Conservation laws are the foundation of science and apply to the conservation of energy, conservation of mass, etc. Heres a fun example. This is a mixing problem where there are two things mixed together to make one. It often helps to represent the problem with a *before* and *after* drawing.

According to their labels, Cereal A contains 38% sugar and Cereal B contains 46% sugar.

- What percentage of sugar is in a mixture of 12 ounces (oz.) of A and 24 oz. of B?
- How many ounces of each cereal are needed to make 20 ounces with 40% sugar?

You must use a conservation law to solve this problem. A __conservation law__ means the amount of sugar before equals the amount of sugar afterwards, and assumes that nothing is lost! It assumes no sugar disappeared and all made it into the mixture.

Thus, the 1st Conservation Law is: amount of sugar in A + amount of sugar in B = total sugar in the mix

The 2nd Conservation Law is: amount of cereal before = amount of cereal afterwards (if nothing is lost).

What does 46% sugar mean? Sugar makes up a part of the cereal and grain makes up the rest.

46% sugar = = = 0.46

__Solution to Part A__

Amount of sugar = Sugar A + Sugar B

Sugar A = 38% of 12 ounces → 0.38 12

Sugar B = 46% of 24 ounces → 0.46 24

→ 0.38 12 + 0.46 24 = 15.6 ounces of sugar

Total amount of sugar = 12 ounces + 24 ounces → 36 ounces

15.6 / 36 = 0.4333 = 43.3% sugar

__Solution to Part B__

**Step 1:** Introduce some variables

x = the number of ounces in A; y = # of ounces in B; m = # of ounces in mix.

We know m; m = 20 ounces

__Sugar__: 38% of x + 46% of y = 40% of m

0.38x + 0.46y = 0.40 20

__Amount__: x + y = 20

**Step 2:** Solve for x

x = 20 y

**Step 3: **Plug in 20 y for x in the other equation and solve for y.

0.38(20 y) = 0.46y = 0.40 20

**Step 4:** Distribute and simplify

0.38(20 y) = 0.46y = 0.40 20

→ 7.6 + - 0.38y + 0.46y = 8

→ 7.6 + 0.08y = 8

→ 0.08y = 0.4

→ y = 5

**Step 5:** Plug 5 for y into x + y = 20

→ x + 5 = 20

→ x = 15

** Solution: **Mix 5 ounces of Cereal B and 15 ounces of Cereal A to get 20 ounces of cereal with 40% sugar.

- CyberProf(TM)