Lesson % Problems and Ratios

Source code of '% Problems and Ratios'

This Lesson (% Problems and Ratios) was created by by Nate(3500)

~~~~Ratios~~~~
A ratio is represented by the following:
(a.) 1 : 2
(b.) 1/2
(c.) 1 to 2
This tells us for every one of something there are two of something else.
Let us do an example.
You have have 3 : 4 ratio of dimes to quarters respectively.
For every 3 dimes you have, you have 4 quarters.
Say you had 6 dimes. You would have 8 quarters.
6/8 = 3/4 = 3 : 4
Now, let us do a problem:
A triangle has a perimeter of 36 meters and a ratio of the sides as 2:3:4 .... what lengths are the sides?
2x : 3x : 4x
2x + 3x + 4x = perimeter = 36
9x = 36
x = 4
8 meters, 12 meters, and 16 meters
Now, lets check how far you have gotten:
You have a ratio of 3 : 2 of rubies to sapphires, and you have a ratio of 4 : 6 of sapphires to emeralds. In all you have 24 gems. How many of each do you have?
Define using the variable "x":
Rubies = 3x
Sapphires = 2x
Now, the number of sapphires (2x) can not be (4x) at the same time .... so we need to change 4 to two
(4/2) : (6/2)
2 : 3
Emeralds = 3x
3x + 2x + 3x = 24
8x = 24
x = 3
Rubies: 9
Sapphires: 6
Emeralds: 9
~~~~% Problems~~~~
A tank of gas may have, for example, 82% gasoline in it. That means that the AVERAGE of the liquids mixed together with who knows what percentages equal out to 82%.
Lets start with an equation:
{{{(l(percentage) + m(percentage))/(l + m) = total percentage of mixed liquid}}}
{{{l}}} is an amount of liquid with its respective percentage of alcohol, gasoline, acid, etc.
{{{m}}} is an amount of liquid with its respective percentage of alcohol, gasoline, acid, etc.
Now, let us do a problem:
If 2000 gallons of gas has a percentage of 92% gasoline, how many gallons of PURE gasoline will I need to add to get a 98% solution?
Pure Gasoline is 100% gasoline in respect with {{{x}}} gallons.
(2000(92%) + x(100%))/(2000 + x) = 98%
{{{(2000(0.92) + x(1.0))/(2000 + x) = 0.98}}}
{{{2000(0.92) + x = (2000 + x)(0.98)}}}
{{{1840 + x = 1960 + 0.98x}}}
-120 = -0.02x
6000 = x
6000 gallons of 100% gasoline needs to be added ....
Now, let us see if you could ever get 100% ?
(2000(92%) + x(100%))/(2000 + x) = 100%
1840 + x = 2000 + x
0 = 160
Nope ....
Now, how much of 93% gasoline solution would you need to extract to get a solution of 88% .. 
{{{(2000(0.92) - x(0.93))/(2000 - x) = 0.88}}}
{{{1840 - 0.93x = 1760 - 0.88x}}}
{{{80 = 0.05x}}}
{{{1600 = x}}}