Without evaluating each expression, decide whether eight and one-fourth divided by eight and seven-eighths is greater than or less than eight and one-fourth divided by five and three-eighths.
To do this, let’s consider another example. Imagine we had 12 divided by two. We’re taking 12 and sorting it into two even groups. Each group would have six. 12 divided by two is six. But then imagine that we took 12 and we divided it by four. There will now be four groups. And each group will have only three people in it. 12 divided by four is three. In this situation, we have the same starting amount. On the top, we’re dividing by a smaller amount. On the bottom, we’re dividing by a larger amount. In the first case, we get a larger result. And in the second case, we get a smaller result. What can we make of this? Well, the larger the divisor, what we’re dividing by, the smaller the result will be. We can represent this visually.
If this block is our starting amount, the amount we need to divide, and we divide it by two, we have a piece that is this size. If we divide that same piece by four, then one of those pieces is smaller. And then if we go another step and divide this whole piece by eight, we see that each piece is smaller still. Now, let’s add some math language here. The starting amount is the dividend. What we divide by is the divisor, and the result is a quotient. And so we can say for the same dividend when we’re starting with the same amount, the larger the divisor, the smaller the quotient.
What does that mean for us? Well, both of these division problems start with eight and one-fourth as the dividend. One is being divided by eight and seven-eighths and the other is being divided by five and three-eighths. Which one has the larger divisor. Eight and seven-eighths is larger than five and three-eighths. If we plug this information in to the example we started with, we can see that eight and one-fourth divided by eight and seven-eighths will have a smaller quotient. It will have a smaller quotient because it has a larger divisor. And so we can say that the quotient of eight and one-fourth divided by eight and seven-eighths will be less than the quotient of eight and one-fourth divided by five and three-eighths.