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# Algebra: Mixture and Combination Problems

## Mixture and Combination Problems

The final type of word problem I'll introduce involves mixing two or more things together into one, larger combination. What's actually being mixed doesn't matter, because the process is the same no matter what the ingredients.

Basically, your job will be to multiply the amount of each individual ingredient by some quantity that describes it, like its concentration, and then add those products, like so:

• ingredient 1 · its value + ingredient 2 · its value = total weight · its value

Notice that you set the sum equal to the total combined weight multiplied by its descriptive value.

##### Kelley's Cautions

If the descriptive value for an ingredient is a percentage, you should convert that to a decimal when creating your equation.

Example 5: A bulk bag of trail mix, weighing 5 pounds, consists of 20% raisins by weight. The contents of the bag are mixed together with a smaller, 3-pound bag of trail mix that consists of 10% raisins by weight. What percentage of the resulting mixture is raisins?

##### You've Got Problems

Problem 4: A 10-gallon fish tank currently contains 7 gallons of water with a 1.2% saline concentration. If you want the entire tank to have a saline concentration of 2%, what saline concentration should the final 3 gallons contain?

Solution: If you weren't in algebra class, let's be honest, the correct answer to this question would be "Who cares? I don't even like raisins," but that answer doesn't get any partial credit, and it usually makes the teacher kind of angry.

In this problem the 5-pound bag will be ingredient 1, and its descriptive value is the percentage of raisins, expressed as a decimal: 20% = 0.20. Ingredient 2, then, will be the 3-pound bag, and its descriptive value will be 10% = 0.10. The total combined weight will be 3 + 5 = 8 pounds, but you don't know how many raisins are in the final mixture, so express that percentage with a variable, x. All together, your equation should look like this:

• 5(0.20) + 3(0.10) = 8(x)

Multiply the values on the left side of the equation and add them together.

• 1 + 0.3 = 8x
• 1.3 = 8x

Solve for x by dividing both sides by 8.

• 1.38 = x
• x ≈ 0.1625

To convert this decimal into a percentage, multiply it by 100; you'll get x = 16.25%. Therefore, the giant, unappetizing 8-pound barrel of trail mix is 16.25% raisins. Hope you're hungry. Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.