Jack can clean the house in five hours. But if his sister Sophie helps him, they can clean it in two hours. If the equation one over five plus one over 𝑥 equals one-half models this situation, what does the term one over 𝑥 represent?
We have the equation one over five plus one over 𝑥 equals one over two. We know that it takes them two hours to clean the house. And so, we could say that one-half represents the amount of the house that’s cleaned in one hour. Another way to say this. If it takes two hours to clean the house, in one hour, half the house will be clean. Let’s try to visualize this.
In one hour, half of the house is clean. We could break up the one-half of the house that was cleaned in one hour into two pieces: the amount that Jack cleaned plus the amount that Sophie cleaned. Jack cleaned one-fifth of the house in one hour, and Sophie cleaned a one over 𝑥 piece of the house in one hour. And so, we say that one over 𝑥 represents the amount Sophie cleaned in one hour. It’s her portion of the whole house that was cleaned in one hour.
If one over 𝑥 represents the amount that Sophie cleaned in one hour, one over five is the amount that Jack cleaned in one hour and one over two is how much of the house Jack and Sophie cleaned together in one hour.