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Question 1181832: A rectangular box has a volume of 200cm^3.
(a) find three possible whole number sets of dimensions that give the correct volume
(b) Find three sets of dimensions with at least one non-whole number that give the correct volume.
okay so for (a) I'm experienced enough to get all three sets in my head: 5*4*10, 2*10*10 and 20*2*5, which all equal to 200cm cube.
but for (b) how can I find non-whole numbers like decimals as the dimensions, are there any reliable algebraic calculations for this?
My Ideas:
So for a decimal number right, its a whatever number divide by 10 or 10 to the power of something. So for example to find a decimal number we can randomly generate a number 825 in our brain. To make this number a decimal, we divide it by 10 which equals 82.5 or we can divide by another 10 which = 8.25. Okay, with this knowledge I create these equations n = a/10 and a/10 * b * h = 200. But the things is b and h can be anything we want. So I tried a/10 * 4 * 8 = 200, a = 62.5, therefore 6.25 is the non-whole number I need.
But I realize there is a flaw in my way, because if I try 4 and 2 as b and h, then a/10*4*2=200
a = 250
which makes my number n = 250/10 = 25 which is still a whole number. Do I just use trial and error on (b and h) until I get my non-whole number or are there different ways?
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850)   (Show Source):
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
A useful trick for performing mental multiplication is to multiply one of the numbers by some constant and divide the other by the same constant.
For example, to multiply 24 times 35, you could cut the 24 in half to 12 and double the 35 to 70; then 12 times 70 is easier than 24 times 35 because one of the numbers now ends in 0.
You can use the same kind of process to find as many combinations as you want of three dimensions that give the required volume of 200.
Start with any single set of three dimensions, like your first one: 5*4*10.
You can double the 5 and cut the 4 in half to get 10*2*10; that is your second one.
Next you can double one of the 10s and cut the other in half to get 20*2*5, which is your third.
Continue doing the same kind of thing. If you want a set of dimensions that is not whole numbers, divide one of the measurements by a number that doesn't give a whole number result.
Your third set was 20*2*5. You can divide the 5 by 2 and double the 2 to get 20*4*2.5.
Then you could take that and again divide the 2.5 by 2 and double the 20 to get 40*4*1.25.
And you can continue with that process to get as many sets as you want that all give a volume of 200.
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