Mixture Problems Involving Percent

Wanda the witch returns from her week-long potions refresher course to find the forest around her has been recklessly chopped down! Wanda has a great idea: she'll make a potion to make the trees grow back! A tree regeneration potion calls for a mix of 40% snake skin and 60% nightshade. Wanda's sure she has some snake skin and nightshade mixtures lying around. With the ingredients she has on hand, we can solve mixture problems involving percentages and help replenish the trees. Wanda has a small jar filled with 30% snake skin and 70% nightshade. Hmm, that's a not enough snake skin for the potion. Neverthless she also has a second jar that's 50% larger than the first jar. It is half snake skin and half nightshade. Oh dear, that's too much snake skin. Wanda decides she'll just combine the two jars into one potion mixture! Will that give her the right percentages of snake skin and nightshade? Let's set-up an algebraic equation to check Wanda's reasoning. We don't know the volume of the first jar. So, we can use a variable, 'v', to represent it. We do know that the second jar contains 50% more than the first jar. This means its 1.5 times bigger than the first jar. Therefore, we can represent this as 1.5v. When Wanda combines the two mixtures, how much potion does she have in total? The volume of the resulting mixture is the sum of the first two mixtures. That's 1v plus 1.5v, giving us 2.5v. Therefore, the total volume in the potion is 2.5v, but what are the percentages of our mixture? Let's focus on snake skin first. We know the first mixture contains 30% snake skin, and the second mixture is 50% snake skin. To find the total amount of snake skin in each jar we multiply the percent as a decimal by each whole volume. Then we combined them through addition to get the percentage of the total. So the first jar is 0.30 times its whole volume, v. And the second jar is 0.50 times its whole volume, 1.5v. So to get the combined total snake skin let's simplify and add like terms. This results in 1.05v. So, 1.05v out of the total mixture, 2.5v, found earlier, is the total amount of snake skin. Simplifying, we see that this fraction results in 0.42. This means 42% of the mixture is snake skin. Since the total percentage of anything is 100%, how much is the remaining part, the nightshade? It is 100% minus 42%, which is 58%. Wanda is close, but her recipe needs to be the exact 40% to 60%, snake skin to nightshade! So, she gives her potion mixture another try. Wanda has another jar the same size as the first jar. It contains a mixture of 35% snake skin and 65% nightshade. Oh, the joys of buying matched sets! Because this jar and the first jar are the same size, what variable should we use to represent its volume? Let's use 'v,' again. Wanda mixes this jar with her earlier potion mixture. Therefore, adding 1v and 2.5v gives us 3.5v in volume of the newest mixture. So, how much of the newest potion mixture is now snake skin? Let's set-up our problem using snake skin percentages. So, the percent of volume in jar 1, plus the percent of volume in jar 2 equals the final percent of total volume. Taking the percent, 35%, of the whole volume, 'v', that's a snake skin amount of 0.35v. We already know 42% of her first potion, with volume of 2.5v, is snake skin. This can be expressed as 0.42 times 2.5v. Simplifying and combining like terms, we see Wanda's final potion contains 1.40v of snake skin. What percentage is that in relation to the total volume of 3.5v? It's 1.4v over 3.5v which simplifies to 0.40. That's 40% snake skin. Which means we have 60% nightshade. Wanda has the exact percentages she needs for her tree regeneration potion! But before she gives her brew a final stir, we should review mixture problems. When dealing with mixture problems we should follow these steps: One, identify our variable. Two, set up the problem. That is the percent of volume 1, plus percent of volume 2, equals the percent of the total volume. Lastly, you can solve for the final percentage by taking the percentage of the total volume. Then dividing it by the total volume. Now Wanda's last step is to brew her potion in a hot cauldron or maybe a quick zap in the microwave will do. Hmm. Certainly a very potent potion.