How to add, subtract, multiply and divide fractions, decimals and negatives
TEAS 7 outline: Perform arithmetic with rational numbers
Also covers HESI A2 Math (Whole numbers, Decimals, Fractions)
- Add or subtract fractions only after finding a common denominator.
- To divide fractions, multiply by the reciprocal of the second fraction.
- Change mixed numbers to improper fractions before multiplying or dividing.
- Same signs give a positive product; different signs give a negative one.
- Multiplication and division share a step and go left to right.
Key facts
- To add or subtract fractions, rewrite them with a common denominator (ideally the least common denominator), then add or subtract only the numerators.
- To multiply fractions, multiply the tops and multiply the bottoms, then simplify. To divide, multiply by the reciprocal (flipped version) of the second fraction.
- Change mixed numbers to improper fractions before you multiply or divide them (2 1/3 = 7/3).
Show all 7 key factsShow fewer
- Line up the decimal points to add or subtract decimals; when multiplying, the answer has as many decimal places as the two factors combined.
- Subtracting a number is the same as adding its opposite, so 5 − (−3) = 5 + 3 = 8.
- When multiplying or dividing, same signs give a positive answer and different signs give a negative answer.
- Order of operations: grouping symbols, exponents, then multiplication and division left to right, then addition and subtraction left to right.
Quick check1 question · 2 more at the end
The temperature in a mountain town was −7°F at 6 a.m. and 12°F at noon. By how many degrees did the temperature rise?
Correct answer: A.
The rise is the later temperature minus the earlier one: 12 − (−7). Subtracting a negative is the same as adding its opposite, so 12 − (−7) = 12 + 7 = 19. On a thermometer, it is 7 degrees from −7 up to 0 and 12 more degrees up to 12, for 19 degrees in all.
A Correct: 12 − (−7) = 12 + 7 = 19 degrees.
B This subtracts in the wrong order, −7 − 12, which describes a drop rather than a rise.
C This computes 12 − 7, treating −7 as if it were positive 7.
D This computes −12 + 7, putting the negative sign on the wrong temperature; a rise is never negative.
Full lesson
Each part starts with its main idea. Open a part when you want the details.
The short answer: Add or subtract fractions only after giving them a common denominator. Multiply fractions straight across, and divide by multiplying by the reciprocal. Change mixed numbers to improper fractions before multiplying or dividing. Line up decimal points to add or subtract. Two like signs give a positive product; unlike signs give a negative one. Follow the order of operations.
1 Which rule goes with which kind of number?Each kind of number has one short rule, and a cheat sheet covers them all. 1 min
Each kind of number has one short rule, and a cheat sheet covers them all. A rational number is any number you can write as a fraction: whole numbers, fractions, mixed numbers, most decimals, and their negatives. Find your operation below, then follow the worked example.
| Operation | Fractions | Decimals |
|---|---|---|
| Add, subtract | Get a common denominator, combine the tops | Line up the decimal points |
| Multiply | Tops × tops, bottoms × bottoms, simplify | Count total decimal places in the answer |
| Divide | Multiply by the reciprocal (flip the second fraction) | Move both points until the divisor is whole |
2 How do I add and subtract mixed numbers?Give the fractions a common denominator first, then borrow a whole if the top fraction is too small. 1 min
Give the fractions a common denominator first, then borrow a whole if the top fraction is too small. The least common denominator (LCD) is the smallest number both denominators divide into.
Example 1. A bag holds 5 1/4 cups of rice. A recipe uses 2 2/3 cups. How much is left?
- LCD of 4 and 3 is 12: 5 1/4 = 5 3/12 and 2 2/3 = 2 8/12.
- 3/12 is smaller than 8/12, so borrow 1 whole (12/12): 5 3/12 = 4 15/12.
- Subtract fractions: 15/12 − 8/12 = 7/12. Subtract wholes: 4 − 2 = 2.
- Answer: 2 7/12 cups.
Classic wrong turn: flipping the fractions to subtract 3/12 from 8/12 and getting 3 5/12.
3 How do I multiply and divide fractions and mixed numbers?Turn every mixed number into an improper fraction before you multiply or divide. 1 min
Turn every mixed number into an improper fraction before you multiply or divide. Multiply the whole number by the denominator and add the numerator: 2 1/3 = (2 × 3 + 1)/3 = 7/3. To divide, keep the first fraction, change ÷ to ×, and flip the second.
Example 2. A bottle holds 22 1/2 teaspoons of liquid vitamin. One serving is 3/4 teaspoon. How many servings are in the bottle?
- Change to an improper fraction: 22 1/2 = 45/2.
- Divide by multiplying by the reciprocal: 45/2 × 4/3.
- Multiply: 180/6 = 30 servings.
Classic wrong turn: multiplying 22 1/2 × 3/4 and getting about 17. Each serving is less than a teaspoon, so there must be more servings than teaspoons.
4 How do negative numbers change the rules?Negatives follow a few sign rules, and the arithmetic itself stays the same. 1 min
Negatives follow a few sign rules, and the arithmetic itself stays the same.
| Situation | Rule | Example |
|---|---|---|
| Add, same signs | Add, keep the sign | −3 + (−5) = −8 |
| Add, different signs | Subtract the sizes, keep the sign of the number farther from 0 | −9 + 4 = −5 |
| Subtract | Add the opposite | 2 − (−6) = 2 + 6 = 8 |
| Multiply or divide | Same signs → positive, different → negative | (−6) × (−2) = 12 |
5 What order do I do the steps in?Always work in the order of operations, never just left to right. 1 min
Always work in the order of operations, never just left to right. Do grouping symbols first, then exponents. Next do multiplication and division together from left to right, then addition and subtraction together from left to right. Watch exponents: −4² means −(4 × 4) = −16, while (−4)² = 16.
Example 3. A lab freezer reads −18°C. It warms 2.5°C per hour for 4 hours. What does it read?
- Write it out: −18 + 2.5 × 4.
- Multiply first: 2.5 × 4 = 10.
- Add: −18 + 10 = −8°C.
Classic wrong turn: adding first, (−18 + 2.5) × 4 = −62. A freezer that warms up cannot end colder.
On the TEAS 7's basic calculator, enter one step at a time and write each result down. Fraction steps like finding an LCD are usually faster by hand.
Common mistakes
The traps students fall into most. Read these before you practice.
Watch out: Add fractions with a common denominator, not by adding tops and bottoms (1/2 + 1/3 is 5/6, not 2/5).
Watch out: Multiply mixed numbers as improper fractions, not part by part (1 1/2 × 2 1/2 is 3 3/4, not 2 1/4).
Watch out: Multiplication does not always come before division: they share a step and go left to right (12 ÷ 3 × 2 is 8, not 2).
Watch out: A negative sign in front of an exponent is not squared unless it is in parentheses: −4² is −16, but (−4)² is 16.
Watch out: When borrowing in mixed-number subtraction, 1 whole becomes the denominator over itself (like 12/12), not 10.
How to remember it
The one rule to remember is to make the numbers the same kind before you combine them. Fractions need a common denominator, mixed numbers become improper fractions, and decimals line up.
Self-check:
- Find 3 1/3 − 1 3/4 using borrowing.
- Divide 2/5 by 1/10 and explain why the answer is bigger than 2/5.
- Simplify 10 − 2 × (−3) and say which step comes first.
Show answers
- 1 7/12.
- LCD of 3 and 4 is 12: 3 1/3 = 3 4/12 and 1 3/4 = 1 9/12.
- 4/12 is smaller than 9/12, so borrow 1 whole (12/12): 3 4/12 = 2 16/12.
- Subtract fractions: 16/12 − 9/12 = 7/12. Subtract wholes: 2 − 1 = 1.
- Answer: 1 7/12.
- 4.
- Keep 2/5, change ÷ to ×, and flip 1/10: 2/5 × 10/1.
- Multiply: 20/5 = 4.
- Why bigger: you are counting how many tenths fit into 2/5. Since 2/5 = 4/10, four tenths fit.
- 16. Multiply first.
- Multiplication comes before subtraction: 2 × (−3) = −6.
- Subtract by adding the opposite: 10 − (−6) = 10 + 6 = 16.