Lesson 4: Four Operations with Rational Numbers

Property

The additive inverse (or opposite) of a number $a$ is $-a$. Their sum is the additive identity, $0$.

The multiplicative inverse (or reciprocal) of a non-zero number $a$ is $\frac{1}{a}$. Their product is the multiplicative identity, $1$.

Examples

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:

Examples

Property

To compare or order expressions, first evaluate each one to get a single numerical value.
Then, compare these values on a number line to order them, or compare their absolute values to find which is closest to zero.

For expressions $A$, $B$, and $C$ with values $v_A$, $v_B$, and $v_C$:

• Order (least to greatest): Arrange $v_A, v_B, v_C$ as they appear on a number line from left to right.
• Closest to zero: Find the minimum of their absolute values: $\min(|v_A|, |v_B|, |v_C|)$.

Property

Solve real-world and mathematical problems involving the four operations with rational numbers.
This requires translating a real-world scenario into a mathematical expression using addition, subtraction, multiplication, or division of rational numbers.

Examples

• A baker has $4\frac{1}{2}$ pounds of flour. A cake recipe requires $2\frac{3}{4}$ pounds. How much flour is left? $\frac{9}{2} - \frac{11}{4} = \frac{18}{4} - \frac{11}{4} = \frac{7}{4}$, or $1\frac{3}{4}$ pounds.
• A submarine at the surface dives $20\frac{1}{2}$ meters, then rises $8\frac{1}{4}$ meters. What is its new depth? $-20\frac{1}{2} + 8\frac{1}{4} = -\frac{41}{2} + \frac{33}{4} = -\frac{82}{4} + \frac{33}{4} = -\frac{49}{4}$, or $-12\frac{1}{4}$ meters.
• Three friends share a pizza. Anna eats $\frac{1}{4}$, and Ben eats $\frac{1}{3}$. What fraction of the pizza did they eat combined? $\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}$ of the pizza.

Explanation

Rational numbers are used to handle everyday tasks like measuring ingredients, tracking distances, or splitting bills. The first step is to read the problem carefully and decide which of the four basic operations is needed to solve it.