3H Adding and Subtracting Mixed Numbers

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A chef has $3 and 1 fourth$ pounds of custard. If she uses $1 and 4 fifths$ pounds of custard in making desserts, how many pounds of custard does she have left? Applications like this are why we need to review how to add and subtract mixed numbers.

Remember, a mixed number is a whole number added to a fraction. So when we add or subtract mixed numbers, we need to add or subtract the fraction parts and add or subtract the whole parts.

For example, look at the following example.

$6 and 4 fifths plus 2 and 2 thirds$ . To add the fraction parts we are going to need a common denominator. The smallest number that 3 and 5 divide evenly is 15. Converting both fractions into equal fractions with a common denominator of 15 gives the following.

$6 \frac{4}{5} + 2 \frac{2}{3} = 6 \frac{12}{15} + 2 \frac{10}{15}$ . Now add the whole parts and the fraction parts. Remember when adding fractions, add the numerators and keep the denominator the same.

$6 \frac{4}{5} + 2 \frac{2}{3} = 6 \frac{12}{15} + 2 \frac{10}{15} = 8 \frac{22}{15}$ . We are almost done. The answer is not a mixed number since the fraction part is improper. We can fix this by converting the improper fraction 22/15 back into a mixed number and then adding the 8.

$8 \frac{22}{15} = 8 + 1 \frac{7}{15} = 8 + 1 + \frac{7}{15} = 9 \frac{7}{15}$ .

Let’s look at the custard example from the beginning of this section. Since she used up some custard, the problem is subtraction.

$3 and 1 fourth minus 1 and 4 fifths$ . Again, we need a common denominator in order to subtract the fraction parts. The smallest number that 4 and 5 divide evenly is 20, so let’s make the fractions into equal fractions with a denominator of 20. We can multiply the ¼ top and bottom by 5. We can multiply the 4/5 top and bottom by 4.

$3 \frac{1}{4} - 1 \frac{4}{5} = 3 \frac{5}{20} - 1 \frac{16}{20}$ . Now we have a problem. Did you notice? We cannot take 16/20 away if we only have 5/20. We will need to borrow. Borrow one from the 3 and add it to the ¼. Then convert into an improper fraction. Now we can subtract.

$3 \frac{5}{20} - 1 \frac{16}{20} = (2 + 1 \frac{5}{20}) - 1 \frac{16}{20} = 2 \frac{25}{20} - 1 \frac{16}{20} = 1 \frac{9}{20}$

Try a few examples with your instructor.

Practice Problems Section 3H

For #1-21, perform the indicated operation and write your answers in simplest form. If an answer is improper, convert it into a mixed number.

$3 and 2 fifths plus 7 and 1 fifth$

Check answer

$10 and 3 fifths$
$8 and 1 seventh plus 3 and 6 sevenths$
$4 \frac{1}{5} - 3 \frac{8}{15}$

Check answer

$\frac{2}{3}$
$4 and 11 twelfths minus 1 and 1 fourth$
$7 \frac{8}{13} - 5 \frac{1}{2}$

Check answer

$2 \frac{3}{26}$
$4 and 1 fifth minus 1 and 3 fifths$
$10 and 1 third plus 6 and 4 fifths$

Check answer

$17 \frac{2}{15}$
$4 + 8 \frac{4}{13}$
$7 \frac{5}{6} + \frac{3}{4}$

Check answer

$8 and 7 twelfths$
$8 and 2 fifths plus 9 and 3 fourths$
$4 and 1 sixth minus 3 and 2 sevenths$

Check answer

$\frac{37}{42}$
$6 \frac{3}{5} + 7 \frac{13}{15}$
$6 \frac{7}{18} - 2 \frac{1}{2}$

Check answer

$3 and 8 ninths$
$8 and 5 ninths minus 7 and 5 sixths$
$4 \frac{7}{10} - 1 \frac{3}{14}$

Check answer

$3 \frac{17}{35}$
$12 and 4 sevenths plus 3 and 2 thirds$
$16 plus 8 and 3 elevenths$

Check answer

$24 and 3 elevenths$
$5 \frac{5}{6} + \frac{5}{7}$
$5 and 3 elevenths minus 1 and 7 twelfths$

Check answer

$3 \frac{91}{132}$
$8 minus 3 and 1 fifth$
$9 - 7 \frac{2}{15}$

Check answer

$1 \frac{13}{15}$
$2 and 3 eighths plus 7 and 11 twelfths$
$12 minus 3 and 5 elevenths$

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$8 and 6 elevenths$
$16 \frac{1}{6} - 7 \frac{14}{15}$
$10 \frac{4}{13} + \frac{9}{10}$

Check answer

$11 \frac{27}{130}$
$14 \frac{7}{15} - 3 \frac{5}{6}$
$24 minus 18 and 7 sixteenths$

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$5 and 9 sixteenths$
$1 \frac{19}{20} + 15 \frac{7}{18}$
$35 + 21 \frac{13}{17}$

Check answer

$56 \frac{13}{17}$
$\frac{15}{16} + 9 \frac{2}{5}$
$5 \frac{1}{10} - 2 \frac{13}{15}$

Check answer

$2 \frac{7}{30}$
$17 - 4 \frac{3}{13}$
$14 minus 9 and 1 sixth$

Check answer

$4 and 5 sixths$
During a triathlon, Jeremy swam $2 and 1 third$ miles, biked $97 and 1 fourth$ miles, and ran $22 and 1 half$ miles. What was the total distance he traveled swimming, biking and running? Write your answer as a mixed number.
In chemistry class, Lacy has a bottle of $2 and 1 fourth$ liters of hydrochloric acid. If she used $1 and 1 third$ liters in a chemical reaction experiment, how much is left in the bottle?

Check answer

$\frac{11}{12}$ Liter
Marcy is a pastry chef at a restaurant. At the start of the day, Marcy had $45 and 1 third$ cups of flour. While baking, she has used $18 and 3 fourths$ cups of flour. How much flour is left?

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.