1.Order of Operations: A Rule Even Calculators Follow
What is 8 − 2 × 3? If you go from left to right, you get 6 × 3 = 18, but the agreed answer is 2. That is because we have agreed to do multiplication and division before addition and subtraction. Without this rule, different people would read the same expression in different ways.
You can remember the order in three steps. First, calculate inside the parentheses. Second, do multiplication and division from left to right. Third, do addition and subtraction from left to right. A point people often miss is that multiplication does not outrank division, or vice versa. 12 ÷ 3 × 2 is worked from the left as 4 × 2 = 8, not 12 ÷ 6 = 2.
- Step 1: Parentheses first. 5 − 2 = 3, so the expression becomes 20 − 4 × 3 + 6 ÷ 3.
- Step 2: Multiplication and division from the left. 4 × 3 = 12 and 6 ÷ 3 = 2. The expression is now 20 − 12 + 2.
- Step 3: Addition and subtraction from the left. 20 − 12 = 8, then 8 + 2 = 10.
- Check: Undo the addition and subtraction in reverse. 10 − 2 = 8 and 8 + 12 = 20 gives back the starting number, so the calculation was done in the right order.
2.Adding and Subtracting Fractions: Common Denominators
A fraction is "some number of pieces of a whole cut into equal parts." 3/4 means 3 of the 4 pieces a whole is cut into. That is why you cannot add pieces of different sizes directly. To add a piece from something cut into quarters to a piece from something cut into thirds, you first have to recut them into pieces of the same size. That is what finding a common denominator does.
It is easiest to use the least common multiple of the two denominators as the common denominator. The least common multiple of 4 and 3 is 12, so 1/4 becomes 3/12 and 2/3 becomes 8/12. Multiplying the top and bottom of a fraction by the same number does not change its size. The most common mistake is adding tops together and bottoms together, as in 1/4 + 2/3 = 3/7. But 3/7 is about 0.43, which is even smaller than 2/3 (about 0.67), so it makes no sense.
- Step 1: Find the least common multiple of 6 and 8. Among the multiples of 6 (6, 12, 18, 24), the first one that is also divisible by 8 is 24.
- Step 2: Rewrite with the common denominator: 5/6 = (5×4)/(6×4) = 20/24 and 3/8 = (3×3)/(8×3) = 9/24.
- Step 3: Subtract the numerators: 20/24 − 9/24 = 11/24. The only common factor of 11 and 24 is 1, so it cannot be simplified further.
- Check: Add it back. 11/24 + 9/24 = 20/24 = 5/6. Checking with decimals too: 5/6 ≈ 0.833 and 3/8 = 0.375, so the difference is about 0.458, and 11/24 ≈ 0.458 matches.
3.Multiplying and Dividing Fractions: Why Flip and Multiply?
To multiply fractions, multiply the numerators together and the denominators together. 2/3 × 3/4 = 6/12 = 1/2. If you read it as "2/3 of 3/4," it also makes sense that the result is smaller than what you started with.
Division asks "how many times does the divisor fit?" 2 ÷ 1/4 asks how many times 1/4 fits into 2. Since 1/4 goes into 1 exactly 4 times, the answer is 8. In other words, dividing by 1/4 is the same as multiplying by 4. The general version of this is the rule "swap the numerator and denominator of the fraction you divide by (its reciprocal) and multiply."
Also remember that dividing by a number smaller than 1 makes the result bigger. The intuition that "dividing makes things smaller" is only true when you divide by a number greater than 1.
- Step 1: Set up the expression. The question is how many times 1/6 fits into 2/3, so it is 2/3 ÷ 1/6.
- Step 2: Multiply by the reciprocal: 2/3 × 6/1 = 12/3.
- Step 3: Simplify: 12/3 = 4.
- Check: 4 cups × 1/6 L = 4/6 L = 2/3 L, the same as the starting amount.
- Step 1: The reciprocal of the divisor 2/5 is 5/2.
- Step 2: 3/4 × 5/2 = 15/8.
- Step 3: 15 ÷ 8 = 1 remainder 7, so 15/8 = 1 7/8.
- Check: See whether quotient × divisor = the number being divided. 15/8 × 2/5 = 30/40 = 3/4. Correct.
4.Converting Between Decimals and Fractions
A decimal is a short way of writing a fraction whose denominator is 10, 100, 1000, and so on. 0.375 is 375/1000, and dividing the top and bottom by their common factor 125 gives 3/8. Going the other way, to turn a fraction into a decimal, divide the numerator by the denominator: 3/8 = 3 ÷ 8 = 0.375.
Not every fraction becomes a tidy decimal. 1/3 = 0.333… becomes a repeating decimal, where the same digit repeats forever. Reduce the fraction to lowest terms and factor the denominator into primes: if the only prime factors are 2 and 5, it is a terminating decimal; if there is any other prime factor, it repeats. Since 8 = 2×2×2, 3/8 terminates, while 1/3, with a 3 in it, repeats.
So writing 0.33 for 1/3 introduces an error. Get into the habit of marking approximations like this with the ≈ sign: 1/3 ≈ 0.33. Computers also work in binary internally and cannot store a number like 0.1 exactly, which is why 0.1 + 0.2 sometimes shows up as 0.30000000000000004.
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333… (≈ 0.33) | about 33.3% |
- Step 1: There are two digits after the decimal point, so 0.65 = 65/100.
- Step 2: The greatest common factor of 65 and 100 is 5. 65 = 5×13 and 100 = 5×20.
- Step 3: Dividing the top and bottom by 5 gives 13/20.
- Check: 13 ÷ 20 = 0.65, the same as the original decimal.
5.Common Mistakes and How to Check
The best check is estimation. Before calculating, say roughly what the answer should be. 1/4 + 2/3 adds about 0.25 and 0.67, so it should be a little less than 1. If you get 11/12 ≈ 0.92, you can relax. If you estimate an AI's calculation the same way first, badly wrong answers get filtered out right away.
- Adding tops together and bottoms together: 1/2 + 1/3 ≠ 2/5. With a common denominator, it is 3/6 + 2/6 = 5/6.
- Ranking multiplication above division (or the reverse): they have the same priority, so work from the left.
- Flipping the first number in a division: the one you turn into its reciprocal is the divisor (the second number).
- Treating an approximation as exact: 0.33 is not 1/3, only an approximation of 1/3.
📌 Key points
- Calculate in this order: parentheses → multiplication and division (from the left) → addition and subtraction (from the left)
- For fractions with different denominators, rewrite them over the least common multiple, then add or subtract the numerators
- To divide by a fraction, multiply by the reciprocal of the divisor; division asks "how many times does it fit?"
- If the denominator of a fraction in lowest terms has only 2s and 5s as prime factors, it is a terminating decimal; any other prime factor makes it repeat
- Estimate before you calculate, and check afterward by working backward
🔁 Unlimited practice
Problems are generated endlessly. Type your answer and press "Check" to have it graded right away, or press "Show solution" to see a step-by-step solution in the same order as the lesson. You can choose the difficulty, and your streak of correct answers is counted.
- Adding and subtracting fractions (common denominators and simplifying)
- Multiplying and dividing fractions (multiplying by the reciprocal)
- Fractions to decimals, and decimals to fractions in lowest terms
- Hard: order of operations in expressions with multiplication
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Answer
Type whole numbers, decimals or fractions (e.g. 12, 0.75, 3/4, −2/3, 1 3/4). A fraction answer also counts as a decimal correct to three places. Your streak resets if you look at the solution first or get one wrong; only your best streak is saved, in this browser.
🤖 Try asking AI like this
Copy a prompt and replace the [ ] parts with your own situation. Don't take the answer on trust — check it against this lesson.
When you want to check whether your fraction solution is right
I'll show you my solution. Don't tell me the answer right away. Instead, mark whether each step is correct, one by one. If a step is wrong, point out only that step and explain why it's wrong. Problem: 5/6 − 3/8. My solution: [paste your solution]
When you need more practice problems
Make 8 problems on adding and subtracting fractions with different denominators and on dividing fractions, from easy to hard. Put the answers and solutions together at the very bottom under the heading "Answers." I'll look at them after I've solved everything.
Before you trust a calculation an AI has done
Check the calculation you just did one more time by converting it to decimals. Also show me a rough estimate of what the answer should be before calculating.
- Standard elementary and middle school math textbook content (fractions and decimals, rational numbers)
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Storage is unavailable in this browser, so this lasts only for this page.📐 Math Basics
- 1Numbers and Operations: Fractions and Decimals Revisited
- 2Ratios and Rates: Reading Percentages Correctly
- 3Equations: A Balance for Finding Unknown Numbers
- 4Functions and Graphs: An Eye for Change
- 5Exponents and Logarithms: A World That Grows by Multiplying
- 6Geometry Basics: Area and Pythagoras
- 7Probability: Putting Numbers on Uncertainty
- 8Statistics: Mean, Median, and Variance
- 9Reading Data: The Traps in Graphs
- 10The Math for Understanding AI