# Upper and Lower Bounds

#### mathhomework

Frances and Daniel are playing with some balance scales. Daniel wants to use in balancing the scales is 50 g. When Daniel places 600 g in weights, the scale falls, but not when he has 550 g. Illustrate on a number line the upper and lower bounds for the total mass of the cars.

#### Ragnarok

I think you didn't quite copy the question correctly...it's hard to understand what the problem is.

An upper bound is the "tipping point", so to speak: it's a value which is greater than or equal to everything else in the set. For this problem it sounds like the upper bound is 600 g. Since you can put anything up to (but not including) 600 g on the scale without it tipping, we call 600 g the upper bound (actually it is the least upper bound).

A lower bound is the opposite. It sounds like the lower bound is 50 g, but it's hard to tell from the problem. If Daniel wants to use at least 50 g, then anything above or equal to 50 g is allowable, but anything lower is not. We call 50 g the greatest lower bound.

#### mathhomework

You are right! I didn't copy the question correctly.Sorry. Here is the correct one.

Frances and Daniel are playing with some balance scales. Daniel wants to find the total mass of ten of his toy cars. The smallest weight he can find to use in balancing the scales is 50 g. When Daniel places 600 g in weights, the scale falls, but not when he has 550 g. Illustrate on a number line the upper and lower bounds for the total mass of the cars.

#### Ragnarok

I'm sorry for the late response, but maybe it will still be useful to you.

In this case the lower bound is 550 g, and the upper bound is 600 g, because you know that the mass of the car lies somewhere between 550 and 600. We know it can't be 600, because the scale falls at that point instead of balancing. (The question is a little unclear on whether the mass can be 550, but I will assume that it is not.) Because he doesn't have any smaller weights, he can't determine the weight of the car to any greater degree of accuracy than that.

The way you graph this on a number line is you draw the line segment from 550 to 600, but instead of closed circles on the endpoints, you draw open circles. Closed circles would mean that the mass could be 550 or 600 in addition to all the points between them. But we already decided that the mass can't have those values, so we draw open circles to indicate that it can be anything between them, but not the endpoints themselves.

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