Add & Subtract Rational Numbers Unit TestJoey Walked East For $15 \frac{2}{3}$ Meters From Home. Then, He Walked West For $21 \frac{3}{4}$ Meters. How Far Was Joey From Home? Write Your Answer As A Mixed Number Only.Joey Was

Understanding Rational Numbers

Rational numbers are a type of real number that can be expressed as the quotient or fraction of two integers, where the denominator is non-zero. They can be added, subtracted, multiplied, and divided, just like integers. In this article, we will focus on adding and subtracting rational numbers, specifically in the context of a unit test.

What is a Unit Test?

A unit test is a small test that checks a specific piece of code or a specific mathematical concept. It is a way to ensure that a particular concept or function is working correctly. In the context of this article, the unit test is to add and subtract rational numbers.

Adding Rational Numbers

To add rational numbers, we need to follow these steps:

1. Like Terms: If the denominators are the same, we can add the numerators directly.
2. Unlike Terms: If the denominators are different, we need to find the least common multiple (LCM) of the denominators and then convert both fractions to have the LCM as the denominator.

Example 1: Adding Rational Numbers with Like Terms

Let's consider the following example:

$$\frac{1}{2} + \frac{1}{2}$$

In this case, the denominators are the same, so we can add the numerators directly:

$$\frac{1}{2} + \frac{1}{2} = \frac{1+1}{2} = \frac{2}{2} = 1$$

Example 2: Adding Rational Numbers with Unlike Terms

Let's consider the following example:

$$\frac{1}{2} + \frac{1}{3}$$

In this case, the denominators are different, so we need to find the LCM of the denominators, which is 6. Then, we convert both fractions to have the LCM as the denominator:

$$\frac{1}{2} = \frac{3}{6}$$

$$\frac{1}{3} = \frac{2}{6}$$

Now, we can add the fractions:

$$\frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{5}{6}$$

Subtracting Rational Numbers

To subtract rational numbers, we need to follow these steps:

1. Like Terms: If the denominators are the same, we can subtract the numerators directly.
2. Unlike Terms: If the denominators are different, we need to find the LCM of the denominators and then convert both fractions to have the LCM as the denominator.

Example 1: Subtracting Rational Numbers with Like Terms

Let's consider the following example:

$$\frac{1}{2} - \frac{1}{2}$$

In this case, the denominators are the same, so we can subtract the numerators directly:

$$\frac{1}{2} - \frac{1}{2} = \frac{1-1}{2} = \frac{0}{2} = 0$$

Example 2: Subtracting Rational Numbers with Unlike Terms

Let's consider the following example:

$$\frac{1}{2} - \frac{1}{3}$$

In this case, the denominators are different, so we need to find the LCM of the denominators, which is 6. Then, we convert both fractions to have the LCM as the denominator:

$$\frac{1}{2} = \frac{3}{6}$$

$$\frac{1}{3} = \frac{2}{6}$$

Now, we can subtract the fractions:

$$\frac{3}{6} - \frac{2}{6} = \frac{3-2}{6} = \frac{1}{6}$$

Joey's Walk

Now, let's go back to Joey's walk. Joey walked east for $15 \frac{2}{3}$ meters from home. Then, he walked west for $21 \frac{3}{4}$ meters. We need to find out how far Joey is from home.

To do this, we need to subtract the distance Joey walked west from the distance he walked east. We can convert both mixed numbers to improper fractions:

$$15 \frac{2}{3} = \frac{47}{3}$$

$$21 \frac{3}{4} = \frac{87}{4}$$

Now, we can subtract the fractions:

$$\frac{47}{3} - \frac{87}{4}$$

To subtract these fractions, we need to find the LCM of the denominators, which is 12. Then, we convert both fractions to have the LCM as the denominator:

$$\frac{47}{3} = \frac{188}{12}$$

$$\frac{87}{4} = \frac{261}{12}$$

Now, we can subtract the fractions:

$$\frac{188}{12} - \frac{261}{12} = \frac{188-261}{12} = \frac{-73}{12}$$

To convert this improper fraction to a mixed number, we need to divide the numerator by the denominator:

$$\frac{-73}{12} = -6 \frac{1}{12}$$

So, Joey is $-6 \frac{1}{12}$ meters away from home.

Conclusion

Q: What is the difference between adding and subtracting rational numbers?

A: The difference between adding and subtracting rational numbers is the operation we are performing. When we add rational numbers, we are combining them to find a total or a sum. When we subtract rational numbers, we are finding the difference between them.

Q: How do I add rational numbers with like terms?

A: To add rational numbers with like terms, we need to follow these steps:

1. Like Terms: If the denominators are the same, we can add the numerators directly.
2. Example: $\frac{1}{2} + \frac{1}{2} = \frac{1+1}{2} = \frac{2}{2} = 1$

Q: How do I add rational numbers with unlike terms?

A: To add rational numbers with unlike terms, we need to follow these steps:

1. Find the LCM: Find the least common multiple (LCM) of the denominators.
2. Convert Fractions: Convert both fractions to have the LCM as the denominator.
3. Add Fractions: Add the fractions.

Example: $\frac{1}{2} + \frac{1}{3}$

1. Find the LCM: The LCM of 2 and 3 is 6.
2. Convert Fractions: $\frac{1}{2} = \frac{3}{6}$ and $\frac{1}{3} = \frac{2}{6}$
3. Add Fractions: $\frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{5}{6}$

Q: How do I subtract rational numbers with like terms?

A: To subtract rational numbers with like terms, we need to follow these steps:

1. Like Terms: If the denominators are the same, we can subtract the numerators directly.
2. Example: $\frac{1}{2} - \frac{1}{2} = \frac{1-1}{2} = \frac{0}{2} = 0$

Q: How do I subtract rational numbers with unlike terms?

A: To subtract rational numbers with unlike terms, we need to follow these steps:

1. Find the LCM: Find the least common multiple (LCM) of the denominators.
2. Convert Fractions: Convert both fractions to have the LCM as the denominator.
3. Subtract Fractions: Subtract the fractions.

Example: $\frac{1}{2} - \frac{1}{3}$

1. Find the LCM: The LCM of 2 and 3 is 6.
2. Convert Fractions: $\frac{1}{2} = \frac{3}{6}$ and $\frac{1}{3} = \frac{2}{6}$
3. Subtract Fractions: $\frac{3}{6} - \frac{2}{6} = \frac{3-2}{6} = \frac{1}{6}$

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, we need to follow these steps:

1. Multiply the Whole Number: Multiply the whole number by the denominator.
2. Add the Numerator: Add the numerator to the product.
3. Write the Improper Fraction: Write the result as an improper fraction.

Example: $15 \frac{2}{3}$

1. Multiply the Whole Number: $15 \times 3 = 45$
2. Add the Numerator: $45 + 2 = 47$
3. Write the Improper Fraction: $\frac{47}{3}$

Q: How do I convert an improper fraction to a mixed number?

A: To convert an improper fraction to a mixed number, we need to follow these steps:

1. Divide the Numerator: Divide the numerator by the denominator.
2. Write the Mixed Number: Write the result as a mixed number.

Example: $\frac{47}{3}$

1. Divide the Numerator: $47 \div 3 = 15 \frac{2}{3}$
2. Write the Mixed Number: $15 \frac{2}{3}$

Q: How do I find the distance between two points on a number line?

A: To find the distance between two points on a number line, we need to follow these steps:

1. Find the Difference: Find the difference between the two points.
2. Write the Answer: Write the answer as a mixed number or an improper fraction.

Example: Find the distance between $15 \frac{2}{3}$ and $21 \frac{3}{4}$

1. Find the Difference: $21 \frac{3}{4} - 15 \frac{2}{3}$
2. Convert Mixed Numbers to Improper Fractions: $\frac{87}{4} - \frac{47}{3}$
3. Find the LCM: The LCM of 4 and 3 is 12.
4. Convert Fractions: $\frac{87}{4} = \frac{261}{12}$ and $\frac{47}{3} = \frac{188}{12}$
5. Subtract Fractions: $\frac{261}{12} - \frac{188}{12} = \frac{261-188}{12} = \frac{73}{12}$
6. Convert Improper Fraction to Mixed Number: $\frac{73}{12} = 6 \frac{1}{12}$

So, the distance between $15 \frac{2}{3}$ and $21 \frac{3}{4}$ is $6 \frac{1}{12}$ units.