Mary wants to make 20 gallons of a 52% acid solution by mixing together a 84% acid solution and a 4% acid solution. how much of each solution must she use?

Step-by-step explanation:

I like to use an "X-diagram" to solve mixture problems. On the left side are the constituents of the mix; in the middle is the result of the mix; and on the right side are the differences between the numbers on each diagonal. These differences are the ratio numbers for the mix.

Here, that means the ratio of 84% solution to 4% solution is ...

  48 : 32 = 12 : 8

Note that the last two "ratio numbers" were chosen so their sum is 20, hence they represent the number of gallons of the corresponding constituent in the mix. (The sum of the first two ratio numbers is 48+32=80, so to get a sum of 20, we divide each by 4.)

Mary must use ...

You may note that this solution takes much longer to explain than to do. The math here can all be done without a calculator.

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Check

12 × 84% + 8 × 4% = 10.40 = 20 × 52%

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Usual Solution

A more conventional approach would be to assign x to the amount of 84% solution needed. Then the number of gallons of acid in the mix is ...

  0.84x + 0.04(20 -x) = 0.52(20)

  0.80x + 0.8 = 10.4 . . . . simplify

  0.80x = 9.6 . . . . . . . . . . subtract 0.8; next, divide by 0.8

  x = 9.6/0.8 = 12 . . . . gallons of 84% acid

  20-x = 8 . . . . . . . . . . gallons of 4% acid

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: