Multiplying fractions and whole numbers word problems
You're having a movie marathon. Each movie lasts about 1 and 3/4 hours. If you want to watch 3 movies, how long will your movie marathon last? You can use the number line below to help you find your answer. So each movie lasts 1 and 3/4 hours long. So let's say the first movie happens to be Star Wars, which I probably would include in any movie marathon. So it's going to be 1 and 3/4 hours long. So this is 1,. 1, and then we're going to have 3/4 more. So each of these are a fourth because we have 4 sections between 1 and 2-- 1, 2, 3, 4. So we're going to watch 3/4. 1, 2, 3. So that's how long my first movie is. And then, my second movie is going to be another 1 and 3/4. So 1 is 4/4. So 1, 2, 3, 4. And then we're going to have another 3/4. 1, 2, 3. And just as the way I said that, 4/4 and 3/4 is 7/4. So 1 and 3/4, that's the same thing as 7/4. And you see that over here. When I did the 1 and 3/4 in blue, I actually marked off 7/4. Same thing here, here in magenta. And now, let's do it in green. So Gandhi is actually a good bit longer than 1 and 3/4, but it's a good movie, so I put it here just because it's one of my favorites. And so let's do another 1 and 3/4 hours. Or we could do that as 4/4 plus 3/4 hours, or 7/4. 1, 2, 3, 4, 5, 6, 7. So that gets us to 5. So this point right over here is 5 and 1/4 hours. Now, let's see if we get that same result when we do it mathematically. I guess this is mathematical as well. But every movie lasts 1 and 3/4 hours. We already said that 1 and 3/4 is the same thing as 1 plus 3/4. 1 is the same thing as 4/4. So it's 4/4 plus 3/4, which is the same thing as 7/4. And another way to convert this mixed number into an improper fraction, you say, hey, look. 4 times 1 is 4-- and that's really where you're getting the 4/4 from-- plus 3 is 7. 7/4. Now, we're going to do 7/4 of an hour three times. So we're going to multiply 7/4 times 3. Now, there are several ways that you could think about this. You could literally view this as having three 7/4 and adding them together. So 7/4 plus 7/4 plus 7/4, which would be equal to 21/4. And if you wanted to convert this into a mixed number, 21/4 is the same thing as-- well, 4 goes into 20 evenly. So you could call this 20/4 plus 1/4. And 20/4 is the same thing as 5. So it's 5 and 1/4, which is exactly what we saw here. Another way of thinking about it, you could say 7/4. And then we can multiply it by 3/1, times 3/1, just to make it clear that we're multiplying the numerators and multiplying the denominators. So in the numerator, I'm to have 7 plus 7 plus 7. That's 7 times 3. And in the denominator, I'm going to have 4 times 1, which is just 4. Well, this is just 7 times 3 over 4, or 21/4, which we've already seen is 5 and 1/4.