3 = 14.
3. Find (4 + 2)2 + (7 − 2)3.
Explanation: 62 = 36 and 53 = 125. 36 + 125 = 161.
4. Find √(16 + 9).
Explanation: Simplify under the radical first: √25 = 5. (7 is √16 + √9, a different expression.)
5. Find 72 ÷ (11 − 4) + (9 + 14).
Explanation: 49 ÷ 7 = 7, and 9 + 14 = 23. 7 + 23 = 30.
6. Find 15 + (4)(3) − 22.
Explanation: Multiply and take the exponent: 12 and 4. Then 15 + 12 − 4 = 23.
7. Find −32 + (−3)2.
Explanation: −32 = −9 (the exponent applies only to 3), and (−3)2 = 9. Sum = 0.
5. Order of Operations in Word Problems and Expressions
Word problems often ask you to choose an expression or to say which operation is done last. To write an expression, build it in pieces and use parentheses so that the operations happen in the intended order. For example, “a 10% discount on the total of 3 shirts at $18 and 2 pants at $35” needs parentheses around the total before the discount is applied: 0.9(3 × 18 + 2 × 35). To find the last operation, follow the order of operations: addition and subtraction happen last (left to right) unless a grouping symbol or fraction bar encloses them. The last operation is the final step that combines the whole expression.
Worked Example: A foundation pays 75% of the cost of infant care classes, which cost $30 per person. The hospital covers the rest for 28 parents. Write and evaluate an expression for what the hospital pays.
The hospital pays the other 25%, or (1 − 0.75), of 28 × $30. The expression is (1 − 0.75)(28)(30) = 0.25 × 840 = $210.
1. A foundation pays 75% of the cost of classes that cost $30 per person for 28 people. The hospital pays the rest. Which expression gives the hospital’s cost in dollars?
Explanation: The foundation covers 75%, so the hospital pays 1 − 0.75 = 25% of the total cost 28 × 30.
2. A store sells 3 shirts at $18 each and 2 pairs of pants at $35 each. Which expression finds the total after a 10% discount on everything?
Explanation: The discount must apply to the whole total, so put the total in parentheses and multiply by 0.9 (the 90% you pay).
3. What is the last operation to perform in 9 − 2(3 + 4) + 52?
Explanation: The order: parentheses (7), exponent (25), multiplication (14), then subtract (9 − 14 = −5), and finally add 25 to get 20.
4. Find 10 + 4[(12 − 3) ÷ (3(7 − 4))].
Explanation: 12 − 3 = 9, and 3(7 − 4) = 9, so 9 ÷ 9 = 1. Then 4(1) = 4 and 10 + 4 = 14.
5. To convert 25°C to Fahrenheit, use 9/5 × C + 32. What is the temperature?
Explanation: Multiply first: 9/5 × 25 = 45. Then 45 + 32 = 77.
6. Which expression equals 16?
Explanation: (2 + 6)(4 − 2) = 8 × 2 = 16. The others give 24, 14, and 24.
Common Mistakes to Avoid
- Working strictly left to right. Multiplication and division come before addition and subtraction, so 3 + 4 × 5 is 23, not 35.
- Treating multiplication as always before division (or addition before subtraction). Each pair is one level; go left to right.
- Not simplifying the numerator and denominator separately when a fraction bar is present.
- Doing a nested group from the outside in. Start with the innermost parentheses.
- Applying an exponent to the wrong thing: −32 = −9, but (−3)2 = 9.
- Dropping a negative sign when subtracting a negative or multiplying negatives. Rewrite each step.
- Taking the square root of each term separately. √(16 + 9) is √25, not √16 + √9.
- Forgetting parentheses when writing an expression for a word problem (for example, a discount on a total).
Frequently Asked Questions
What is the order of operations?
Do grouping symbols first, then exponents and roots, then multiplication and division from left to right, then addition and subtraction from left to right.
Why do I go left to right for multiplication and division?
Because they are at the same level. In 18 ÷ 3 × 2 you divide first (the leftmost operation) to get 6 and then multiply to get 12.
How do I handle nested parentheses?
Start with the innermost set and work outward, following the order of operations inside each set.
Is the fraction bar a grouping symbol?
Yes. Simplify everything above the bar and everything below the bar before dividing.
How do I use the GED calculator for these problems?
Use the parentheses keys so the calculator follows the same order: put each numerator and denominator in parentheses, and use the change-sign key for negative numbers.
How do I know which operation is last?
Find the lowest-level operation that is not inside a group: usually the final addition or subtraction. Operations inside parentheses or above/below a fraction bar happen earlier.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.