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QUESTION: Hello:

How is the solution determined for the following?

If 12 horses eat 20 bushels of grain in 8 days.  24 horses eat 16 bushels of grain in how many days?

I think that 1 horse eats 1.666... bushels of grain in 8/12 days.

The best way to go is to find the rates step by step.
So, if 12 horses eat 20 bushels of grain in 8 days, it will take them a lesser time of (8/20 = 0.4) days to eat a bushel. Now, it's easy to see that it will take one horse a longer time of (0.4 x 12 = 4.8) days to eat a bushel of grain, and subsequent rates can be easily found.
For instance, it will take one horse (4.8 x 16 = 76.8) days to eat 16 bushels, meaning that it will then take 24 horses a lesser time of (4.8 x 16/24 = 3.2 ) days to eat the same amount of grains.

Note that it would be wrong to say that 1 horse eats 20/12 bushels of grain in 8/12 days, this is equivalent to saying that 1 horse would eat 20 bushels of grain in 8 days which is clearly incorrect. You can't divide through in that manner because some of the quantities are inversely proportional.

Regards

---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for the reply.

If all of the quantities are directly proportional, could I divide through as I did incorrectly in my first answer?

I thank you again for your assistance.

Hi Kenneth,
In cases like this it would be unlikely that all the quantities would be directly proportional, but it would be correct to perform appropriately similar operations on those that are directly proportional while keeping the odd one(s) constant depending on the situation. For instance, given that;
12 horses eat 20 bushels of grain in 8 days
We can say that;
12 horses eat 20x bushels of grain in 8x days, where x is any positive quantity, integer or fractional.

Care has to be taken though in figuring out the proportionality type between the quantities, to avoid confusion.

Regards
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