Farmer Joe says "I'll trade with you 3 horses for 20 pigs, 4 frogs for 1 duck, 3 cows for 8 pigs and 35 ducks for 4 horses." how many frogs for 1 cow?

Question

I forget how I can get the answer by converting all at once

Answer 1

Hi, Mathalina.

Rewriting each of these conversions as fractions will help you see the process better.

3 horses         4 frogs        3 cows       35 ducks
 20 pigs          1 duck         8 pigs         4 horses

They are asking for the number of frogs, given 1 horse.  Start by writing 1 horse over 1 (as a fraction).  Then find a fraction in the list above that has either a horse, frog, or duck in its expression.  Keep in mind, you want to have frogs at the end of the problem.  Also, it's okay to write the fractions "upside down."

1 horse         35 ducks         4 frogs
    1               4 horses        1 duck   

Now multiply these fractions together:

1 horse    •     35 ducks   •     4 frogs
    1                4 horses          1 duck

Do you see that horse cancels with horses and ducks cancels with duck?  
This leaves us with:

1   •    35    •    4 frogs
1         4              1  

When we multiply these (yes, you may cancel the 4's), we get 35 frogs.

In chemistry, this is called the "factor-label method."  I suppose it works for farm animals too. ;-)