1. Adding and Subtracting Fractions

Fractions are numbers expressed as the quotient of two integers. 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 mixed number consists of a whole number and a fraction. 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

When working with integers, 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), find a common denominator before combining:

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

4. Adding and Subtracting Radicals

Radicals (such as square roots) can be added or subtracted only when they have the same radicand (the expression under the radical sign). For example:

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

However, if the radicands are different, such as:

\( \sqrt{a} + \sqrt{b} \) (with \( a \neq b \)),

they cannot be combined into a single radical.

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.

Subtraction with regrouping works similarly.

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

You can easily find free printable worksheets and online practice exercises by searching for the relevant keywords.