07+Weighted+Averages

The **weighted average** of a set of data is the sum of the product of the number of units and the value per unit divided by the sum of the number of units.
 * Mixture problems ** are problems in which two or more parts are combined into a whole. They are solved using weighted averages.

** Example 1: ** Assorted dried fruit sells for $5.50 per pound. How many pounds of mixed nuts selling for $4.75 per pound should be mixed with 10 pounds of dried fruit to obtain a trail mix that sells for $4.95 per pound?

Let x = pounds of mixed nuts
 * || ** Number of pounds ** || ** Price per pound ** || ** Total price ** ||
 * ** Dried Fruit ** || 10 || 5.50 || 55.0 ||
 * ** Mixed Nuts ** || x || 4.75 || 4.75x ||
 * ** Trail Mix ** || 10 + x || 4.95 || 4.95(10 + x) ||

55 + 4.75x = 49.5 + 4.95x 5.5 = .2x x = 27.5 27.5 pounds of nuts should be used to obtain the mixture.

Motion problems are another application of weighted averages. **Uniform motion** problems are problems where an object moves at a certain speed, or rate. The formula **//d = rt//** is used to solve these problems. In the formula, //d// represents distance, //r// represents rate, and //t// represents time.

There are four different scenarios we encounter when solving uniform motion problems.
 * ** Opposite direction ** || Add distances and set equal to distance apart. ||
 * ** Same direction ** || Subtract distances and set equal to distance apart. ||
 * ** Same distance ** || Set distances equal to each other. ||
 * ** Average speed ** || Total distance divided by total time. ||

· First, determine which type of uniform motion problem it is. //This is an average speed problem//. · The formula for average speed is   ·  In the example above, the total distance traveled is 90 miles (45 miles in both directions). The total time spent traveling is 3 hours. Now, use the formula. **//Average speed = 30 miles per hour//**
 * Example 2: ** On Alberto’s drive to his aunt’s house, the traffic was light, and he drove the 45 mile trip in one hour. However, the return trip took him two hours. What was his average speed for the round trip?

Example 3: Two trains leave Pittsburgh at the same time, one traveling east and the other traveling west. The eastbound train travels at 40 miles per hour, and the westbound train travels at 30 miles per hour. In how many hours will the trains be 245 miles apart?  ·  First, determine which type of uniform motion problem it is. This is an opposite direction problem. 40t + 30t = 245 70t = 245 t = 3.5 The two trains will be 245 miles apart in 3.5 hours.
 * || **Rate** || **Time** || **Distance** ||
 * **Eastbound** || 40 || t || 40t ||
 * **Westbound** || 30 || t || 30t ||

Let x = GCF between the 4 piles of M&Ms 2x + 7x + 9x + 11x = 464 29x = 464 x = 16 2(16) = 32 M&Ms 7(16) = 112 M&Ms 9(16) = 144 M&Ms 11(16) = 176 M&Ms
 * Example 4: Lauren has 464 M&Ms. She needs to split them up into four piles in the ratio of 2:7:9:11. How many M&Ms will be in each pile?  **

**Helpful Links:**

Uniform Motion Mixture Problems

Weighted Averages - Math Words

**Practice Problems:**