Adding Negative Fractions
My son gets a behavior report every day from school. Based on how well he did in school, he gets rewarded with free time to play with his friends or to watch TV. If his report is poor, he owes me time by doing extra chores around the house.
Last week was an especially challenging week for him. On Monday, he had 3 negative comments from his teacher, so he owed me 30 minutes of chores. Tuesday was a little better, but he still owed 15 more minutes of chores. Mathematically, we would represent his total time owed as:
Total time owed in minutes
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We would add the total minutes and the result would be -45 minutes, meaning that he owes me a total of 45 minutes of chores. But this lesson is about fractions, so let's re-write our equation using hours as our units. Our equation now becomes:
Total time owed in fractions of an hour
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Notice that I placed the negative sign in the numerator. With negative fractions, the negative sign can be placed in the front of the fraction, in the numerator, or in the denominator. I find it most helpful to keep it in the numerator when doing calculations.
Let's continue with our calculation. It is important to remember that when adding or subtracting fractions, they must have a common denominator, which means that the number on the bottom of both of the fractions have to be the same. We can change the first fraction to -2/4 so that both fractions have a denominator of 4.
Fractions written with common denominators
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Next, we add the numerators and keep the denominator the same.
Example 1 completed
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Our total is -3/4 hours. So, for Monday and Tuesday, my son owed me 3/4 of an hour of chores. This is equal to the -45 minutes that we calculated earlier.
Now let's take a look at how to subtract negative fractions.
Subtracting a Negative Fraction
My college roommate was a math major, and she insisted on turning everything into a math challenge. One day I asked to borrow $10.84 for lunch. She wrote it down to make sure that I did not forget to pay her back. But instead of writing it as dollars and cents, she wrote it as a fraction.
Note showing money owed written as a fraction
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Later that week, she needed some help doing laundry. In return for helping her with her laundry, I would owe $7/2 less of what I owed her for lunch. But of course, she insisted that I work out the calculation as a fraction to figure out how much was left to pay her back. First, I needed to remember that money owed means negative values. I also needed to remember that in math, less means subtract. Next, I had to set up my equation.
Equation for example 2
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The least common multiple between 25 and 2 is 50. So, I need to multiply the first fraction by 2/2 and the second fraction by 25/25 to make 50 the common denominator.
Example 2 with common denominators
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Subtracting a negative changes to adding a positive
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I finish the calculation by adding the numerators together and keeping the denominator the same. The result is -367/50.
Let's see if this makes sense. I started out owing her $10.84 and she took away $7/2 (which is $3.50) of the debt, which means I should owe only $7.34. To convert the fraction 367/50 to a decimal, I divide 367 by 50. I end up with 7.34. I get the same answer either way, so it worked!
Lesson Summary
Working with negative numbers, or numbers that are less than zero, can be tricky and calculations with negative fractions are especially challenging. Thinking of negative values as amounts owed and remembering to find a common denominator for fractions can take away some of that frustration. Also, remember that subtracting a negative value is the same as adding a positive value.
