Problem 5 - Basic Engineering Mathematics

Question

The numbers of various types of dwellings sold by a company annually over a three-year period are as shown below. Draw percentage component bar charts to present these data. \(\begin{array}{lccc} & \text { Year } 1 & \text { Year } 2 & \text { Year } 3 \\\ \text { 4-roomed bungalows } & 24 & 17 & 7 \\ \text { 5-roomed bungalows } & 38 & 71 & 118 \\ \text { 4-roomed houses } & 44 & 50 & 53 \\ \text { 5-roomed houses } & 64 & 82 & 147 \\ \text { 6-roomed houses } & 30 & 30 & 25\end{array}\)

Step-by-Step Solution

Answer

Create percentage bar charts for each year based on calculated values.

Step 1: Understand the Data

The table presents the number of various types of dwellings sold over three years, categorized as 4-roomed bungalows, 5-roomed bungalows, 4-roomed houses, 5-roomed houses, and 6-roomed houses. Each column corresponds to a year, and each row represents a dwelling type.

Step 2: Calculate Total Sales for Each Year

Add up the numbers in each column to get the total dwellings sold for each year. - Year 1: 24 + 38 + 44 + 64 + 30 = 200 - Year 2: 17 + 71 + 50 + 82 + 30 = 250 - Year 3: 7 + 118 + 53 + 147 + 25 = 350

Step 3: Calculate Percentages for Each Dwelling Type

For each type of dwelling, divide the number of sales by the total sales in that year, then multiply by 100 to get the percentage. - Year 1: - 4-roomed bungalows: (24 / 200) * 100 = 12% - 5-roomed bungalows: (38 / 200) * 100 = 19% - 4-roomed houses: (44 / 200) * 100 = 22% - 5-roomed houses: (64 / 200) * 100 = 32% - 6-roomed houses: (30 / 200) * 100 = 15% - Year 2: - 4-roomed bungalows: (17 / 250) * 100 = 6.8% - 5-roomed bungalows: (71 / 250) * 100 = 28.4% - 4-roomed houses: (50 / 250) * 100 = 20% - 5-roomed houses: (82 / 250) * 100 = 32.8% - 6-roomed houses: (30 / 250) * 100 = 12% - Year 3: - 4-roomed bungalows: (7 / 350) * 100 = 2% - 5-roomed bungalows: (118 / 350) * 100 = 33.7% - 4-roomed houses: (53 / 350) * 100 = 15.1% - 5-roomed houses: (147 / 350) * 100 = 42% - 6-roomed houses: (25 / 350) * 100 = 7.1%

Step 4: Create the Percentage Component Bar Chart

Draw three separate bars, one for each year, each divided into sections representing the percentages calculated in the previous step. The first bar (Year 1) will show segments of 12%, 19%, 22%, 32%, and 15%. The second bar (Year 2) will have segments of 6.8%, 28.4%, 20%, 32.8%, and 12%. The third bar (Year 3) will have segments of 2%, 33.7%, 15.1%, 42%, and 7.1%. Each segment should be labeled clearly, and a legend should be included to indicate what each color or pattern represents.

Key Concepts

Understanding Calculating Percentages

Calculating percentages is a crucial skill in analyzing and interpreting data. It's a simple yet powerful tool to express a number as a fraction of 100, making it easier to compare different data points. For instance, when you need to find the percentage of sales for a specific dwelling type in a year, you start by dividing the number of houses sold by the total sales for that year.
Then, multiply the result by 100 to convert it into a percentage. This process uncovers the proportion that each category contributes to the whole. With percentages, you can compare categories across different years and discern changes and trends more clearly. In our example, percentages help you see how sales of various dwelling types change over the three-year period.

Exploring Data Visualization

Data visualization is the representation of data through graphical formats. It enables quicker comprehension of large amounts of data and helps in recognizing patterns and trends.
One popular method of visualization is the use of bar charts. By visualizing data, you transform it from numbers to a story that becomes more engaging and accessible to a wider audience.
Percentage component bar charts are particularly useful because they not just show data but highlight relationships between different data series in uniform proportions. They help you look at data from a high-level view, grasp discrepancies quickly, and effectively communicate your findings.

Bar Charts Essentials

Bar charts are one of the simplest and most effective forms of data visualization.
They use horizontal or vertical bars to represent data values in comparison to one another. Each bar's length corresponds to the size or frequency of the value it represents in the dataset.
Bar charts are excellent for representing categorical data. With clear markers like differing colors or patterns, it's straightforward to separate and compare different categories or groups within the data.

• Ideal for comparing discrete categories.
• Easy to read and interpret.
• Visually appealing and informative.

Conducting Categorical Data Analysis

Categorical data analysis involves examining data that can be divided into specific categories. These categories are distinct and do not overlap.
Within the context of our exercise, the types of dwellings are the categorical variables. Analyzing them helps you understand how each category contributes to overall trends.
You'll often employ techniques such as calculating frequencies and percentages or using graphical representations like bar charts to perform this analysis. Effective categorical data analysis aids decision-making as it provides detailed insights into distinct group behaviors, performances, and trends over time.

Mastering Mathematical Problem Solving

Mathematical problem solving is about finding methods and strategies to tackle math-related challenges. It's a vital skill that transcends math and is applicable in real-world situations.
In our context, problem solving involves calculating totals, determining percentages, and creating bar charts. Each step builds on the previous one, leading to a comprehensive understanding of the whole problem.
A step-by-step approach to problem solving helps you understand data better and make informed conclusions based on quantitative insights. Key strategies include breaking down the problem into smaller, manageable parts, verifying calculations, and interpreting the results accurately.

• Identify the problem clearly.
• Decompose it into simpler tasks.
• Validate your results.

These strategies will enhance your capacity to tackle and solve mathematical problems effectively.