Multiplying Mixed Fractions A Step By Step Guide With Examples

In mathematics, the multiplication of mixed fractions is a fundamental skill that builds upon the understanding of both fractions and multiplication. Mixed fractions, which combine a whole number and a proper fraction, require a specific approach when being multiplied. This article aims to provide a comprehensive guide on how to confidently and accurately multiply mixed fractions, complete with explanations and examples.

Understanding Mixed Fractions

Before diving into the multiplication process, it's crucial to understand what mixed fractions are. A mixed fraction is a number written as a whole number and a proper fraction combined. For instance, $2 \frac{4}{6}$ is a mixed fraction where 2 is the whole number part and $\frac{4}{6}$ is the fractional part. The fractional part is always less than one, making it a proper fraction. To effectively multiply mixed fractions, we first need to convert them into improper fractions. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

Converting Mixed Fractions to Improper Fractions

The process of converting a mixed fraction to an improper fraction involves a simple two-step procedure:

1. Multiply the whole number by the denominator of the fractional part.
2. Add the numerator of the fractional part to the result obtained in step 1. This sum becomes the new numerator, and the denominator remains the same.

For example, let’s convert the mixed fraction $2 \frac{4}{6}$ into an improper fraction:

1. Multiply the whole number (2) by the denominator (6): $2 \times 6 = 12$
2. Add the numerator (4) to the result: $12 + 4 = 16$

So, the improper fraction is $\frac{16}{6}$. This means that $2 \frac{4}{6}$ is equivalent to $\frac{16}{6}$. Converting to improper fractions is a critical first step because it allows us to apply the standard rules of fraction multiplication.

Practice Converting Mixed Fractions

Let's practice converting a few more mixed fractions to solidify our understanding. Consider the mixed fraction $1 \frac{3}{8}$. Following the steps:

1. Multiply the whole number (1) by the denominator (8): $1 \times 8 = 8$
2. Add the numerator (3) to the result: $8 + 3 = 11$

Thus, $1 \frac{3}{8}$ is equivalent to $\frac{11}{8}$. This conversion process ensures that we are working with a single fraction, making multiplication straightforward. Similarly, if we have $2 \frac{5}{9}$, the conversion would be:

1. Multiply the whole number (2) by the denominator (9): $2 \times 9 = 18$
2. Add the numerator (5) to the result: $18 + 5 = 23$

Therefore, $2 \frac{5}{9}$ converts to $\frac{23}{9}$. Consistent practice with these conversions is essential for mastering mixed fraction multiplication.

The Multiplication Process

Once the mixed fractions are converted into improper fractions, the multiplication process becomes quite simple. The rule for multiplying fractions is: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. In mathematical terms:

$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

This rule applies universally to all fractions, including those derived from mixed numbers. Let's illustrate this with an example. Suppose we want to multiply $2 \frac{4}{6}$ by $1 \frac{3}{8}$. We’ve already converted these to improper fractions: $2 \frac{4}{6} = \frac{16}{6}$ and $1 \frac{3}{8} = \frac{11}{8}$. Now, we multiply the improper fractions:

$\frac{16}{6} \times \frac{11}{8} = \frac{16 \times 11}{6 \times 8} = \frac{176}{48}$

So, the result of multiplying $\frac{16}{6}$ and $\frac{11}{8}$ is $\frac{176}{48}$. This fraction can be simplified, which is often the next step in solving these problems.

Simplifying Improper Fractions

After multiplying improper fractions, it’s important to simplify the result. Simplifying a fraction means reducing it to its lowest terms. This is done by finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by the GCD. In our example, we have $\frac{176}{48}$. To simplify this, we first find the GCD of 176 and 48.

The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 176 are: 1, 2, 4, 8, 11, 16, 22, 44, 88, and 176.

The greatest common divisor (GCD) of 176 and 48 is 16. Now, we divide both the numerator and the denominator by 16:

$\frac{176 \div 16}{48 \div 16} = \frac{11}{3}$

So, the simplified improper fraction is $\frac{11}{3}$. This fraction can also be converted back into a mixed fraction for a more intuitive understanding of the result. To convert $\frac{11}{3}$ back to a mixed fraction, we divide 11 by 3:

$11 \div 3 = 3$ with a remainder of $2$.

This means that $\frac{11}{3}$ is equal to $3 \frac{2}{3}$. Simplifying fractions, both improper and mixed, is a key aspect of mastering fraction multiplication.

Examples of Multiplying Mixed Fractions

Let’s work through some additional examples to further illustrate the process. Consider the multiplication of $1 \frac{1}{7}$ and $3 \frac{2}{9}$.

Example 1: $1 \frac{1}{7} \times 3 \frac{2}{9}$

First, we convert each mixed fraction to an improper fraction:

$1 \frac{1}{7} = \frac{(1 \times 7) + 1}{7} = \frac{8}{7}$

$3 \frac{2}{9} = \frac{(3 \times 9) + 2}{9} = \frac{29}{9}$

Next, we multiply the improper fractions:

$\frac{8}{7} \times \frac{29}{9} = \frac{8 \times 29}{7 \times 9} = \frac{232}{63}$

Now, we simplify the improper fraction $\frac{232}{63}$. The GCD of 232 and 63 is 1, which means the fraction is already in its simplest form. Finally, we convert the improper fraction to a mixed fraction:

$232 \div 63 = 3$ with a remainder of $43$.

Thus, $\frac{232}{63}$ is equal to $3 \frac{43}{63}$.

Example 2: $2 \frac{5}{9} \times 1 \frac{4}{5}$

Convert the mixed fractions to improper fractions:

$2 \frac{5}{9} = \frac{(2 \times 9) + 5}{9} = \frac{23}{9}$

$1 \frac{4}{5} = \frac{(1 \times 5) + 4}{5} = \frac{9}{5}$

Multiply the improper fractions:

$\frac{23}{9} \times \frac{9}{5} = \frac{23 \times 9}{9 \times 5} = \frac{207}{45}$

Simplify the improper fraction $\frac{207}{45}$. The GCD of 207 and 45 is 9. Divide both numerator and denominator by 9:

$\frac{207 \div 9}{45 \div 9} = \frac{23}{5}$

Convert the simplified improper fraction to a mixed fraction:

$23 \div 5 = 4$ with a remainder of $3$.

Therefore, $\frac{23}{5}$ is equal to $4 \frac{3}{5}$.

Example 3: $2 \frac{2}{3} \times 1 \frac{1}{5}$

Convert the mixed fractions to improper fractions:

$2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3}$

$1 \frac{1}{5} = \frac{(1 \times 5) + 1}{5} = \frac{6}{5}$

Multiply the improper fractions:

$\frac{8}{3} \times \frac{6}{5} = \frac{8 \times 6}{3 \times 5} = \frac{48}{15}$

Simplify the improper fraction $\frac{48}{15}$. The GCD of 48 and 15 is 3. Divide both numerator and denominator by 3:

$\frac{48 \div 3}{15 \div 3} = \frac{16}{5}$

Convert the simplified improper fraction to a mixed fraction:

$16 \div 5 = 3$ with a remainder of $1$.

Thus, $\frac{16}{5}$ is equal to $3 \frac{1}{5}$.

Example 4: $2 \frac{1}{6} \times 5 \frac{8}{15}$

Convert the mixed fractions to improper fractions:

$2 \frac{1}{6} = \frac{(2 \times 6) + 1}{6} = \frac{13}{6}$

$5 \frac{8}{15} = \frac{(5 \times 15) + 8}{15} = \frac{83}{15}$

Multiply the improper fractions:

$\frac{13}{6} \times \frac{83}{15} = \frac{13 \times 83}{6 \times 15} = \frac{1079}{90}$

Simplify the improper fraction $\frac{1079}{90}$. The GCD of 1079 and 90 is 1, so the fraction is already in its simplest form. Convert the improper fraction to a mixed fraction:

$1079 \div 90 = 11$ with a remainder of $89$.

Therefore, $\frac{1079}{90}$ is equal to $11 \frac{89}{90}$.

Conclusion

Mastering the multiplication of mixed fractions involves converting them into improper fractions, multiplying the numerators and denominators, simplifying the resulting fraction, and, if necessary, converting back to a mixed fraction. This process, while it may seem multi-step at first, becomes straightforward with practice. By working through examples and understanding the underlying principles, you can confidently tackle any mixed fraction multiplication problem. Remember, the key is to break down the problem into manageable steps and to practice consistently. This comprehensive guide should provide you with the tools and knowledge needed to excel in this area of mathematics. Consistent application of these steps will build your proficiency and confidence in handling mixed fraction multiplications.