Mastering Weighted Averages for Competitive Exams
Welcome to the module on Weighted Averages! This concept is a crucial building block in quantitative aptitude for competitive exams like the CAT. Unlike a simple average, a weighted average assigns different levels of importance (weights) to different values. Understanding this distinction is key to solving a variety of problems efficiently.
What is a Weighted Average?
A weighted average is an average where each value in a dataset is multiplied by a predetermined weight before summing them up. The sum of these weighted values is then divided by the sum of the weights. This is particularly useful when some data points contribute more significantly to the overall outcome than others.
Weighted average accounts for varying importance of data points.
Imagine calculating your final grade. Exams might be worth 50%, homework 30%, and participation 20%. These percentages are the weights. A weighted average ensures that your exam performance has a greater impact on your final grade than your participation.
The formula for a weighted average is: (\text{Weighted Average} = \frac{\sum (\text{value} \times \text{weight})}{\sum \text{weight}}). In the grade example, if you scored 80 on exams (weight 0.5), 90 on homework (weight 0.3), and 70 in participation (weight 0.2), your weighted average grade would be: (\frac{(80 \times 0.5) + (90 \times 0.3) + (70 \times 0.2)}{0.5 + 0.3 + 0.2} = \frac{40 + 27 + 14}{1} = 81). This is different from a simple average of 80, 90, and 70, which would be 80.
Key Applications in Competitive Exams
Weighted averages appear in various contexts within quantitative aptitude sections:
- Mixtures and Alligations: When combining two or more ingredients with different prices or properties, the resulting mixture's average price or property is a weighted average.
- Percentage and Profit/Loss: Calculating average profit percentage when different quantities of goods are sold at different profit margins.
- Averages of Groups: When combining averages of different groups, where the size of each group acts as the weight.
The weight represents the relative importance or contribution of each value to the overall average.
Illustrative Example: Mixtures
Consider a shopkeeper who mixes two types of tea. Type A costs 40/kg. He mixes them in a ratio of 2:3. What is the average cost per kg of the mixture?
To solve this, we can consider a hypothetical quantity, say 5 kg of mixture (2 kg of Type A and 3 kg of Type B). The total cost of Type A tea is (2 \text{ kg} \times $30/\text{kg} = $60). The total cost of Type B tea is (3 \text{ kg} \times $40/\text{kg} = $120). The total cost of the mixture is (60 + \120 = $180). The total weight of the mixture is (2 \text{ kg} + 3 \text{ kg} = 5 \text{ kg}). Therefore, the average cost per kg of the mixture is (180 / 5 \text{ kg} = \36/\text{kg}). Here, the quantities (2 kg and 3 kg) act as the weights.
Text-based content
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Combined Average = (A1n1 + A2n2) / (n1 + n2)
Common Pitfalls and Tips
A common mistake is to treat all values equally when they have different weights. Always identify what the 'weights' are in a problem. They might be quantities, percentages, time durations, or even frequencies. Ensure the weights sum up correctly in the denominator.
Think of weights as multipliers that amplify the impact of a value on the final average.
Practice Problems
Practice is key! Work through various problems involving weighted averages, focusing on identifying the values and their corresponding weights. Pay attention to how different question types frame the concept of weighting.
Learning Resources
This blog post provides a clear explanation of weighted averages with examples relevant to competitive exams.
Toppr offers a concise explanation of weighted averages, including their formula and practical examples.
This article delves into weighted averages specifically for CAT aspirants, covering common question types.
Cracku provides a detailed breakdown of the weighted average concept with illustrative examples.
BYJU'S explains the definition and formula of weighted average with clear examples.
This page offers practice questions on averages, including those that involve weighted average concepts.
A visual tutorial explaining the concept of weighted average with practical examples.
This video focuses on weighted averages in the context of CAT quantitative aptitude preparation.
IndiaBIX provides a comprehensive explanation and solved examples for weighted averages.
Vedantu explains the definition, formula, and various applications of weighted averages.