Estimate eight-ninths times six-sevenths by rounding to zero, a half, or one.
Now we could do this calculation accurately by doing eight-ninths times six-sevenths. We might then spot that three is going to nine three times and three is going to six twice, so I’ve got eight times two over three times seven, which is sixteen over twenty-one.
But that’s not what this question is asking us to do. It’s asking us to estimate eight-ninths times six-sevenths, and it’s giving us a specific method of calculating that estimate. So this unfortunately isn’t gonna get us any marks in this question.
What we’ve got to do is take each component of the calculation and do some rounding on it before we actually carry out the calculation. So for example, eight-ninths, is it closer to zero or a half or one? And six-sevenths, is it closer to zero or a half or one?
Let’s think about eight-ninths first. So I’ve drawn a little number line here and I’ve got zero, a half, and one. Now I’m gonna express those as ninths. So zero is zero-ninths, and one is nine-ninths, and a half of nine is four and a half, so a half is four and a half ninths. Now you shouldn’t normally write fractions within fractions, but we’re gonna let that go just for now.
Now eight-ninths is nearly one; it’s clearly bigger than a half; it’s clearly not very close to zero. And I strongly suspect it’s closer to nine-ninths than it is to four and a half ninths; it’s closer to one than it is to a half, but let’s just check that.
The distance from eight-ninths to nine-ninths is one-ninth, and the distance from eight-ninths back to four and a half ninths is minus three and a half ninths. So yes it seems that it’s closer to nine-ninths or closer to one than it is to a half. So instead of doing the calculation involving eight-ninths, I’m gonna round eight-ninths to one.
So eight-ninths is approximately equal to one; that’s what that sign there means, approximately equal to. Now let’s do the same kind of logic for six-sevenths; we’ve got to round that. Six-sevenths is nearly one; it’s nowhere near zero; it’s somewhere between a half because a half will be three and a half sevenths and one which is seven-sevenths.
So being between a half and one, what I’ve gotta work out is, is it closer to a half or is it closer to one? Well the distance from six-sevenths to seven-sevenths is plus one-seventh, and the distance from six-sevenths back to three and a half sevenths is minus two and a half sevenths. So it looks like it’s closer to one than it is to a half. It’s only one-seventh away from one, whereas it’s two and a half sevenths away from a half.
So we’re gonna use the approximation of one for six-sevenths, so this calculation, eight-ninths times six-sevenths, is approximately equal to one times one and one times one is one.
So we rounded eight-ninths to one and we rounded six-sevenths to one. And then when we completed the calculation, we found that eight-ninths times six-sevenths is roughly equal to one.