The results of a survey show that six-sevenths of the population of a certain town jog regularly. Of those, a half jog along the river. Which of the following expressions represents the fraction of people that jog along the river? And the choices we’ve got are A) six-sevenths plus a half, B) six-sevenths minus a half, C)
six seven [six-sevenths] times a half, or D) six-sevenths divided by a half.
Now if we represent the population of that town by this rectangle and we divide it into seven equal parts, an astonishing six-sevenths of the people in the town regularly jog.
Well that’s one-seventh two-sevenths three-sevenths four-sevenths five-sevenths six-sevenths.
So six-sevenths is nearly all of the population. If I added a half to that, that’s another three and a half boxes, that would take me over the entire population of the town, so that’s not right.
Now if we look at six-sevenths divided by a half, that’s really saying how many times does a half go into six-sevenths. Well since six-sevenths is bigger than a half, there’s more than one half in six-sevenths, so the fraction that we’d get is actually bigger than one. That’s more than the whole population of the town again, so that can’t be right.
So that means we’ve got to choose between six-sevenths minus a half and six-sevenths times a half. Now we look at the question again; we know that six-sevenths of the population, of the whole population, of the town jog regularly, and then it says of those a half jog along the river. So we’re looking for a half of the six-sevenths.
So the pink shaded region represents that proportion who jog along the river. Let’s look at option B six-sevenths minus a half. So we’re talking about six-sevenths of the whole population minus a half of the whole population. Well that’s not quite right. We only want to get rid of the half of the six-sevenths who don’t jog along the river, so that was not gonna be right either.
Now option C perfectly describes our situation. We’ve identified the six-sevenths of the population who regularly jog, and we want to identify half of those.
So multiplying by a half is the same as dividing by two, and that’s exactly what we’ve done here. We want to take half of the six-sevenths, not half of the whole population, so there’s our answer, C.