Add and Subtract

1. Adding and Subtracting Fractions

Fractions are numbers expressed as the quotient of two integers with a nonzero denominator. All denominators below must be nonzero. When adding or subtracting fractions, the method depends on whether the denominators are the same or different.

1.1 Fractions with Like Denominators

When the denominators are the same, simply add or subtract the numerators:

\( \frac{a}{d} + \frac{b}{d} = \frac{a+b}{d} \)

Similarly, for subtraction:

\( \frac{a}{d} - \frac{b}{d} = \frac{a-b}{d} \)

1.2 Fractions with Unlike Denominators

When fractions have different denominators, find a common denominator (typically the product or least common multiple of the denominators) and adjust the numerators accordingly. For example:

\( \frac{a}{b} + \frac{c}{d} = \frac{a\cdot d + c\cdot b}{b\cdot d} \)

And for subtraction:

\( \frac{a}{b} - \frac{c}{d} = \frac{a\cdot d - c\cdot b}{b\cdot d} \)

1.3 Adding Mixed Numbers

A positive mixed number consists of a whole-number part and a proper fraction. In the symbolic examples below, b and d are positive, 0 ≤ a < b and 0 ≤ c < d. To add mixed numbers, convert them to improper fractions or add the whole numbers and fractions separately (after finding a common denominator for the fractional parts). For example, if we have:

\( 3\frac{a}{b} + 2\frac{c}{d} \)

You can write each as an improper fraction:

\( 3\frac{a}{b} = \frac{3b+a}{b} \) and \( 2\frac{c}{d} = \frac{2d+c}{d} \)

Then add them using the method for unlike denominators.

2. Adding and Subtracting Integers and Decimals

Addition and subtraction are equally important when working with integers and decimals.

2.1 Adding and Subtracting Integers

In these sign rules, a and b are nonnegative magnitudes. Opposite numbers sum to zero when a = b. Subtracting a number means adding its opposite. Remember the following rules:

• \( (+a) + (+b) = +(a+b) \)
• \( (-a) + (-b) = -(a+b) \)
• \( (+a) + (-b) = a-b \) (if \( a > b \)) or \( -(b-a) \) (if \( b > a \))
• \( (-a) - (-b) = -a + b \)

2.2 Adding and Subtracting Decimals

When adding decimals, align the decimal points and add digit by digit:

For example, if you want to add:

\( 12.34 + 5.678 = 18.018 \)

Subtraction follows the same rule:

\( 12.34 - 5.678 = 6.662 \)

Worksheets on adding and subtracting decimals can help solidify these concepts.

3. Adding and Subtracting Polynomials and Rational Expressions

Polynomials are algebraic expressions consisting of variables and coefficients. To add or subtract polynomials, combine like terms.

For example, consider:

\( (3x^2 + 2x + 1) + (5x^2 - 4x + 7) = (3+5)x^2 + (2-4)x + (1+7) = 8x^2 - 2x + 8 \)

Similarly, subtracting polynomials involves subtracting the coefficients of like terms:

\( (3x^2 + 2x + 1) - (5x^2 - 4x + 7) = (3-5)x^2 + (2+4)x + (1-7) = -2x^2 + 6x - 6 \)

For rational expressions (fractions with polynomials in the numerator and denominator), require Q(x) ≠ 0 and S(x) ≠ 0, and find a common denominator before combining. These original restrictions remain even if factors later cancel:

\( \frac{P(x)}{Q(x)} + \frac{R(x)}{S(x)} = \frac{P(x) \cdot S(x) + R(x) \cdot Q(x)}{Q(x) \cdot S(x)} \)

4. Adding and Subtracting Radicals

For real square roots, radicands must be nonnegative. Simplify each radical first, then combine like radical terms with the same index and radicand by adding or subtracting their coefficients. For example, for a ≥ 0:

\( \sqrt{a} + 2\sqrt{a} = 3\sqrt{a} \)

Different starting radicands do not by themselves determine whether terms combine. For example:

\( \sqrt{8} + \sqrt{18} = 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2} \).

By contrast, \(\sqrt{2}+\sqrt{3}\) has unlike simplified radical parts, so these are not like terms. Do not add radicands: \(\sqrt{2}+\sqrt{3}\neq\sqrt{5}\). For subtraction, \(3\sqrt{12}-\sqrt{27}=6\sqrt{3}-3\sqrt{3}=3\sqrt{3}\).

5. Addition and Subtraction with Regrouping

In multi-digit arithmetic, regrouping (or carrying/borrowing) is necessary when the sum of digits in a place value exceeds 9, or when subtracting a larger digit from a smaller digit in a given place.

For example, consider the addition:

\( \begin{array}{r} \ \ 57 \\ + 68 \\ \hline \end{array} \)

Adding the ones: \( 7 + 8 = 15 \) (write 5 and carry 1). Then add the tens with the carried digit: \( 5 + 6 + 1 = 12 \). The final answer is 125.

For 402 − 187, regroup one hundred into ten tens, then one ten into ten ones. The top number becomes 3 hundreds, 9 tens and 12 ones. Subtract by column: 12 − 7 = 5, 9 − 8 = 1, and 3 − 1 = 2, giving 215. Check: 215 + 187 = 402.

6. Practice Worksheets and Word Problems

Practicing addition and subtraction through worksheets helps solidify these concepts. Here are some popular topics:

• Adding fractions with unlike denominators
• Adding and subtracting decimals
• Adding and subtracting integers (including positive and negative numbers)
• Adding and subtracting mixed numbers
• Adding and subtracting polynomials
• Adding and subtracting rational expressions
• Word problems involving addition and subtraction

Try these short problems, then check the answers below. The quiz above practices whole-number addition and nonnegative subtraction; the lesson also covers the wider topics listed here.

1. Find \(\frac{2}{3}+\frac{1}{4}\).
2. Find \(2\frac{1}{3}-1\frac{3}{4}\).
3. A temperature is −3°C and rises by 8°C. What is the new temperature?
4. A 20 m ribbon is cut into pieces of 7.85 m and 4.6 m. How much remains?

Answers: \(\frac{8}{12}+\frac{3}{12}=\frac{11}{12}\); \(\frac{7}{3}-\frac{7}{4}=\frac{28}{12}-\frac{21}{12}=\frac{7}{12}\); 5°C; \(20-(7.85+4.6)=7.55\) m.