Business Mathematics Past Paper PDF 2022-2023
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Mapúa University Senior High School
2022
Mapúa University
Monteiro, Charlotte N.
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Summary
This is a past Business Mathematics exam paper from Mapúa University Senior High School for the 2022-2023 academic year. The paper covers various topics related to data presentation, including textual, tabular, and graphical methods. It also includes course outcome bulletins, objectives, subject matters, and learning competencies.
Full Transcript
**MATHEMATICS and ABM CLUSTER** SCHOOL YEAR 2022 - 2023 **[Business Mathematics]\ **MATH02 **Course Outcome 4** Quarter 02 Prepared by:\ Monteiro, Charlotte N. **COURSE OUTCOME BULLETIN** **Objective:** Use appropriate data presentation tools. **Subject Matters:** Lesson 10. Presentation of...
**MATHEMATICS and ABM CLUSTER** SCHOOL YEAR 2022 - 2023 **[Business Mathematics]\ **MATH02 **Course Outcome 4** Quarter 02 Prepared by:\ Monteiro, Charlotte N. **COURSE OUTCOME BULLETIN** **Objective:** Use appropriate data presentation tools. **Subject Matters:** Lesson 10. Presentation of Data Lesson 10.1. Textual Presentation Lesson 10.2. Tabular Presentation Lesson 10.3. Graphical Presentation Lesson 10.4. Application in MS Excel Lesson 10.5. Frequency Distribution Table Lesson 10.6. Histogram Lesson 10.7. Ogive Lesson 10.8. Pareto Diagram **Learning Competencies:** I. To present data in three different ways II. To present data using MS Excel III. To construct a frequency distribution table, histogram, ogive and pareto diagram **Evaluation:** *Performance Task***:** Blackboard Exam (Week 12) *Written Work:* Blackboard Exam (Week 12) What do I know? (Pre-test) ========================== 1. Ungrouped data are data that are organized, or if arranged, could only be from highest to lowest or vice versa. 2. Grouped data are data that are not organized and arranged into different classes or categories. 3. Tabulation is the process of evaporating and classifying data and arranging them in a table. 4. Tabular presentation is much concise than the text statement. 5. A stem-and-leaf plot is a table which sorts data according to a certain pattern. 6. The stem consists of the first digit number. 7. The leaf consists of the last digit. 8. The column graph consists of patterned rectangles displayed along a baseline called the y-category or the horizontal axis. 9. The bar graph is basically a column graph in which the rectangles are arranged horizontally along the y-axis. 10. The line graph connects data points with lines. What is it? =========== In any research or survey, collected data must be organized to show significant characteristics. They can be presented in different forms such as textual, tabular and graphical. Textual form, when data are presented in paragraph form. Tabular form, when data are presented in rows or columns. Graphical form, when data are presented in visual form. Data can be classified as grouped or ungrouped. ***Ungrouped*** data are data that are **not organized**, or if arranged, could only be from highest to lowest or vice versa. ***Grouped*** data are data that are **organized** and arranged into different **classes** or **categories**. 10.1. Textual Presentation -------------------------- Many people have difficulty in understanding data presented in tabular form unless there is a written explanation. In the data presentation, the author can emphasize or highlight the importance of some figures. **[Example: ]** You are asked to present the performance of your section in the Statistics test. The following are the test scores of your class. First, arrange the data in order for you to identify the important characteristics. This can be done in two ways: rearranging from lowest to highest. ![](media/image2.png) With the rearranged data, pertinent data worth mentioning can be easily recognized. The following is one way of presenting data in textual form. *In the Statistics class of 40 students, 3 obtained the perfect score of 50. 16 students got a score of 40 and above, while only 3 got 19 and below. Generally, the students performed well in the test with 23 or 70% getting a passing score of 38 and above.* Notice how the data is presented. For the first sentence, the total population (number of students) was mentioned, as well as the highest data value which is the perfect score. Then, the extremes (upper 80% of the score, 40 over 50, and lower percentage of the scores) were mentioned. Lastly, for the conclusion of the paragraph, the overall behavior of the data was noted. 10.2. Tabular Presentation -------------------------- +-----------------------------------------------------------------------+ | ***Tabulation*** is the process of **condensing** and **classifying | | data** and **arranging** them in a table. | | | | ***Tabular presentation*** is much **concise** than the text | | statement. | +-----------------------------------------------------------------------+ Here's the parts of a table. One type of tabular presentation is the stem-and-leaf plot. A ***stem-and-leaf plot*** is a table which sorts data according to a certain pattern. It involves **separating a number into two parts**. In a two-digit number, the **stem** consists of the first digit number (depends on the given data, can include the first two digits, three digits and so on), and the **leaf** consists of the last digit (place value: ones). In a one-digit number, the stem is zero. **[Example:]** Using the given example for the textual presentation: ![](media/image2.png) Separating the stem (tens digit) from the leaves (ones digit): Utilizing the stem-and-leaf plot, we can readily see the order of the data. Thus, we can say that the top ten got scores 50, 50, 50, 49, 48, 46, 46, 46, 45, and 45 and the ten lowest scores are 9, 17, 18, 20, 23, 23, 24, 25, 26, and 27. 10.3. Graphical Presentation ---------------------------- Apart from diagrams, graphic presentation is another way of the presentation of data and information. Usually, graphs are used to present time series and frequency distributions. The graphic presentation of data and information offers a quick and simple way of understanding the features and drawing comparisons. Further, it is an effective analytical tool. ### A. Column Graph The column graph consists of patterned **rectangles** displayed along a baseline called the x-category or the **horizontal** **axis**. The height of the rectangle represents the amount of data, which is evaluated using the y-category or the vertical axis. The left-to-right bias most people possess makes column graphs more **appropriate for time-series data** than bar graphs. Column graphs best show changes in data over a short time span and relationships between two or more data series. The parts of a column graph is shown in the first example. **[Examples:]** Using the following data set, construct a column graph for: 1\. Company A sales, 2011-2015 2\. Company B sales vs. net income, 2011-2015 3\. Company A vs. B vs. C net income, 2011-2013 ![](media/image5.png) 1. Company A sales, 2011-2015 2. Company B sales vs. net income, 2011-2015 ![](media/image7.png) 3. Company A vs. B vs. C net income, 2011-2013 ### Bar Graph The bar graph is basically a column graph in which the **rectangles** are **arranged horizontally along the y-axis**. The length of each rectangle represents its value, which is evaluated using the x-axis values. Bar graphs best show data series and data comparisons in competition and with **no natural order**, such as according to time, etc. The parts of a bar graph are shown in the first example. **[Examples:]** Using the earlier data set, construct a bar graph for:\ 1. Company D sales and itemized expenses, 2014\ 2. Sales per Company, 2015\ 3. Sales and net income per Company, 2013 Solution: 1. Company D sales and itemized expenses, 2014 ![](media/image10.png) 2. Sales per Company, 2015 3. Sales and net income per Company, 2013\ (Net Income is computed by subtracting cost of sales, commissions, and salaries/wages from sales) ![](media/image12.png) ### Line Graph The line graph connects data points with **lines**; different series are given different line markings (for example, dashed or dotted) or different tick marks. Line graphs best show the comparison of long series and is best used when attempting to communicate a **data trend**. **[Examples:]** Using the earlier data set, construct a line graph for: 1\. Company C sales and net income, 2011-2015 2\. Net income per Company, 2011-2015 Solution: 1. 2. ![](media/image14.png) ### Pie Chart The pie chart is a **circle** with radii connecting the center to the edge of the circle. The area between two radii is called a slice. The proportions of the data values in the pie chart to the whole are reflected in the areas of the slices. Pie charts best show the **composition** or **breakdown** of a whole. **[Examples:]** Using the earlier data set, construct a pie chart for: 1\. Breakdown of Company E expenses, 2014 2\. Breakdown of Company A total sales from 2011-2015 Solution: 1. Given the data below from 2011 (1^st^ column) to 2015 (last column). To solve for the percentages, add first all the expenses: Cost of Sales, Commissions, and Salaries. Then, divide each expense by the sum you computed previously. For example, cost of sales is equal to [\$\\frac{77,625}{121,495} = 0.64 = 64\\%\$]{.math.inline}. ![](media/image16.png) 2. Given the data below from 2011 (1^st^ column) to 2015 (last column). To solve for the percentages, add first all the sales per year. Then, divide each sales by the sum you computed previously. For example, percentage of the sales in 2012 is equal to [\$\\frac{120,000}{744,160} = 0.16 = 16\\%\$]{.math.inline}. 10.4. Application in MS Excel ----------------------------- MS Excel is an important tool for constructing data graphs for presentation. The key steps in creating graphs in MS Excel are selecting the data and creating a chart. A. **Selecting the data** First, select the cells that contain the values you want shown in the chart. Click and drag the cursor from the top left cell to the bottom right cell of the worksheet -- including column and row headings when possible. Non-contiguous rows and columns of cells can be selected by pressing and holding the Ctrl key while selecting each group of cells. A group of related data points is called a data series. Typically, each data series will be represented by a different color and will be included in the legend. B. **Creating a chart** After selecting the cells, click the Insert tab and click the chart type from the Charts section of the ribbon. A chart sub-type menu will appear. Then, click the desired chart sub-type to make the chart appear on the worksheet. **[Example:]** Create a column chart showing Company A sales from 2011-2015. 1. Plot the data series as follows: ![](media/image19.png) 2. Select all cells (including the blank cell at the top-left corner), then click on the Insert tab (boxed in red). 3. Then click the Column button and the first chart sub-type under 2-D Column (boxed in red). ![](media/image21.png) 4. A graph will appear on the worksheet 10.5. Frequency Distribution Table ---------------------------------- A frequency distribution table is a table which shows the data arranged into different classes (or categories) and the number of cases (or frequencies) which fall into each class. The following is an illustration of a frequency distribution table for ungrouped data: ![](media/image23.png) Here's an illustration a frequency distribution table for grouped data: **Lower class limits** are the smallest numbers that can actually belong to different classes. ![](media/image25.png) **Upper class limits** are the largest numbers that can actually belong to different classes. ![](media/image27.png)**Class boundaries** are the numbers used to separate classes, but without the gaps created by class limits. The **class mark** or **class midpoint** is the respective average of each class limits **Class width** is the difference between two consecutive lower class limits or two consecutive class boundaries. ![](media/image29.png) A **relative frequency table** is a type of frequency table with the relative percentages. The formula is, \ [\$\$relative\\ frequency = \\frac{\\text{class\~frequency}}{\\text{sum\~of\~all\~frequencies}}\$\$]{.math.display}\ *An example is given below.* A **cumulative frequency table** is a table with columns allotted for the cumulative frequencies. [ \ *cf*]{.math.inline}, start with the total frequency as your first row input. For the second row input, subtract the 1^st^ row frequency from the first row input. For the third row input, subtract the 2^nd^ row frequency from the second row input, and so on, until you reach the last row. ![](media/image31.png) **Guidelines for Frequency Distribution Tables:** 1. Be sure that the classes are mutually exclusive. 2. Include all classes, even if the frequency is zero. 3. Try to use the same width for all classes. Odd numbered class widths is recommended to easily find the midpoint. 4. Select convenient numbers for class limits. 5. Use between 5 to 20 classes ([*k*]{.math.inline}). 6. Use the [2^*k*^]{.math.inline} rule, where [2^*k*^]{.math.inline}≥𝑛 7. The sum of the class frequencies must equal the number of original data values. **Steps in Constructing the Frequency Distribution Table:** 1. Decide on the number of classes, [*k*]{.math.inline}. Use the [2^*k*^]{.math.inline} rule. 2. Determine the class width by dividing the range by the number of classes Range = Highest score -- lowest score (then round up the answer if not exact) Class Width = [\$\\frac{\\text{Range}}{\\text{No}\\text{.\~}\\text{of}\\ \\text{classes}}\$]{.math.inline} (round up if not exact) 3. Determine the class width by dividing the range by the number of classes 4. Add the class width to the starting point to get the second lower class limit, add the width to the second lower limit to get the third, and so on. 5. List the lower class limits in a vertical column and enter the upper class limits. 6. Represent each score by a tally mark in the appropriate class. Total tally marks to find the total frequency for each class. 7. Compute the relative frequency or percentage frequency, and the less than cumulative frequency (\cf). **[Example:]** Construct an FDT for the given data Solution: ![](media/image32.png) 10.6. Histogram --------------- A frequency distribution shows how often each different value in a set of data occurs. A histogram is the most commonly used graph to show frequency distributions. It is **similar to a vertical bar graph**. However, a histogram, unlike a vertical bar graph, **shows no gaps between the bars**. **[Example:]** The following is a list of prices (in dollars) of birthday cards found in various drug stores: 1.45 2.20 0.75 1.23 1.25 1.25 3.09 1.99 2.00 0.78 1.32 2.25 3.15 3.85 0.52 0.99 1.38 1.75 1.22 1.75 Make a histogram for this data. **Step 1:** Group the data into classes. You can use the same rule in identifying the number of classes and class width as the frequency distribution table. Decide on the number of classes, [*k*]{.math.inline}. Use the [2^*k*^]{.math.inline} rule, where [2^*k*^]{.math.inline}≥𝑛. Determine the class width by dividing the range by the number of classes **Step 2:** Count the frequencies. **Intervals (in dollars)** **Frequency** ---------------------------- --------------- 0.50 - 0.99 4 1.00 - 1.49 7 1.50 - 1.99 3 2.00 - 2.49 3 2.50 - 2.99 0 3.00 - 3.49 2 3.50 - 3.99 1 Total 20 **Step 3:** Proceed with the graph. Same procedure with column graph but there is no spaces in between the columns. The left end of the bar should be placed in the lower class limit and the right end of the bar is placed in the lower class limit of the next class. ![](media/image34.png) 10.7. Ogive ----------- Cumulative histograms, also known as **ogives**, are graphs that can be used to **determine how many data values lie above or below a particular value in a data set**. An ogive looks like a line graph going upwards since it is cumulative. **[Example:]** Determine the cumulative frequencies of the following quiz scores in Math and complete the table below. Use the table to draw an ogive of the data. **Step 1:** Compute for the cumulative frequencies. ![](media/image36.png) **Step 2:** Plot dots of the cumulative frequencies. Place them above the lower limit of their class. Then, connect the dots with a line. 10.8. Pareto Diagram -------------------- A Pareto diagram provides facts needed for setting priorities. It organizes and displays information to show the relative importance of various problems or causes of problems. It is a form of a vertical bar chart that puts items in order (from the highest to the lowest) relative to some measurable effect of interest: frequency, cost or time. The chart is based on the Pareto principle, which states that when several factors affect a situation, a few factors will account for most of the impact. The Pareto principle describes a phenomenon in which 80 percent of variation observed in everyday processes can be explained by a mere 20 percent of the causes of that variation. **How to make a Pareto diagram?** Step 1: Tally, for each item, how often it occurred (or cost or total time it took). Then, add these amounts to determine the grand total for all items. Find the percent of each item. ![](media/image39.png) **Step 2:** List the items being compared in decreasing order and compute for the cumulative percentages. **Step 3:** List the items on the horizontal axis of a graph from highest to lowest. Label the left vertical axis with the numbers (frequency, time or cost), then label the right vertical axis with the cumulative percentages (the cumulative total should equal 100 percent). Draw in the bars for each item. **Step 4:** Draw a line graph of the cumulative percentages. The first point on the line graph should line up with the top and center of the first bar. Excel offers simple charting tools you can use to make your graphs, or you can do them with paper and pencil. **Step 5:** Analyze the diagram by identifying those items that appear to account for most of the difficulty. Follow the 80-20 rule. 80% of the data is explained by the bars outside the rectangle drawn (blue lines) and 20% of the data is explained by the bars enclosed inside the rectangle. ![](media/image41.png) What's more? ============ 1. Twenty-five army inductees were given a blood test to determine their blood type. The data set is Table Description automatically generated Construct a frequency distribution for the ungrouped data. 2. These data represent the record high temperatures in degrees Fahrenheit (°F) for each of the 50 states. Construct a grouped frequency distribution for the data, using 7 classes. Include only the class, class boundaries, and frequencies. ![Table Description automatically generated](media/image43.png) What I have learned =================== In five sentences, write down the challenges you have experienced in presenting data in different formats? How did you overcome these challenges? What I can do ============= 1. Construct a histogram to represent the data shown for the record high temperatures for each of the 50 states. Table Description automatically generated 2. The table shows the average money spent by first-year college students. Draw a horizontal and vertical bar graph for the data. ![Graphical user interface, text, application Description automatically generated](media/image45.png) 3. This frequency distribution shows the number of pounds of each snack food eaten during the Super Bowl. Construct a pie graph for the data. Table Description automatically generated Assessment (Post-test) ====================== 1. Apart from diagrams, graphic presentation is another way of the presentation of data and information. 2. The graphic presentation of data and information offers a quick and simple way of understanding the features and drawing comparisons. 3. Bar graphs best show changes in data over a short time span and relationships between two or more data series. 4. Line graphs best show data series and data comparisons in competition and with no natural order, such as according to time, etc. 5. Column graphs best show the comparison of long series and is best used when attempting to communicate a data trend. 6. The pie chart is a circle with radii connecting the center to the edge of the circle. The area between two radii is called a pie. 7. Pie charts best show the composition or breakdown of a whole. 8. MS Word is an important tool for constructing data graphs for presentation. 9. A frequency distribution table is a table which shows the data arranged into different classes (or categories) and the number of cases (or frequencies) which fall into each class. 10. Lower class limits are the smallest numbers that can actually belong to different classes. References ========== **Teaching Guide for Senior High School: Business Mathematics by The Commission on Higher Education in collaboration with the Philippine Normal University, 2016** Practical Business Math Procedures. Slater. 2017. McGraw-Hill Education.