# What does “maximum should be at least” mean?

I found this phrase used in a question in a competitive examination.

The maximum value of X should be at least ...

("X" and the rest of the question isn't relevant here, so I am omitting it.)

What does this statement mean? It seems inherently inconsistent (How could something be the least and the maximum at the same time?), but I could be wrong.

• What kind of the competition is it? Were you supposed to answer that "at least" value? (If that's the case, then it would sound like a quick estimation contest to me.) – Damkerng T. Jan 18 '14 at 13:34

It does sound rather odd, but it is actually correct. Imagine that you are using a car to travel through the desert. On most compact cars (in the US, anyway) the fuel tank is 10-12 gallons (~37 - ~45 litres.) Generally speaking, this is enough to travel ~350 miles (560 km.) Thus, the maximum amount of fuel will take the car a certain distance.

But let us say that the desert is 500 miles (800 km) across, with no fueling stations anywhere in that distance. The maximum amount of fuel the car can hold will not be enough to make it across.

Therefore, in order to have even a chance of successfully crossing the desert, the car's maximum amount of fuel must be at least 15 gallons (55 litres.) If the maximum does not meet this minimum standard, the car does not have even the possibility of a successful crossing.

Because of the confusion that can result from this sentence construction, it's usually better to phrase it something like this:

The maximum must be larger/greater than. . .

1) at least is very differently than the least. At least means bigger than a specific value, while the least means the smallest.
2) In your task, you have to show "the maximum should be at least". This might occur in several possible situations: the maximum is impossible to be calculated (science is unable to do that), the maximum can be calculated but the exam takers do not have the ability to do it (e.g. too advanced), the exam takers have ability but there is no time to do it during the exam period. Since you yet want to gain some information from the hypothesis, even if you cannot calculate the maximum, you show at least that it is bigger than a certain value.
Another possibility is the maximum can be calculated in a relative time, but the exam wants to test the ability to apply a certain analytic procedure (e.g. known inequality), a procedure which will take less time to apply than calculating the maximum.

It a nutshell, your maximum value should -also- be higher than a specified threshold, before being taken into account or being considered as useful.

Here is a simple example of how this can make sense:

Choose three numbers from the range 0 to 100 such that the minimum is at least 9 and the maximum is at least 50.

10, 20, and 75 would be a good set. 10, 20, and 30 would not.

To start the answer of this question, I'll begin by giving you this well known quote. "Is the glass half full or Is it half empty." It's a simple concept of understanding the very distinct difference in the way you look at a particular situation. For example, the maximum cost for a 10 piece candy bar is \$10 and I have have \$20. But there are only 10 piece's left. The least I can buy is 10 piece's. But if I word it this way, at least I have \$10 to buy the maximum stock of 10 pieces. Than the maximum should be at least your minimal \$10.

• Your example has nothing to do with the actual question, and I also wonder if it was really necessary 4 years after the question has been answered. – Masked Man Mar 17 '18 at 15:22