Evaluate and simplify three-eighths times nine sixteenths plus five sixteenths times one-half plus three-eighths times one-eighth plus five sixteenths times negative one-eighth using the distributive property.
The first thing we do is copy down this expression, and then we think about the distributive property. In order to use the distributive property, you’ll need to find a common factor in your terms. As we scan the terms, we see that three-eighths occurs twice. We also see a one-eighth and a negative one-eighth. As well, we see two terms that have five sixteenths as a factor. But there aren’t any factors that are shared by all four terms. And that means we might have to use the distributive property more than once. So let’s use three-eighths and five sixteenths to see if this can help us.
Before we use the distributive property here, though, we want to regroup these terms. We want to group the terms that have the three-eighths together and the terms that have the five sixteenths together. At this stage, all we’ve done is rearrange the terms. So we’ve switched the position of the second and third terms. And now, we’re ready to use the distributive property. Our first two terms had a factor of three-eighths. If we undistribute that three-eighths, we’ll be left with nine sixteenths plus one-eighth. Following that same procedure, we now want to undistribute five sixteenths from the third and fourth term. That will give us five sixteenths times one-half plus negative one-eighth. Make sure you have that negative with your one-eighth.
At this stage, we can add or subtract what’s in the parentheses. We have nine sixteenths plus one-eighth. In order to add nine sixteenths and one-eighth, we need to rewrite one-eighth as two sixteenths. And in order to add one-half to negative one-eighth, we’ll rewrite one-half as four-eighths. And instead of adding a negative one-eighth, we can just write this as subtract one-eighth. This gives us three-eighths times eleven sixteenths plus five sixteenths times three-eighths. Four-eighths minus one-eighth is three-eighths.
At this point, we can actually use the distributive property a third time. Three-eighths is still a factor of both of our terms. And if we undistribute this three-eighths, we’ll have three-eighths times eleven sixteenths plus five sixteenths. Eleven sixteenths plus five sixteenths is sixteen sixteenths, which is one. And three-eighths times one is three-eighths. This means that the expression we started with after it’s been simplified completely and evaluated is three-eighths.