Learn on PengiBig Ideas Math, Course 2Chapter 2: Rational Numbers

Lesson 2: Adding Rational Numbers

In this Grade 7 lesson from Big Ideas Math, Course 2 (Chapter 2: Rational Numbers), students learn how to add rational numbers — including fractions, mixed numbers, and decimals — by applying the same sign rules used for integers. The lesson covers finding sums using a least common denominator, adding positive and negative decimals, and evaluating algebraic expressions with rational number values. Students also apply these skills to real-life problems involving gains and losses represented as rational numbers.

Section 1

Adding Fractions with Common Denominators

Property

If $a$ , $b$ , and $c$ are numbers where $c \neq 0$ , then

\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}

To add fractions with the same denominator, add the numerators and place the result over the common denominator.

Examples

Section 2

Adding Fractions with Different Denominators

Property

To add fractions with different denominators, first find the least common denominator (LCD). The LCD is the least common multiple (LCM) of the denominators. Then, convert each fraction into an equivalent fraction with the LCD. Finally, add the numerators and place the result over the common denominator. Do not simplify the equivalent fractions before combining them, or you will lose the common denominator.

Examples

Section 3

Adding Decimals with Different Signs

Property

To add decimals with different signs, we apply the same rules as adding integers, then align decimal points for computation:

If the signs are the same, add the absolute values and keep the common sign.
If the signs are different, subtract the smaller absolute value from the larger absolute value and use the sign of the number with the larger absolute value.
Line up the decimal points vertically and use zeros as placeholders as needed.
Add or subtract the numbers as if they were whole numbers, then place the decimal point in the answer.

Examples

Section 4

Evaluating Algebraic Expressions with Rational Numbers

Property

To evaluate an algebraic expression with rational numbers, substitute the given rational number values for the variables and perform the operations using the rules for adding rational numbers:

\text{If } x = a \text{ and } y = b, \text{ then evaluate the expression by replacing variables with their values}

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Adding Fractions with Common Denominators

Property

If $a$ , $b$ , and $c$ are numbers where $c \neq 0$ , then

\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}

To add fractions with the same denominator, add the numerators and place the result over the common denominator.

Examples

Section 2

Adding Fractions with Different Denominators

Property

Examples

Section 3

Adding Decimals with Different Signs

Property

To add decimals with different signs, we apply the same rules as adding integers, then align decimal points for computation:

If the signs are the same, add the absolute values and keep the common sign.
If the signs are different, subtract the smaller absolute value from the larger absolute value and use the sign of the number with the larger absolute value.
Line up the decimal points vertically and use zeros as placeholders as needed.
Add or subtract the numbers as if they were whole numbers, then place the decimal point in the answer.

Examples

Section 4

Evaluating Algebraic Expressions with Rational Numbers

Property

To evaluate an algebraic expression with rational numbers, substitute the given rational number values for the variables and perform the operations using the rules for adding rational numbers:

\text{If } x = a \text{ and } y = b, \text{ then evaluate the expression by replacing variables with their values}

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Adding Fractions with Common Denominators

Property

If $a$ , $b$ , and $c$ are numbers where $c \neq 0$ , then

\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}

To add fractions with the same denominator, add the numerators and place the result over the common denominator.

Examples

Section 2

Adding Fractions with Different Denominators

Property

Examples

Section 3

Adding Decimals with Different Signs

Property

To add decimals with different signs, we apply the same rules as adding integers, then align decimal points for computation:

If the signs are the same, add the absolute values and keep the common sign.
If the signs are different, subtract the smaller absolute value from the larger absolute value and use the sign of the number with the larger absolute value.
Line up the decimal points vertically and use zeros as placeholders as needed.
Add or subtract the numbers as if they were whole numbers, then place the decimal point in the answer.

Examples

Section 4

Evaluating Algebraic Expressions with Rational Numbers

Property

To evaluate an algebraic expression with rational numbers, substitute the given rational number values for the variables and perform the operations using the rules for adding rational numbers:

\text{If } x = a \text{ and } y = b, \text{ then evaluate the expression by replacing variables with their values}