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Averages and weighted average problems

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Learning objective
Compute averages and solve weighted average problems.

Introduction to Averages and Weighted Averages

In everyday life and competitive exams, understanding how to find an average is essential. The average is a way to find a single value that represents a group of numbers. It helps us understand the "central" or "typical" value in a data set. For example, if you want to know the average marks scored by students in a class or the average price of fruits bought, calculating the average gives a quick summary.

There are two main types of averages we will study:

  • Simple Average (Mean): When all data points are equally important.
  • Weighted Average: When some data points have more importance or weight than others.

Knowing the difference between these two and how to calculate each will help you solve a variety of problems, especially in exams like the Meghalaya Police SI written test.

Average (Mean)

The average, often called the mean, is calculated by adding all the values in a data set and then dividing by the number of values.

Mathematically, if you have numbers \(x_1, x_2, x_3, \ldots, x_n\), the average is:

Simple Average

\[\text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}}\]

Add all values and divide by their count

Sum of all observations = Total of all values
Number of observations = Count of values

Let's see how this works with an example.

Table 1: Calculation of Average Marks
Subject Marks Obtained
Mathematics75
English80
Science70
History65
Geography85
Total375

Here, the total marks are 375, and there are 5 subjects.

So, the average marks = \( \frac{375}{5} = 75 \).

This means the student scored an average of 75 marks across all subjects.

Weighted Average

Sometimes, not all data points contribute equally. For example, in exams, some subjects may carry more weight (importance) than others. In such cases, we use the weighted average.

The weighted average takes into account the value of each data point and its corresponding weight.

The formula for weighted average is:

Weighted Average

\[\text{Weighted Average} = \frac{\sum (\text{Value}_i \times \text{Weight}_i)}{\sum \text{Weight}_i}\]

Multiply each value by its weight, sum all, then divide by total weights

\(Value_i\) = Individual data value
\(Weight_i\) = Weight of the value 
graph TD    A[Start] --> B[Multiply each value by its weight]    B --> C[Sum all the weighted values]    C --> D[Sum all the weights]    D --> E[Divide sum of weighted values by sum of weights]    E --> F[Result is Weighted Average]

Let's understand this with an example:

  • Suppose a student scores 80 marks in a subject with weight 3, and 70 marks in another subject with weight 2.
  • The weighted average = \( \frac{(80 \times 3) + (70 \times 2)}{3 + 2} = \frac{240 + 140}{5} = \frac{380}{5} = 76 \).

This means the average marks, considering the importance of each subject, is 76.

Why Use Weighted Average?

Weighted averages are useful when data points have different levels of significance. For example:

  • Calculating GPA where courses have different credit hours.
  • Finding the average price of goods bought in different quantities.
  • Determining average speed when distances traveled at different speeds vary.

Worked Examples

Example 1: Simple Average of Marks Easy
Calculate the average marks obtained by a student in 5 subjects: 60, 70, 80, 90, and 100.

Step 1: Add all the marks: \(60 + 70 + 80 + 90 + 100 = 400\).

Step 2: Count the number of subjects: 5.

Step 3: Calculate the average: \( \frac{400}{5} = 80 \).

Answer: The average marks are 80.

Example 2: Weighted Average of Exam Scores Medium
A student scores 85 in Mathematics (weight 4), 75 in English (weight 3), and 90 in Science (weight 5). Find the weighted average score.

Step 1: Multiply each score by its weight:

  • Mathematics: \(85 \times 4 = 340\)
  • English: \(75 \times 3 = 225\)
  • Science: \(90 \times 5 = 450\)

Step 2: Sum the weighted scores: \(340 + 225 + 450 = 1015\).

Step 3: Sum the weights: \(4 + 3 + 5 = 12\).

Step 4: Calculate weighted average: \( \frac{1015}{12} \approx 84.58 \).

Answer: The weighted average score is approximately 84.58.

Example 3: Data Interpretation Using Averages Medium
The monthly sales (in INR) of a shop for 6 months are as follows:
Month Sales (INR)
January50,000
February55,000
March60,000
April65,000
May70,000
June75,000

Step 1: Add all sales: \(50,000 + 55,000 + 60,000 + 65,000 + 70,000 + 75,000 = 375,000\).

Step 2: Number of months = 6.

Step 3: Calculate average monthly sales: \( \frac{375,000}{6} = 62,500 \) INR.

Answer: The average monthly sales are INR 62,500.

Example 4: Weighted Average with Multiple Weights Hard
A trader buys 20 kg of rice at INR 40 per kg, 30 kg at INR 45 per kg, and 50 kg at INR 50 per kg. Find the average price per kg of the rice.

Step 1: Multiply each price by the quantity (weight):

  • \(40 \times 20 = 800\)
  • \(45 \times 30 = 1350\)
  • \(50 \times 50 = 2500\)

Step 2: Sum of total cost: \(800 + 1350 + 2500 = 4650\) INR.

Step 3: Total quantity: \(20 + 30 + 50 = 100\) kg.

Step 4: Calculate weighted average price: \( \frac{4650}{100} = 46.5 \) INR per kg.

Answer: The average price per kg of rice is INR 46.5.

Example 5: Average Speed Problem Using Weighted Average Hard
A car travels 60 km at 40 km/h and then 90 km at 60 km/h. What is the average speed for the entire journey?

Step 1: Calculate time taken for each part:

  • Time for first part = \( \frac{60}{40} = 1.5 \) hours
  • Time for second part = \( \frac{90}{60} = 1.5 \) hours

Step 2: Total distance = \(60 + 90 = 150\) km.

Step 3: Total time = \(1.5 + 1.5 = 3\) hours.

Step 4: Average speed = \( \frac{\text{Total distance}}{\text{Total time}} = \frac{150}{3} = 50 \) km/h.

Answer: The average speed for the entire journey is 50 km/h.

Formula Bank

Simple Average
\[ \text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
where: Sum of all observations = total of all values; Number of observations = count of values.
Weighted Average
\[ \text{Weighted Average} = \frac{\sum (\text{Value}_i \times \text{Weight}_i)}{\sum \text{Weight}_i} \]
where: Value_i = individual data value; Weight_i = corresponding weight of the value.

Tips & Tricks

Tip: Use total weight as the denominator in weighted average calculations.

When to use: Whenever weights are given, always sum the weights for the denominator instead of the number of observations.

Tip: Check if the problem can be simplified by assuming a common base or total.

When to use: In weighted average problems involving ratios or proportions.

Tip: For average speed problems, use the formula for weighted average speed based on distances, not time.

When to use: When speeds are given for different distances.

Tip: Always verify units and convert if necessary before calculation.

When to use: In problems involving measurements or currency.

Tip: Use elimination method to find missing values in average problems.

When to use: When the average changes after adding or removing data points.

Common Mistakes to Avoid

❌ Using the number of observations as denominator in weighted average instead of sum of weights.
✓ Always divide by the sum of weights, not the count of observations.
Why: Students confuse weighted average with simple average and ignore weights.
❌ Adding averages directly without considering the number of data points or weights.
✓ Calculate total sums before finding the combined average.
Why: Misunderstanding that average is not additive.
❌ Mixing up weighted average speed formula with simple average speed.
✓ Use distance-based weighted average formula for speeds, not arithmetic mean.
Why: Assuming speeds can be averaged like normal numbers.
❌ Ignoring units or currency conversions in word problems.
✓ Convert all quantities to the same unit before calculation.
Why: Leads to incorrect results due to inconsistent units.
❌ Not interpreting data tables correctly leading to wrong averages.
✓ Carefully read data and identify values and their frequencies or weights.
Why: Rushing through data interpretation questions.
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