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Operations with Rational Numbers

Add, subtract, multiply and divide integers, fractions, mixed numbers and decimals — and use the order of operations to get one correct answer.

Practice this skill 30 min Add, subtract, multiply and divide rational numbers Leer en español
Key idea

What you'll learn

In plain words: you will learn to do the four basic operations with any kind of number, in the right order.

In this lesson you will learn to add, subtract, multiply and divide every kind of number the GED uses:

  • integers (whole numbers and their negatives, like -8 and 15),
  • fractions and mixed numbers (like \tfrac{3}{4} and 2\tfrac{1}{2}),
  • decimals (like 0.75, or prices such as $12.49 and $3.50),

and to combine them in the right order using the order of operations.

Why it matters on the GED: almost every math question uses these skills somewhere. Some questions test them directly. These often come in the short no-calculator part at the start of the test. Many more hide them in word problems about money, recipes or temperatures. Careful arithmetic stops the small slips that cost easy points.

In real life you use these skills to split a bill, track a bank balance, or double a recipe.

Words to know

  • Integer — a whole number or its negative: \dots, -2, -1, 0, 1, 2, \dots (for example -15 or 7, but not 2.5).
  • Rational number — any number you can write as a fraction of two integers, \tfrac{a}{b} with b \ne 0. Integers, fractions and mixed numbers are all rational. So are decimals that stop or repeat (e.g. 0.75 = \tfrac{3}{4}, -3 = \tfrac{-3}{1}).
  • Absolute value — the distance of a number from 0; it is never negative: \lvert -6 \rvert = 6.
  • Numerator / denominator — the top / bottom of a fraction: in \tfrac{3}{8}, the numerator is 3, the denominator is 8.
  • Common denominator — a shared bottom number used to add or subtract fractions: \tfrac{1}{4} and \tfrac{1}{6} can both be written in twelfths.
  • Least common denominator (LCD) — the smallest common denominator: for 4 and 6 it is 12.
  • Mixed number — a whole number plus a fraction: 2\tfrac{1}{2} = 2 + \tfrac{1}{2}.
  • Improper fraction — numerator bigger than or equal to the denominator: \tfrac{5}{2}.
  • Reciprocal — a fraction flipped upside down: the reciprocal of \tfrac{3}{4} is \tfrac{4}{3}; the reciprocal of 5 is \tfrac{1}{5}.
  • Order of operations — the agreed order for doing several operations. First parentheses, then exponents, then multiply/divide (left to right), then add/subtract (left to right).

All Math words in the glossary

1. Adding and subtracting integers

In plain words: think of money. A positive number is cash you have. A negative number is money you owe.

If you owe 8 dollars and earn 5 dollars, you still owe 3 dollars. That is -8 + 5 = -3. On a number line, adding a positive moves you right. Adding a negative moves you left.

Rule 1 — same signs: add the absolute values and keep the sign.

  • 4 + 9 = 13
  • -4 + (-9) = -13

Rule 2 — different signs: subtract the absolute values. Keep the sign of the number with the bigger absolute value.

  • -8 + 5: 8 - 5 = 3, and -8 is "bigger", so the answer is -3.
  • 12 + (-7): 12 - 7 = 5, and 12 is bigger, so the answer is 5.

Rule 3 — subtracting means adding the opposite. Change the minus to plus. Then flip the sign of the number after it ("keep, change, change"):

  • 6 - 10 = 6 + (-10) = -4
  • -3 - (-8) = -3 + 8 = 5 (two negatives next to each other become a plus)
  • -5 - 4 = -5 + (-4) = -9
Number line from -10 to 10. Starting at 0, an arrow moves 8 units left to -8; a second arrow moves 5 units right from -8 to -3, which is marked with a dot. This shows -8 + 5 = -3.−10−9−8−7−6−5−4−3−2−1012345678910−8+5−8 + 5 = −3

Multiplying and dividing signed numbers uses one short rule:

SignsResultExample
+ and +positive6 \times 3 = 18
- and -positive(-6)(-3) = 18
+ and -negative6 \times (-3) = -18
- and +negative-18 \div 3 = -6

Same signs → positive. Different signs → negative. With many factors, count the negatives. An even number of negatives gives a positive answer. An odd number gives a negative answer. Example: (-1)(-2)(-3) = -6 (three negatives → negative).

2. Fractions and mixed numbers

In plain words: you can only add pieces that are the same size.

Think of a pizza. A quarter slice and a third slice are different sizes. Cut both into twelfths first. Then you can count the pieces.

Adding and subtracting fractions — you need a common denominator.

  1. Find the LCD.
  2. Rewrite each fraction with that denominator (multiply top and bottom by the same number).
  3. Add or subtract the numerators. Keep the denominator.
  4. Simplify.

Example: \tfrac{3}{4} + \tfrac{2}{3} = \tfrac{9}{12} + \tfrac{8}{12} = \tfrac{17}{12} = 1\tfrac{5}{12}.

Fraction bars cut into twelfths. Three quarters is shaded as 9 twelfths (blue) and two thirds as 8 twelfths (red). Together they fill one whole bar (12 twelfths) plus 5 twelfths of a second bar, so 3/4 + 2/3 = 17/12 = 1 5/12.3/4 + 2/3 = 9/12 + 8/12 = 17/12 = 1 5/123/4 = 9/129 twelfths2/3 = 8/128 twelfthssum1 whole+ 5/12

Multiplying fractions — no common denominator needed. Multiply straight across, top × top and bottom × bottom. You may cancel common factors first:

\frac{2}{3} \times \frac{9}{10} = \frac{2 \times 9}{3 \times 10} = \frac{18}{30} = \frac{3}{5}

Dividing fractions — keep, change, flip: keep the first fraction, change \div to \times, flip the second fraction (use its reciprocal).

\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} = 1\tfrac{1}{4}

Mixed numbers.

  • To multiply or divide, first change each mixed number to an improper fraction: 2\tfrac{1}{3} = \tfrac{2 \times 3 + 1}{3} = \tfrac{7}{3}.
  • To add or subtract, you can work with whole parts and fraction parts separately. But you may have to borrow. Example: 5\tfrac{1}{4} - 2\tfrac{3}{4}. You can't take \tfrac{3}{4} from \tfrac{1}{4}, so borrow 1 from the 5: 5\tfrac{1}{4} = 4\tfrac{5}{4}. Then 4\tfrac{5}{4} - 2\tfrac{3}{4} = 2\tfrac{2}{4} = 2\tfrac{1}{2}. (You can also change both to improper fractions. This always works: \tfrac{21}{4} - \tfrac{11}{4} = \tfrac{10}{4} = 2\tfrac{1}{2}.)
Six bars, each cut into quarters. Five whole bars and one quarter are shaded, 21 quarters in all, showing 5 1/4. The last 11 shaded quarters are crossed out in red to take away 2 3/4. The 10 quarters left, two whole bars and two quarters, show 2 1/2.5 1/4 − 2 3/4 = 21/4 − 11/4 = 10/4 = 2 1/2123456left: 10 quarters = 2 1/2taken away: 11 quarters = 2 3/4Shaded: 5 wholes + 1 quarter = 21 quarters (one whole is "borrowed" as 4 quarters)

Negative fractions follow the same sign rules as integers: -\tfrac{1}{2} \times \tfrac{4}{5} = -\tfrac{2}{5}, and -\tfrac{3}{4} \div \left(-\tfrac{1}{2}\right) = \tfrac{3}{4} \times 2 = \tfrac{3}{2}.

3. Decimals

Add or subtract: line up the decimal points. Fill empty places with zeros.

12.6 + 3.45 = 12.60 + 3.45 = 16.05 \qquad 20 - 7.35 = 20.00 - 7.35 = 12.65

In plain words: decimals work like whole numbers. You just have to keep track of the point.

Multiply: first multiply as if there were no decimal points. Then count the decimal places in both numbers. Put that many places in the answer.

0.4 \times 0.03: 4 \times 3 = 12; there are 1 + 2 = 3 decimal places, so the answer is 0.012.

Divide by a decimal: move the point in the number you divide by (the divisor) until it is a whole number. Move the point in the other number (the dividend) the same number of places.

4.5 \div 0.15 = 450 \div 15 = 30

Sense check: dividing by a number less than 1 makes the answer bigger. How many half-dollars are in 10 dollars? Twenty: 10 \div 0.5 = 20.

Money is rounded to the cent: \$43.5834 \approx \$43.58.

Converting: fraction → decimal means divide top by bottom: \tfrac{3}{8} = 0.375.

4. The order of operations

In plain words: when a problem has several steps, everyone does them in the same order. That way there is only one right answer. Remember PEMDAS ("Please Excuse My Dear Aunt Sally"):

  1. Parentheses and other grouping symbols — brackets [\;], the bar of a fraction, a square-root sign. Work from the innermost group out.
  2. Exponents (and roots).
  3. Multiplication and Division — together, from left to right. Multiplication does not always come before division.
  4. Addition and Subtraction — together, from left to right.

Example: 18 - 4 \times 3 + 6 \div 2

  • Multiply and divide first: 4 \times 3 = 12 and 6 \div 2 = 3 → 18 - 12 + 3
  • Then left to right: 18 - 12 = 6, then 6 + 3 = 9.
Step diagram for 18 − 4 × 3 + 6 ÷ 2. First multiply and divide: 4 × 3 = 12 and 6 ÷ 2 = 3, giving 18 − 12 + 3. Then add and subtract from left to right: 18 − 12 = 6, giving 6 + 3, and finally 9.18 − 4 × 3 + 6 ÷ 2× and ÷ first: 4 × 3 = 12, 6 ÷ 2 = 318 − 12 + 3left to right: 18 − 12 = 66 + 3add: 6 + 3 = 99

The fraction bar is a grouping symbol. \dfrac{10 + 6}{2 - 6} means (10 + 6) \div (2 - 6) = 16 \div (-4) = -4.

Watch the negative sign with exponents. -3^2 = -(3 \times 3) = -9, but (-3)^2 = (-3)(-3) = 9. The exponent only touches what is directly under it.

Formula

Rules to memorize (NOT on the formula sheet)

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd} \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} a - b = a + (-b) \qquad a - (-b) = a + b

Here a, b, c, d are integers, no denominator is 0, and c \ne 0 when dividing.

Sign rule: same signs → positive; different signs → negative (for \times and \div).

Order of operations: grouping → exponents/roots → \times and \div left to right → + and - left to right.

None of these rules are on the GED formula sheet. You must know them by heart.

Worked example

Example 1 (easy): integers

Evaluate -7 + 12 - (-4).

Number line from −8 to 10. Start at −7, a blue arrow moves 12 to the right to 5, then a green arrow moves 4 more to the right to 9. This shows −7 + 12 − (−4) = −7 + 12 + 4 = 9.−7 + 12 − (−4) = −7 + 12 + 4 = 9−8−7−6−5−4−3−2−1012345678910+12+4startanswer
  1. Rewrite the subtraction as adding the opposite: -7 + 12 + 4.
  2. Left to right: -7 + 12 = 5 (different signs: 12 - 7 = 5, keep the sign of 12).
  3. 5 + 4 = 9.

Answer: 9

Check: On a number line, start at -7, move 12 right to 5, then 4 more right to 9. ✓

Worked example

Example 2 (word problem): dividing mixed numbers

A ribbon is 5\tfrac{1}{4} yards long. A crafter cuts it into pieces that are each \tfrac{3}{4} yard long. How many pieces does she get?

A ribbon 5 1/4 yards long drawn to scale over a ruler marked 0 to 5 yards. Dashed lines mark every quarter yard (21 quarters). Thick lines cut the ribbon into 7 numbered pieces, each 3/4 yard long, so 5 1/4 ÷ 3/4 = 7.5 1/4 yd ÷ 3/4 yd = 7 pieces12345670 yd1 yd2 yd3 yd4 yd5 ydwhole ribbon: 5 1/4 ydEach piece is 3/4 yd = 3 quarter-yards; 21 quarter-yards make 7 pieces.
  1. "How many pieces of size \tfrac{3}{4} fit into 5\tfrac{1}{4}" means divide: 5\tfrac{1}{4} \div \tfrac{3}{4}.
  2. Change the mixed number: 5\tfrac{1}{4} = \tfrac{5 \times 4 + 1}{4} = \tfrac{21}{4}.
  3. Keep, change, flip: \tfrac{21}{4} \times \tfrac{4}{3} = \tfrac{84}{12}.
  4. Simplify: \tfrac{84}{12} = 7.

Answer: 7 pieces

Check: 7 \times \tfrac{3}{4} = \tfrac{21}{4} = 5\tfrac{1}{4} yards. ✓

Worked example

Example 3 (word problem): decimals and money

Dana buys 3 notebooks at $2.35 each and a calculator for $14.99. She pays with a $25 bill. How much change does she get?

Bar model drawn to scale for 25 dollars paid. The bar is split into 3 notebooks for 7.05 dollars (blue), a calculator for 14.99 dollars (red) and the change of 2.96 dollars (green). A brace under the first two parts shows the total cost of 22.04 dollars, so the change is 25.00 dollars − 22.04 dollars = 2.96 dollars.Paid $25.00 = $7.05 + $14.99 + change$25.00 paid3 notebooks$7.05calculator$14.99$2.96changetotal cost: $7.05 + $14.99 = $22.04Change = $25.00 − $22.04 = $2.96
  1. Cost of notebooks: 3 \times 2.35 = 7.05.
  2. Total cost: 7.05 + 14.99 = 22.04.
  3. Change: 25.00 - 22.04 = 2.96.

Answer: $2.96

Check: Add the change back: 22.04 + 2.96 = 25.00 ✓.

Worked example

Example 4 (GED-hard): order of operations with negatives

Evaluate 4 - 3(2 - 5)^2 \div 9.

Step diagram for 4 − 3(2 − 5)² ÷ 9. Parentheses: 2 − 5 = −3, giving 4 − 3(−3)² ÷ 9. Exponent: (−3)² = 9, giving 4 − 3 · 9 ÷ 9. Multiply: 3 · 9 = 27, giving 4 − 27 ÷ 9. Divide: 27 ÷ 9 = 3, giving 4 − 3. Subtract: the answer is 1.4 − 3(2 − 5)² ÷ 9Parentheses: 2 − 5 = −34 − 3(−3)² ÷ 9Exponent: (−3)² = (−3)(−3) = 94 − 3 · 9 ÷ 9Multiply (left to right): 3 · 9 = 274 − 27 ÷ 9Divide: 27 ÷ 9 = 34 − 3Subtract: 4 − 3 = 11
  1. Parentheses: 2 - 5 = -3. Now 4 - 3(-3)^2 \div 9.
  2. Exponent: (-3)^2 = (-3)(-3) = 9. Now 4 - 3 \cdot 9 \div 9.
  3. Multiply and divide, left to right: 3 \cdot 9 = 27, then 27 \div 9 = 3. Now 4 - 3.
  4. Subtract: 4 - 3 = 1.

Answer: 1

Check: Typing 4 − 3 ( 2 − 5 ) x² ÷ 9 into the calculator also gives 1 ✓. (Treating (-3)^2 as -9 would wrongly give 7.)

Worked example

Example 5 (work backwards): find the missing number

What number, multiplied by -\tfrac{3}{4}, gives 6?

Number line from −10 to 8 with points at −8 (blue, labelled n) and 6 (red). A red arrow from −8 to 6 is labelled times negative three fourths: the sign flips and the size shrinks to 3/4 of 8, which is 6. A dashed blue arrow back from 6 to −8 shows the undo step, 6 ÷ (−3/4) = 6 × (−4/3) = −8.n × (−3/4) = 6 so n = 6 ÷ (−3/4) = −8−10−8−6−4−202468n× (−3/4): sign flips, size shrinks to 3/4 of 8undo: 6 ÷ (−3/4) = 6 × (−4/3) = −8
  1. Write it as an equation: n \times \left(-\tfrac{3}{4}\right) = 6.
  2. Undo multiplication by dividing: n = 6 \div \left(-\tfrac{3}{4}\right).
  3. Keep, change, flip: n = 6 \times \left(-\tfrac{4}{3}\right) = -\tfrac{24}{3}.
  4. Simplify: n = -8.

Answer: -8

Check: -8 \times \left(-\tfrac{3}{4}\right) = \tfrac{24}{4} = 6 ✓ (two negatives give a positive).

Calculator tip

Using the TI-30XS MultiView

  • Negative numbers: use the (−) key (bottom row) for a negative sign. Do not use the subtraction key. To enter -8 + 5, press (−) 8 + 5 enter. Using the minus key at the start can give an error or a wrong answer.
  • Fractions: press n/d and type the numerator. Press the down arrow and type the denominator. Then press the right arrow to leave the fraction. For a mixed number use 2nd + n/d (the U\tfrac{n}{d} function).
  • Fraction ↔ decimal: after you get an answer, press toggle to switch between fraction and decimal form, e.g. \tfrac{17}{12} \leftrightarrow 1.416666667.
  • Order of operations: the calculator follows it by itself. But you must type any parentheses the problem has. This includes parentheses around a negative being squared: ( (−) 3 ) x².

When NOT to use it: the first few questions of the GED math test are calculator-free. They often test exactly this skill (integer signs, simple fractions, order of operations). Practice doing these by hand until they feel automatic. Even in the calculator section, a quick mental estimate catches typing mistakes.

Common mistake

Common mistakes

  • Adding denominators. Wrong: \tfrac{1}{2} + \tfrac{1}{3} = \tfrac{2}{5}. Right: rewrite with a common denominator, \tfrac{3}{6} + \tfrac{2}{6} = \tfrac{5}{6}.
  • Forgetting that minus a negative is plus. Wrong: -8 + 5 - (-6) = -8 + 5 - 6 = -9. Right: -8 + 5 + 6 = 3.
  • Dividing fractions without flipping. Wrong: \tfrac{2}{3} \div \tfrac{4}{9} = \tfrac{8}{27}. Right: \tfrac{2}{3} \times \tfrac{9}{4} = \tfrac{3}{2}.
  • Working left to right and ignoring the order of operations. Wrong: 18 - 4 \times 3 = 14 \times 3 = 42. Right: 18 - 12 = 6.
  • Multiplying before dividing. Wrong: 24 \div 4 \times 2 = 24 \div 8 = 3. Right: left to right, 6 \times 2 = 12.
  • Mixed-number subtraction without borrowing. Wrong: 5\tfrac{1}{4} - 2\tfrac{3}{4} → "\tfrac{3}{4} - \tfrac{1}{4} = \tfrac{2}{4}", giving 3\tfrac{1}{2}. Right: borrow, 4\tfrac{5}{4} - 2\tfrac{3}{4} = 2\tfrac{1}{2}.
Test strategy

GED test strategy

  • Estimate first. Before calculating, round: 8\tfrac{1}{4} - 3\tfrac{5}{6} is about 8 - 4 = 4. Any answer choice near 12 or near 1 can be crossed out.
  • Check the sign before the size. Many wrong choices are the right number with the wrong sign. First use the sign rules to decide "positive or negative?" Then cross out choices.
  • Plug the answer back in. For "what number…" questions, test each choice in the original statement — often faster than solving.
  • Read the question twice. Is it asking for the amount left, the change, or the total?

Check yourself

Add, subtract, multiply and divide rational numbersNo calculator

What is the value of -8 + 5 - (-6)?

Answer choices
Hint

Subtracting a negative number is the same as adding a positive number.

Check yourself

Add, subtract, multiply and divide rational numbersNo calculatorFill in

A baker uses \tfrac{3}{4} cup of sugar for a batch of muffins and \tfrac{2}{3} cup of sugar for a loaf of bread. How many cups of sugar does she use in all?

Enter your answer as a fraction or a decimal rounded to the nearest hundredth.

cups

Type your answer in the box. Fractions like 3/4 are fine.

Hint

To add fractions, first rewrite them so they have the same denominator. What is the smallest number that both 4 and 3 divide into?

Summary

Recap

  • Add/subtract integers: same signs → add and keep the sign. Different signs → subtract and keep the sign of the larger absolute value. Subtracting a number = adding its opposite.
  • Multiply/divide signs: same signs → positive; different signs → negative.
  • Fractions: common denominator to add or subtract. Straight across to multiply. Keep-change-flip to divide. Change mixed numbers to improper fractions to multiply or divide.
  • Decimals: line up points to add or subtract; count decimal places to multiply; shift points to divide.
  • Order of operations: grouping → exponents → \times\div left to right → +- left to right.
  • None of this is on the formula sheet — practice it by hand.
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