Mixed Number to Improper Fraction — Conversion & Practice

(Video) Mixed Numbers

by mathman1024

Here’s a video by mathman1024 that explains converting a mixed number to a fraction and converting a fraction to a mixed number.

$\large{\text{Summary}}$

Converting from a mixed number into an improper fraction means we have to break down the mixed number. $$\begin{align} 3\frac12&=3+\frac12\\ &=\frac3{{\color{#29A603}{1}}}+\frac12 \end{align}$$ This is just two fractions with unlike denominators being added. So, we have to find the common denominator, which in this example is $2$. We find the equivalent fractions to add with the denominator of $2$. $$\begin{align} \frac31+\frac12&={\color{#29A603}{\frac62}}+\frac12\\ &=\frac{6+1}2\\ &=\frac72 \end{align}$$

But there's actually a trick to quickly convert mixed numbers to improper fractions.

1. Take the whole number and multiply it by the denominator.
2. Add the numerator to this product.
3. Put this value over the denominator. $$\frac{\text{Whole}\cdot\text{Denominator}+\text{Numerator}}{\text{Denominator}}$$

So, for $3\frac12$ we can convert it using the formula. $$3\frac12=\frac{3\cdot2+1}{2}=\frac72$$

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When dealing with negative mixed numbers, the process is the same; we just tack on the negative at the end. So if we want to convert $-4\frac13$ into a fraction, we: $$\begin{align} -4\frac13&=-\left(4\frac13\right)\\ &=-\left(\frac{4\times3+1}{3}\right)\\ &=-\left(\frac{12+1}{3}\right)\\ &=-\left(\frac{13}{3}\right)\\ &=\boxed{-\frac{13}3} \end{align}$$

Try converting this mixed number to an improper fraction on your own. Then, watch the video to see the answer!

1. $5\frac23$

Converting Mixed Numbers to Fractions

Mixed numbers are very good at giving us quick, clear information about a number. The amount that $3\frac25$ represents is easy to visualize just from a glance at the number. However, mixed numbers are not as easy to work with as improper fractions. Sometimes solving a problem will require us to change mixed numbers into improper fractions. This new type of fraction is a fraction with bigger numerator than denominator.

Example: $3\frac25$

We're going to convert $3\frac25$ into an improper fraction. We need to first multiply the whole number by the denominator of the fraction. $$3\cdot 5 = 15$$ Now, we're going to add the numerator to this number. $$15+2=17$$ The last step is to put our answer over the denominator. $$\frac{17}{5}$$

Practice Problem 1

We're going to try some practice problems to check our understanding.

Let's go through each of the steps. First, we're going to multiply the whole number by the denominator. $$5 \cdot 3 = 15$$ Next, we're going to add the numerator to our answer. $$15+1=16$$ Finally, we are going to put this number over our denominator. $$\frac{16}{3}$$ Our final answer is $\frac{16}{3}$.

Example Problem 2: Bigger Numbers

Bigger numbers can make problems look a little scary, but we still use the same process to convert from a mixed number to an improper fraction.

Our first step is to multiply the whole number by the denominator. $$11 \cdot 9 = 99$$ Now we're going to add the numerator to our answer. $$99+7=106$$ The last step is to put this number over the denominator from the original mixed number. $$\frac{106}{9}$$

We can convert a mixed number (a.k.a. a mixed fraction) into an improper fraction. Recall these fractions have numerator greater than or equal to the denominator. Then, the fractional part will be a proper fraction.

Here are the steps you need to take:

1. Multiply the whole number by the denominator (the bottom number in the fraction)
2. Add the total to the numerator (the top number in the fraction)
3. Replace the numerator above the denominator.

After you convert the mixed number, you will always get an improper fraction. You can think of this problem in terms of containers. The whole number in a mixed number represents how many containers are completely filled up. The numerator of an improper fraction shows how many things we have. The denominator shows how much one container can hold. When we convert a mixed number to a fraction, we're pouring out containers.

Let's try it out

For example, if I wanted to turn $5\frac{1}{4}$ into a fraction, here are the steps I would take:

Step 1: Multiply the whole number by the denominator

In the number $5\frac{1}{4}$, $5$ is the whole number and $4$ is the denominator.

$4 \times 5$ is $20$.

Step 2: Add the total to the numerator

Our total is $20$.

The numerator is $1$, so $20 + 1 = 21$.

Step 3: Place your new numerator in its place above the denominator.

The final fraction is $\frac{21}{4}$.

(Video) Mixed Numbers

by Math Antics

This video by Math Antics covers improper fractions, mixed numbers, converting fractions to mixed numbers, and converting mixed numbers to fractions.

$\large{\text{Summary}}$

We can convert mixed numbers into fractions. Multiply the integer by the denominator and add it to the numerator. $$\frac{\text{whole number}\times\text{denominator}+\text{numerator}}{\text{denominator}}$$

To convert $2\frac18$ we multiply $2$ by $8$ and add that to $1$. $$2\frac18=\frac{2\times8+1}{8}=\frac{17}{8}$$