1. ## Chinese Math PRoblem

There are 2 piles of coins. One pile contains 8 gold coins and the other 12 silver coins. The 2 piles of coins weigh the same. One coin is taken from each pile and put into the other. It is now found that the pile of mainly gold coins weighs 2 units more than the pile of mainly silver coins. Find the weight of a silver coins and a gold coin.

2. You can see that there are 7 gold coins and 1 silver coin in one stack, and 11 silver coins and 1 gold coin in another stack, now, if we call the weight of 1 gold coin x and the weight of 1 silver coin y, we get:

$\displaystyle 7x + y = 11y + x + 2$

Now, we can combine like terms:

$\displaystyle 6x -10y = 2$

And, we also know that 8 gold coins have the same weight as 12 silver coins, so we get:

$\displaystyle 8x = 12y$

Which gives us:

$\displaystyle 8x - 12y = 0$

Now we have a system of equations:

$\displaystyle 6x - 10y = 2$
$\displaystyle 8x - 12y = 0$

Solve for x and y and you will have the weights of the gold and the silver coins.

3. I keep getting a negative number

4. I keep getting a negative number - that cant be right can it?

5. Notice that you get:

x = -3 and y = -2

As answers, but also notice this:

$\displaystyle 8x = 12y$

If x and y are -3 and -2, they solve the equation, but now lets assume that x = 3 and y = 2. The equation is still solved, which says that:

$\displaystyle 8(-x) = 12(-y)$
$\displaystyle -8x = -12y$
$\displaystyle 8x = 12y$

Which means:

$\displaystyle [8(-x) = 12(-y)] = (8x = 12y)$

So, even though we get negative numbers, they give you the actual answer in positive numbers.

Using x = -3 and y = -2, we get:

$\displaystyle 8(-3) = 12(-2)$

$\displaystyle -8(3) = -12(2)$

$\displaystyle 8(3) = 12(2)$

x = 3, y = 2.