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Mixture Problem in Mathematics

(Mula sa http://www.purplemath.com/modules/mixture.htm ang halimbawa)


Ang mixture problem sa Math ay itinuturing na mahirap na sagutin ng mga mag-aaral. Ngunit kung pag-aaralan lamang mabuti at maraming pagsasanay, napakadali ng problemang ito. Tunghayan ang halimbawa sa ibaba.


Suppose you work in a lab. You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30% solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix 10% solution with 30%solution, to make your own 15% solution. You need 10 liters of the 15% acid solution. How many liters of 10% solution and 30% solution should you use?

Let x stand for the number of liters of 10% solution, and let y stand for the number of liters of 30%solution. (The labeling of variables is, in this case, very important, because "x" and "y" are not at all suggestive of what they stand for. If we don't label, we won't be able to interpret our answer in the end.) For mixture problems, it is often very helpful to do a grid:


Nakatutulong na mabuti ang pagsasaayos ng mga elemento sa isang grid tulad nito:

	liters sol'n	percent acid	total liters acid
10% sol'n	x	0.10	0.10x
30% sol'n	y	0.30	0.30y
mixture	x + y = 10	0.15	(0.15)(10) = 1.5

Since x + y = 10, then x = 10 – y. Using this, we can substitute for x in our grid, and eliminate one of the variables:   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

	liters sol'n	percent acid	liters acid
10% sol'n	10 – y	0.10	0.10(10 – y)
30% sol'n	y	0.30	0.30y
mixture	x + y = 10	0.15	(0.15)(10) = 1.5

When the problem is set up like this, you can usually use the last column to write your equation: The liters of acid from the 10% solution, plus the liters of acid in the 30% solution, add up to the liters of acid in the 15% solution. Then:

0.10(10 – y) + 0.30y = 1.5  1 – 0.10y + 0.30y = 1.5  1 + 0.20y = 1.5  0.20y = 0.5  y = ^0.5/_0.20 = 2.5

Then we need 2.5 liters of the 30% solution, and x = 10 – y = 10 – 2.5 = 7.5 liters of the 10%solution. (If you think about it, this makes sense. Fifteen percent is closer to 10% than to 30%, so we ought to need more 10% solution in our mix.)

Quadratic Equation: Completing the Square

One way of solving for x in a quadratic equation is by completing the square. Here's how:

x^2 + 6x - 7 = 0 ===> ax^2 +bx + c = 0

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1) Move the constant to the right side. In the equation above, the constant is 7. You can do this by adding the negative value of the constant on both sides of the equation;

x^2 + 6x -7 + (+7) = 0 + (+7)

x^2 + 6x = 7

2) Take half the value of the x-term or b ( 6).

6/2 = 3

3) Square it.

3^2 = 9

4) Add the result to both sides.

x^2 + 6x + 9 = 7 + 9

5) Convert the left side to squared form and simplify the right side.

( x + 3)^2 = 16

6) Take the square root of both sides.

x + 3 = +/- 4

7) Solve for x

x + 3 = 4

x = 4 - 3

x = 1

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x + 3 = -4

x = -4 - 3

x = -7

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Answers:

x = 1 and -7