32. Average and total calculations problems

32. Average and total calculations problems - step by step solution procedure and detailed explanation for each problem.  Give me 10 examples for each variety of problem.

Below, I’ll provide a detailed explanation of how to solve average and total calculation problems, including step-by-step solution procedures for each. I’ll then present 10 example problems for average calculations and 10 example problems for total calculations, each with a detailed solution. The explanations will be clear, concise, and structured to ensure understanding, while avoiding unnecessary complexity.

---

Average Calculation Problems

Definition and Formula

The average (or arithmetic mean) of a set of numbers is calculated by summing all the numbers and dividing by the count of numbers.

Formula:

\text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}}

Step-by-Step Solution Procedure

1. Identify the data: List all the numbers or values provided in the problem.
2. Count the values: Determine how many numbers are in the set.
3. Calculate the sum: Add all the values together.
4. Compute the average: Divide the sum by the number of values.
5. Simplify or round: If necessary, simplify the result (e.g., convert to a decimal or round as specified).
6. Verify: Double-check the sum and division to ensure accuracy.

10 Example Problems with Solutions

Example 1

Problem: Find the average of the numbers 4, 8, 12, 16, and 20.

Solution:

1. Data: 4, 8, 12, 16, 20.
2. Count: 5 numbers.
3. Sum:

   4 + 8 + 12 + 16 + 20 = 60

   .
4. Average:

   \frac{60}{5} = 12

   .
5. Simplify: The result is already simplified.
6. Verify: Sum = (60), count = (5),

   60 \div 5 = 12

   .

Answer: The average is (12).

---

Example 2

Problem: A student scored 85, 90, 78, and 92 on four tests. What is their average score?

Solution:

1. Data: 85, 90, 78, 92.
2. Count: 4 scores.
3. Sum:

   85 + 90 + 78 + 92 = 345

   .
4. Average:

   \frac{345}{4} = 86.25

   .
5. Simplify: The result is (86.25) (decimal form, as it’s a score).
6. Verify: Sum = (345), count = (4),

   345 \div 4 = 86.25

   .

Answer: The average score is (86.25).

---

Example 3

Problem: The temperatures over 5 days were 72°F, 75°F, 68°F, 70°F, and 73°F. What was the average temperature?

Solution:

1. Data: 72, 75, 68, 70, 73.
2. Count: 5 temperatures.
3. Sum:

   72 + 75 + 68 + 70 + 73 = 358

   .
4. Average:

   \frac{358}{5} = 71.6

   .
5. Simplify: (71.6) (rounded to one decimal place for temperature).
6. Verify: Sum = (358), count = (5),

   358 \div 5 = 71.6

   .

Answer: The average temperature is (71.6°F).

---

Example 4

Problem: Find the average of 10, 20, 30, 40, 50, and 60.

Solution:

1. Data: 10, 20, 30, 40, 50, 60.
2. Count: 6 numbers.
3. Sum:

   10 + 20 + 30 + 40 + 50 + 60 = 210

   .
4. Average:

   \frac{210}{6} = 35

   .
5. Simplify: The result is (35).
6. Verify: Sum = (210), count = (6),

   210 \div 6 = 35

   .

Answer: The average is (35).

---

Example 5

Problem: A car traveled 50 miles, 60 miles, and 70 miles over three days. What is the average distance traveled per day?

Solution:

1. Data: 50, 60, 70.
2. Count: 3 distances.
3. Sum:

   50 + 60 + 70 = 180

   .
4. Average:

   \frac{180}{3} = 60

   .
5. Simplify: The result is (60).
6. Verify: Sum = (180), count = (3),

   180 \div 3 = 60

   .

Answer: The average distance is (60) miles per day.

---

Example 6

Problem: The prices of 4 items are $5.50, $7.25, $3.75, and $8.00. What is the average price?

Solution:

1. Data: 5.50, 7.25, 3.75, 8.00.
2. Count: 4 prices.
3. Sum:

   5.50 + 7.25 + 3.75 + 8.00 = 24.50

   .
4. Average:

   \frac{24.50}{4} = 6.125

   .
5. Simplify: (6.13) (rounded to two decimal places for currency).
6. Verify: Sum = (24.50), count = (4),

   24.50 \div 4 = 6.125

   .

Answer: The average price is ($6.13).

---

Example 7

Problem: A team scored 120, 150, 130, 140, and 110 points in 5 games. What is their average score per game?

Solution:

1. Data: 120, 150, 130, 140, 110.
2. Count: 5 scores.
3. Sum:

   120 + 150 + 130 + 140 + 110 = 650

   .
4. Average:

   \frac{650}{5} = 130

   .
5. Simplify: The result is (130).
6. Verify: Sum = (650), count = (5),

   650 \div 5 = 130

   .

Answer: The average score is (130) points per game.

---

Example 8

Problem: The weights of 3 packages are 2.5 kg, 3.2 kg, and 4.1 kg. What is the average weight?

Solution:

1. Data: 2.5, 3.2, 4.1.
2. Count: 3 weights.
3. Sum:

   2.5 + 3.2 + 4.1 = 9.8

   .
4. Average:

   \frac{9.8}{3} \approx 3.2667

   .
5. Simplify: (3.27) (rounded to two decimal places for weight).
6. Verify: Sum = (9.8), count = (3),

   9.8 \div 3 \approx 3.2667

   .

Answer: The average weight is (3.27) kg.

---

Example 9

Problem: A worker earned $200, $250, $300, and $350 over 4 days. What is the average daily earning?

Solution:

1. Data: 200, 250, 300, 350.
2. Count: 4 earnings.
3. Sum:

   200 + 250 + 300 + 350 = 1100

   .
4. Average:

   \frac{1100}{4} = 275

   .
5. Simplify: The result is (275).
6. Verify: Sum = (1100), count = (4),

   1100 \div 4 = 275

   .

Answer: The average daily earning is ($275).

---

Example 10

Problem: The times to complete a task were 15, 20, 25, 30, and 35 minutes. What is the average time?

Solution:

1. Data: 15, 20, 25, 30, 35.
2. Count: 5 times.
3. Sum:

   15 + 20 + 25 + 30 + 35 = 125

   .
4. Average:

   \frac{125}{5} = 25

   .
5. Simplify: The result is (25).
6. Verify: Sum = (125), count = (5),

   125 \div 5 = 25

   .

Answer: The average time is (25) minutes.

---

Total Calculation Problems

Definition and Formula

The total is the sum of all values in a set. Total calculation problems often involve finding the sum directly or using the average to find the total when the number of items and the average are given.

Formula (when finding total from average):

\text{Total} = \text{Average} \times \text{Number of values}

Formula (when summing directly):

\text{Total} = \text{Value}_1 + \text{Value}_2 + \dots + \text{Value}_n

Step-by-Step Solution Procedure

1. Identify the problem type:
   • If values are given, sum them directly.
   • If average and number of values are given, multiply to find the total.
2. List the data: Write down all values or the average and count.
3. Perform the calculation:
   • For direct summing: Add all values.
   • For average-based: Multiply average by the number of values.
4. Simplify or round: Ensure the result is in the appropriate format (e.g., dollars for money, whole numbers if required).
5. Verify: Recalculate the sum or multiplication to confirm accuracy.

10 Example Problems with Solutions

Example 1

Problem: Find the total of the numbers 5, 10, 15, and 20.

Solution:

1. Type: Direct summing.
2. Data: 5, 10, 15, 20.
3. Calculation:

   5 + 10 + 15 + 20 = 50

   .
4. Simplify: The result is (50).
5. Verify:

   5 + 10 = 15

   ,

   15 + 15 = 30

   ,

   30 + 20 = 50

   .

Answer: The total is (50).

---

Example 2

Problem: A student scored an average of 80 points on 5 tests. What was the total score?

Solution:

1. Type: Average-based.
2. Data: Average = 80, number of tests = 5.
3. Calculation:

   80 \times 5 = 400

   .
4. Simplify: The result is (400).
5. Verify:

   80 \times 5 = 400

   .

Answer: The total score is (400) points.

---

Example 3

Problem: The prices of 3 items are $12.50, $15.75, and $9.25. What is the total cost?

Solution:

1. Type: Direct summing.
2. Data: 12.50, 15.75, 9.25.
3. Calculation:

   12.50 + 15.75 + 9.25 = 37.50

   .
4. Simplify: The result is (37.50).
5. Verify:

   12.50 + 15.75 = 28.25

   ,

   28.25 + 9.25 = 37.50

   .

Answer: The total cost is ($37.50).

---

Example 4

Problem: A team scored an average of 120 points per game over 6 games. What was the total score?

Solution:

1. Type: Average-based.
2. Data: Average = 120, number of games = 6.
3. Calculation:

   120 \times 6 = 720

   .
4. Simplify: The result is (720).
5. Verify:

   120 \times 6 = 720

   .

Answer: The total score is (720) points.

---

Example 5

Problem: A car traveled 100 miles, 150 miles, 200 miles, and 50 miles over 4 days. What is the total distance?

Solution:

1. Type: Direct summing.
2. Data: 100, 150, 200, 50.
3. Calculation:

   100 + 150 + 200 + 50 = 500

   .
4. Simplify: The result is (500).
5. Verify:

   100 + 150 = 250

   ,

   250 + 200 = 450

   ,

   450 + 50 = 500

   .

Answer: The total distance is (500) miles.

---

Example 6

Problem: The average weight of 4 packages is 5.5 kg. What is the total weight?

Solution:

1. Type: Average-based.
2. Data: Average = 5.5, number of packages = 4.
3. Calculation:

   5.5 \times 4 = 22

   .
4. Simplify: The result is (22).
5. Verify:

   5.5 \times 4 = 22

   .

Answer: The total weight is (22) kg.

---

Example 7

Problem: A worker earned $500, $600, and $700 over 3 days. What is the total earning?

Solution:

1. Type: Direct summing.
2. Data: 500, 600, 700.
3. Calculation:

   500 + 600 + 700 = 1800

   .
4. Simplify: The result is (1800).
5. Verify:

   500 + 600 = 1100

   ,

   1100 + 700 = 1800

   .

Answer: The total earning is ($1800).

---

Example 8

Problem: The average time to complete 5 tasks is 30 minutes. What is the total time?

Solution:

1. Type: Average-based.
2. Data: Average = 30, number of tasks = 5.
3. Calculation:

   30 \times 5 = 150

   .
4. Simplify: The result is (150).
5. Verify:

   30 \times 5 = 150

   .

Answer: The total time is (150) minutes.

---

Example 9

Problem: The temperatures over 3 days were 68°F, 72°F, and 70°F. What is the total temperature?

Solution:

1. Type: Direct summing.
2. Data: 68, 72, 70.
3. Calculation:

   68 + 72 + 70 = 210

   .
4. Simplify: The result is (210).
5. Verify:

   68 + 72 = 140

   ,

   140 + 70 = 210

   .

Answer: The total temperature is (210°F).

---

Example 10

Problem: A store sold an average of 50 items per day over 7 days. What is the total number of items sold?

Solution:

1. Type: Average-based.
2. Data: Average = 50, number of days = 7.
3. Calculation:

   50 \times 7 = 350

   .
4. Simplify: The result is (350).
5. Verify:

   50 \times 7 = 350

   .

Answer: The total number of items sold is (350).

---

Summary

• Average problems involve summing values and dividing by the count. They often require careful handling of decimals or rounding, especially for units like money or weight.
• Total problems can involve direct summing or multiplying an average by the number of items. They are straightforward but require attention to units and context.
• Each example above includes a clear step-by-step solution, verification, and a final answer in the appropriate format.