Sophia is painting the fence around her house. She knows that she can paint 69 planks of fence using a gallon of paint. If she has three quarts of paint, how many planks can she paints?
This is the sort of problem where drawing a bar model can really help us understand what we need to do. So, we know that Sophia is painting a fence. And we’re told that a gallon of paint, let’s let this bar represent one gallon, can be used by Sophia to paint 69 planks of her fence. And so, at the moment, our bar model shows one gallon being equal to 69 planks. The question goes on to ask us, how many planks can Sophia paint if she has three quarts of paint?
Now, we know that four quarts are equal to one gallon. They’re literally a quarter of a gallon. And so, we need to use the facts that we already know to calculate how many planks Sophia can paint if she has three quarts of a gallon. Now, there are two ways we could find out the answer here. And both of them involve finding out how many planks Sophia could paint if she had one quart of paint.
Once we know what one of these bars is worth, we could either subtract it from one gallon. And that would leave three quarters, or three quarts. Or we could multiply it by three. And again, that would give us the number of planks that Sophia can paint using three quarts of paint. So, whichever method we choose to use, we’re still gonna have to begin by finding one quarter of 69.
Let’s use short division to help us find the answer. How many fours are there in six? There’s one four in six. And there are two left over, so there’s a remainder of two. Now, how many fours are there in 29? Let’s see how close we can get. Four, eight, 12, 16, 20, 24, 28, that’s seven fours. But we have a remainder of one. Now, let’s think about what that remainder means because we can write this in a way that will help us answer the question.
We were dividing by four, so our remainder of one is one out of a possible four. Now, because we’re talking about planks of wood here, we could actually write one out of a possible four as one-quarter. That would make a little bit more sense and help us to answer the question too. For each quarter of a gallon of paint, Sophia is going to be able to paint 17 and a quarter planks of fence.
Now, as we’ve said already, there are two different ways that we could use this value of 17 and a quarter to help us answer the question. We could either subtract it from 69. This would leave us with the number of planks that Sophia could paint using three-quarters of a gallon. Or we could multiply it by three. Both methods will give us the same answer, but let’s use the second one.
And to help us multiply 17 and a quarter by three, let’s partition this mixed number. We could split it into 10, seven, and then the fraction on the end, the quarter. We know 10 times three equals 30. If we count in seven three times, we can find out the answer to seven times three, seven, 14, 21. And finally, one-quarter times three equals three-quarters. By writing the fraction all the way over on the right there, we can now add up the columns as if it was a column addition. The fraction on the end will be three-quarters. Zero ones plus one one equals one one. And three tens plus two tens equals five tens.
We’ve multiplied each part by three and then combined them all back together again. First of all, we divided 69 by four to find the number of planks that Sophia could paint using one-quarter of a gallon, or one quart. And we said that she could paint 17 and a quarter planks. We then multiplied this amount by three to find the number of planks that Sophia can paint if she has three quarts of paint. The number of planks is 51 and three-quarters.