Question 78962: "If I had cows and you had cows and if I gave you one of my cows, we'd both have the same number of cows. However, if You gave me one of your cows, I'd have twice as many cows as you have. How many cows did each of us have?"
Answer by mathdoc314(58)   (Show Source):
You can put this solution on YOUR website!
well the question is about how many cows "you" have and how many cows "I" have
To avoid a lot of confusion we have to have "you" and "I" in the problem be the same person whoever is saying it. So pretend "I" is me and you on the Internet and "you" is a large alien monster in a green suit that wants cows.
If you see a 'how many' it can often signal that it can be a variable!
let M = how many cows "I" have
Y = how many cows "you" have
Now let us translate the statements in the problem into algebra:
"if I gave you one of my cows, we'd both have the same number of cows"
There is a variable you can add here to make it easier: N the number
of cows we would have after this
"I" would have 1 less cow, and "you" would have one more cow
M - 1 = N
Y + 1 = N
It is simpler to forget about N and say: If I lose a cow and you gain a cow, then our cow counts will be equal
M - 1 = Y + 1
That is the first equation!!!!! We now know Y = M-2, or "you" has two fewer cows than "me".
Next it says: "if You gave me one of your cows, I'd have twice as many cows as you have"
There are two quantities of cow mentioned here
Q1 = Y-1 how many cows You will have
Q2 = M+1 how many cows I will have
Q2 = 2 * Q1 I would have twice as many cows as you
M+1 = 2(Y-1)
Now we have turned the cow language into algebra and we can solve for M
M - 1 = Y + 1
M+1 = 2(Y-1)
Simplify these
M = Y + 2
M = 2Y - 2 - 1 = 2Y-3
Therefore: Y + 2 = 2Y - 3
Subtract Y from each side, and add 3 to each side:
Y - Y + 2 + 3 = 2Y - Y - 3 + 3
5 = Y
That means "You" has 5 cows and so "Me" has 7 cows
If "I" give "you" one then it'll be even, 6 to 6
If "you" give "me" one then it'll be double, 4 to 8
That means 5 and 7 are the right answers.
Hope this makes sense & helps you out.
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