When you're in Year 9 math, you might encounter a fun twist with mixed numbers. They change how we do math with fractions, especially when it comes to adding, subtracting, multiplying, and dividing. So, what exactly are mixed numbers, and why are they important?

What are Mixed Numbers?

Mixed numbers are made up of a whole number and a proper fraction.

For example, $2\frac{3}{4}$ is a mixed number. It means you have 2 whole units and 3/4 of another unit.

Using mixed numbers helps us show amounts that are more than one but still have a part that's less than a whole.

Working with Mixed Numbers

1. Adding and Subtracting:

When you add or subtract mixed numbers, start by handling the whole number and the fraction separately.

Example 1 – Addition: Let’s say we want to add $1\frac{1}{2}$ and $2\frac{3}{4}$.

1. Change them to improper fractions:

   • $1\frac{1}{2} = \frac{3}{2}$
   • $2\frac{3}{4} = \frac{11}{4}$

2. Find a common denominator (in this case, it's 4):

   • Change $\frac{3}{2}$ to $\frac{6}{4}$.

3. Now, add them:

   • $\frac{6}{4} + \frac{11}{4} = \frac{17}{4}$

4. Change it back to a mixed number:

   • $\frac{17}{4} = 4\frac{1}{4}$

Example 2 – Subtraction: For $3\frac{1}{3} - 1\frac{1}{6}$:

1. Change them to improper fractions:

   • $3\frac{1}{3} = \frac{10}{3}$
   • $1\frac{1}{6} = \frac{7}{6}$

2. Find a common denominator (here, it's 6):

   • Change $\frac{10}{3}$ to $\frac{20}{6}$.

3. Now subtract:

   • $\frac{20}{6} - \frac{7}{6} = \frac{13}{6}$

4. Change it back to a mixed number:

   • $\frac{13}{6} = 2\frac{1}{6}$

2. Multiplying:

When you multiply mixed numbers, first convert them into improper fractions.

Example: To multiply $1\frac{2}{3}$ by $2\frac{1}{2}$:

1. Change to improper fractions:

   • $1\frac{2}{3} = \frac{5}{3}$
   • $2\frac{1}{2} = \frac{5}{2}$

2. Now multiply:

   • $\frac{5}{3} \times \frac{5}{2} = \frac{25}{6}$

3. Change it back to a mixed number:

   • $\frac{25}{6} = 4\frac{1}{6}$

3. Dividing:

To divide mixed numbers, start by changing them to improper fractions.

Example: To divide $2\frac{1}{4}$ by $1\frac{3}{5}$:

1. Change to improper fractions:

   • $2\frac{1}{4} = \frac{9}{4}$
   • $1\frac{3}{5} = \frac{8}{5}$

2. Flip the second fraction and multiply:

   • $\frac{9}{4} \div \frac{8}{5} = \frac{9}{4} \times \frac{5}{8} = \frac{45}{32}$

3. Change it back to a mixed number:

   • $\frac{45}{32} = 1\frac{13}{32}$

Wrap Up

Mixed numbers are really helpful in math. They let us mix whole numbers and fractions without any fuss.

Knowing how to add, subtract, multiply, and divide mixed numbers is a great step towards understanding more challenging math later on. So remember, whether you're adding, subtracting, multiplying, or dividing, mixed numbers can make things easier!