Solving a mixture problem takes precision, but you can break it down with mathematical English as well. You have to remember these steps. 1) What is being mixed? 2) Organize the information. 3) Set up gallons of the ingredient for each container. 4) Form a mixture equation. 5) Solve and Check.
Q: How many gallons of a 60% solution must be added to 30 gallons of a 10% solution in order to produce a 20% solution?
We know so far that a 60% solution is being mixed with a 10% solution. We do NOT know how much of the 60% solution we have, so that is “x”. We do have 30 gallons of the 10% solution.
The next step is to ORGANIZE the information. One jar contains the 60% solution (x gallons). Another jar contains the 10% solution (30 gallons). There is a 20% solution jar that combines the two as well (x + 30). The last jar is the place where we want to have the 20% solution created. In the first jar, we have a 60% solution that houses 60% of the gallons of one liquid. In the other jar, we have 40% of the container having other ingredients.
Now, it is time to calculate the gallons of the ingredient in each of the containers by multiplying the total gallons of EACH container by the percentage listed for the container. Next, you have 60% * x, 10% * 30, and 20% * (x+30). The verbal equation is (Ingredient in Jar 1) + (Ingredient in Jar 2) = (Ingredient in Jar 3). The equation using decimals is: .60x + 3 = .20x + 6. You would then subtract .20x from both sides and subtract 3 from both sides to get .40x = 3. Next, you divide both sides by .40 and x = 7.5.
Problem solved and you should check it by plugging the value for x back into the equation. This is how easy math can be using mathematical English. Let’s try this again in a different way.
“What is 75% of 300?”
How are we going to break this problem down? When a problem says “What”, that means “x”. The word “is” ALWAYS means “equals”. “Percentage” always means “out of 100 parts”, so the 75% converts to 75/100. “Of” means multiplication and the number 300 remains just that – a number. So, here is the updated version of this: x = (75/100) * 300. Now, 100 goes into 300 three times. So, you have 3 times 75 which equals 225. Problem solved. This is how easy math can be using mathematical English. Let’s try this again in a different way.
35 is what percent of 80?
We now solve this problem: 80 goes into 100 1.25 times. So, we have 35 = x/1.25. We then cross multiply 35 * 1.25 and get 43.75 which is our answer. X = 43.75%. As you see, this process is again very simple. Visual learners will need to break problems down by drawing pictures or sketching out the process of the problem. Auditory learners just need to hear the correct conversion of the problem. Kinesthetic learners will need to get up and act out the problem or write on the boards themselves. This simple mathematical English process can be used on more advanced problems to provide clarity to the problem solver.
文 | Jay Veal（INC Tutoring輔導公司總裁）