In math, there are lots of ways to do the same thing. This can be a source of frustration or confusion. However, if you embrace the dynamics of math, you can use it to your advantage. When we recognize that there are several methods to arrive at the same solution, we can choose the course of "least resistance". Consider the following algegra problem: (1/2)x + (2/3) = (5/6)x - (1/8)
Yuck! FRACTIONS!! Look at the following two solutions. We get the same answer either way. Does one seem easier to you?
First approach: deal with the fractions
We have variable terms on both sides of the equation. Let's subtract (1/2)x from both sides. (1/2)x - (1/2)x+ (2/3) = (5/6)x - (1/2)x- (1/8)
Now, let's simplify. We need to combine the like variable terms. (5/6)x - (1/2) x = (1/3)x [see below for details]*
This leaves 2/3 = (1/3)x - (1/8)
We are trying to isolate the variable, x. So, let's add (1/8) to both sides.
2/3 + 1/8 = 1/3 x [For details of adding...
A student recently asked me to help with the following problem involving rates: Suppose you sell nuts at $16 per pound and dried fruit at $4 per pound. You want to create 8 pounds of mix that you will sell for $12 per pound. How much should you use of each?
There are several strategies for solving such problems. I think that it is helpful to organize the problem in a table. Keep in mind that to figure the price of a product, we multiply the amount by the rate.
Product Amount * Rate = Price
Nuts x $16/# 16x
Fruit y $4/# 4y
Mix 8 $12/# 8x12
So, we know that the total amount of mix is 8 pound. That implies that if we add the amount of nuts and fruit, we need to end up with 8 pounds total. We can write an equation for the total amount: x + y = 8, and if we solve the equation for y, we have y = 8 -x.
Next, lets write an equation for the total price. For that, we add the price of each component. So, 16x + 4y = Total Price. We have already decided that...