7.06 Add and subtract mixed numbers

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A mixed number such as $2\frac{3}{4}$234​ is made up of a whole part (in this case $2$2) and a fractional part (in this case $\frac{3}{4}$34​). We can think of it as the sum of the two parts, $2+\frac{3}{4}$2+34​. With the properties of addition as well, we can follow a series of steps to add (and subtract) mixed numbers.

Let's look at an example of $2\frac{3}{4}+3\frac{5}{8}$234​+358​:

Step 1. We can rewrite this as four terms, by splitting each mixed number into its parts:

$2\frac{3}{4}+3\frac{5}{8}$234​+358​

$=$=

$2+\frac{3}{4}+3+\frac{5}{8}$2+34​+3+58​

Step 2. We can now add the whole parts together, and separately add the fractional parts together. Adding the wholes first:

$2+\frac{3}{4}+3+\frac{5}{8}$2+34​+3+58​

$=$=

$5+\frac{3}{4}+\frac{5}{8}$5+34​+58​

Step 3. To add the fractional parts together, we will first rewrite them to have a common denominator. In this case we can get a common denominator of $8$8, by multiplying numerator and denominator of $\frac{3}{4}$34​ by $2$2:

$5+\frac{3}{4}+\frac{5}{8}$5+34​+58​

$=$=

$5+\frac{6}{8}+\frac{5}{8}$5+68​+58​

Step 4. We can now add the fractional parts:

$5+\frac{6}{8}+\frac{5}{8}$5+68​+58​

$=$=

$5+\frac{11}{8}$5+118​

Step 5. Since this fractional part $\frac{11}{8}$118​ is larger than one whole, we can't just combine the parts to make a mixed number. We can instead rewrite the improper fraction $\frac{11}{8}$118​ as its own mixed number:

$5+\frac{11}{8}$5+118​

$=$=

$5+1\frac{3}{8}$5+138​

Step 6. Finally, we can combine the whole parts to get:

$5+1\frac{3}{8}$5+138​

$=$=

$6\frac{3}{8}$638​

So we have that $2\frac{3}{4}+3\frac{5}{8}=6\frac{3}{8}$234​+358​=638​.

Sometimes we don't need all of the steps in this process, depending on the particular fractions.

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