Just like whole numbers can be positive or negative, fractions, decimals and percents can also be positive or negative. So far, you’ve probably only seen them as positive, either with a positive sign (+) in front of them, or no sign at all, implying that they’re positive. However, fractions, decimals, and percents can also be negative. The “rules” that we talked about in the pages on adding, subtracting, multiplying, and dividing positive and negative numbers all still apply. We’ll go through several examples of how to work with fractions, decimals, and percents when they are negative.
Negative fractions are dealt with in the same way as whole negative numbers, and can also be calculated on a number line. For example, take the following problem and solution:
This would follow the normal rule for adding positive numbers to negative numbers—start at the negative number, and count forwards (add). So, for this problem, we would find -1/4 on the number line, and count forward 3/4 from there, like this:
When we finish counting forward (adding), we would be at +2/4. Our answer is positive now, because we are on the positive (right) side of zero now. Each space we moved forward on the number line represented 1/4 rather than 1, since we were dealing with fourths. Finally, we end at 2/4 as our answer.
Let’s try another example. This one’s a little bit harder.
This problem follows the same rule as we learned for subtracting a negative number from a positive number. Instead of having subtraction of a negative, we turn the two signs into an addition sign, like this:
Then we have to add the two fractions together, like this:
Thus, we get our answer, 5/5, which simplified equals 1.
Let’s try one last example. This one is tricky so pay close attention. Here’s the problem:
Remember, we turn the minus and negative sign in front of the 2 2/3 into a plus sign, so the new problem says -1 1/3 + 2 2/3.
The red dot above -1 1/3 shows that that’s where we’re starting our problem. The blue arrow counting forward shows that we’re adding 2 2/3 to the original number. After adding 2 2/3, we find our answer to be 1 1/3. This is shown by the blue circle around the 1/3 that comes after the 1 (which means 1 1/3).
Multiplying and dividing fractions also work the same way as multiplying and dividing whole positive and negative numbers.
For example, if you had the problem
You would notice that one fraction is positive and one fraction is negative. Recalling the rules for multiplying and dividing positive and negative numbers, you would realize that a positive number times a negative number results in a negative number. Therefore, you would perform normal fraction multiplication and then make the number negative when you’re done, like this:
Negative decimals also work like whole positive and negative numbers but, as with fractions, they represent parts of numbers. A decimal problem with negative numbers may look like this:
-0.55 + 2.45 = ____
In order to complete this problem, you would follow the usual rules for adding a positive number to a negative number. The rule says that you would count forward (or add) the amount of the positive number. With decimals, this sometimes get tricky, so in order to make it easier on yourself, you can simply subtract the negative number from the positive number, like this:
2.45 – 0.55 = ____
Once you perform normal subtraction, you would get 1.90 as your final answer. This number is positive because the larger number (2.45) was positive.
Here’s another example:
-7.89 – (-3.45) = ____
In order to complete this problem, you would follow the usual rules for subtracting a negative number from another negative number. First, you change the minus and negative sign into an addition sign, so your problem reads:
-7.89 + 3.45 = ____
Normally, after this step, you would count forward (add) the second number to the first number. However, due to the decimals, it would be very difficult to count forward from -7.89. Therefore, you would simply subtract the two numbers, like this:
7.89 – 3.45 = 4.44
Now, go back the equation that says -7.89 + 3.45 = _____. Which number is larger, the negative number or the positive number? As you can see, the negative number is bigger. Therefore, our answer is going to be negative, so our final answer is -4.44.
Let’s try one more decimal problem together. This time, our problem looks like this:
-6.43 – 5.94 = ____.
In order to complete this problem, you would follow the usual rules for subtracting a positive number from a negative number. The rules say that, starting at the first negative number (in this case, -6.43) count backwards (or subtract) the second number (in this case, 5.94). However, with decimals this becomes far more difficult. Instead of following that method, you can simply add the two numbers together (6.43 + 5.94) and then, since they are both negative numbers, you will put a negative sign in front of your answer.
To solve this problem, 6.43 + 5.94 = 12.37, and then put a negative sign in front of the answer, like this: -12.37. Then, you’re done!
Very rarely will you come across negative percents; however, we will still show you a few short examples in case you do see a problem with a negative percent. In real life, negative percents are often used when dealing with money, the stock market, and value of items. For example, negative percents may come to you in word problems, like this:
The AKL stock started at $53.99 this morning. It dropped 1.59% at 10 AM but then rose 2.4% near 2 PM this afternoon. Overall, what percentage did this stock company gain or lose?
Solution: A percent that “dropped” or decreased means you have a negative percentage. A percent that “rose” or increased means you have a positive percentage. To solve this problem, you have to combine the two percentages by writing a numerical equation. This equation would say: -1.59% + 2.4% = ____. This works very similarly do adding and subtracting positive and negative decimal numbers. You would find it very hard to count forward from -1.59, so you can switch the numbers around, and subtract. Your new problem would say 2.4% – 1.59% = ____. After doing the subtraction, you would get 0.81%; this is your final answer.
Let’s try one more example of working with negative percents. Marci’s math grade started at an 80%. It dropped 14% after one test, but at the end of the semester, it rose 20%. What was Marci’s overall gain or loss?
Solution: Notice again that this involves a dropped, or decreased, percent and then an increased percent, so our equation would look like this:
-14% + 20% = ____.
In order to solve this, you can start at -14 and count forward 20, which would give you +6%. So, your answer is a 6% gain in her grade.
Here are a few problems for you to practice working with positive and negative numbers. Do them on your own, and then check them with our answers.
|1. -3/4 + 1/2||2. -4.5 – 2.8||3. 8/9 – (-2/9)|
|4. 80% + (-30%)||5. 5.9 – (-2.34)||6. -90% + 50%|
|7. -7/10 + (-1/10)||8. -9.2 – (-2.4)||9. -3/5 – 4/5|
|10. -0.35 + (-1.35)|
|1. -1/4||2. -7.3||3. 10/9 or 1 1/9|
|4. 50%||5. 8.24||6. -40%|
|7. -8/10||8. -6.8||9. -7/5 or -1 2/5|