GED Mathematical Reasoning: Bar and Line Graphs

Bar graphs

  • Bar graphs organize and present data using bars, hence its name. The bars in a bar graph can run vertically (up and down) or horizontally (side to side).

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  • The title of this graph is: “Favorite Types of Music”. The horizontal (or side to side) axis is labeled “Types of Music” so each bar represents a music type favored by students. The vertical (or up and down) axis is labeled “Number of Students,” so the height of each bar represents a number of students.

 

Line graphs

  • A line graph uses a thin line to connect points that represent data. A line graph is useful for showing changes over time. If the line rises, that represents an increase over time. If the line falls, that represents a decrease.

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  • The title of the graph in example 3 is: “Temperatures in New York City”. The horizontal axis is labeled “Day.” The vertical axis is labeled “Degrees in Fahrenheit.”
  • So each point on the line represents the temperature in New York City on a certain day.

 

Example 1

A group of students were surveyed about their favorite type of music. The results of the survey are given in the bar graph below. Use the bar graph to answer the following questions.

  1. How many students chose “classical” as their favorite type of music?
  2. What is the ratio of the number of students who favored “hip hop” to the number of students who favored “jazz”? Give your answer as a fraction in lowest terms.

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In part (a) we are asked to determine the number of students that chose “classical” as their favorite type of music. To do so, we’ll first locate the bar that represents “classical” music on the horizontal axis. The number of students who favor classical music is represented by the height of that bar, so we’ll follow the top of the bar to the left side of the graph where the vertical axis is labeled.

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Notice that the top of the bar corresponds to the number eight on the vertical axis, which means eight students prefer “classical” music.

To find the ratio requested for part (b), we’ll need to follow that same process to find the number of students who favor “hip hop” and the number who favor “jazz.”

Recall that when stating a ratio, the order matters. We’ll need to place the number of students who favor “hip hop” in the numerator (since that came first in the wording) and we’ll place the number of students who favor “jazz” in the denominator.

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So based on the graph: the bar representing “hip hop” has a height corresponding to 12 students. And the bar representing “jazz” has a height corresponding to 14 students.

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Therefore the ratio of students who prefer “hip hop” to students who prefer “jazz” can be expressed as: \frac{12}{14}.

To reduce the fraction to lowest terms, we’ll divide the numerator and denominator by a common factor of 2, to arrive at a final answer of  \frac{6}{7}. This means that for every 6 students that prefer “hip hop” music, there are 7 students that prefer “jazz” music.

 

Example 2

The double bar graph below represents the number of campers that participated in various camp activities. Use the double bar graph to answer the following questions.

  1. How many total campers participated in tennis?
  2. What is the ratio of the number of boys that participated in crafts to the number of girls that participated in crafts? Give your answer as a fraction in lowest terms.

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Part (a) of example two is asking us to find the number of total campers that participated in tennis. So first, we’ll locate the activity of Tennis on the horizontal axis. Then look at the bars representing the number of campers that participated in tennis. The height of the red bar tells us that 30 girls participated in tennis, while the height of the blue bar tells us that 25 boys participated.

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And the total number of campers that participated in tennis can be found by adding these two numbers together.  30 plus 25 equals 55 campers. So a total of 55 campers participated in tennis.

Part (b) of example 2 reads:  What is the ratio of the number of boys that participated in crafts to the number of girls that participated in crafts?

Based on the height of the blue and red bars over the craft activity, 5 boys participated in crafts and 10 girls participated in crafts.

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So the ratio of the number of boys that participated in crafts to the number of girls that participated in crafts is: \frac{5}{10}.

To reduce the fraction to lowest terms, we’ll divide the numerator and denominator by a common factor of 5, to arrive at a final answer of one-half or \frac{1}{2}. This means that for every 1 boy that participated in crafts, there were 2 girls that participated in crafts.

 

Example 3

The line graph below represents the temperature (in degrees Fahrenheit) of New York City over a 6 day period. During which day (as a 24 hour period) did the temperature increase most rapidly?

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To read the line graph: Start with the horizontal axis on day 5 and follow it up to the point on the line. Using the labeling on the vertical axis, we see that the temperature reading on day 5 was 59 degrees.

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And notice that the point representing this temperature reading is sitting just below the horizontal line labeled 60. The point is conveniently labeled for us on this graph, otherwise we would need to approximate the temperature using the scale on the vertical axis.

A line graph is an effective tool for seeing changes in data over time.

For example, with just a quick glance at the line graph you can see that there is a gradual increase in temperature overall, since the line rises from start to finish. However, there is a slight decrease in temperature from day 2 to day 3 where the line falls.

Let’s go back to our example. Looking at the graph, can you answer the question asked in the example and determine during which day the temperature increased most rapidly? The answer is: From day 1 to day 2.

Visually, notice that the line segment connecting the data point from day 1 to day 2 is the steepest. Numerically speaking, the temperature increase from 43 degrees to 53 degrees is the largest temperature increase represented by the graph.

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You may run across a line graph that contains more than one line and a key will tell you what each line represents. Here is an example.

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This is a double-line graph that represents the working habits of students, ages 14 – 18. Notice that the blue line represents the number of students that did work while the pink line represents the number of student that did not work.

As you can easily see, the blue line rises with age while the pink line falls. So the double line graph allows us to see that based on the data collected, the number of students that worked increased with age, while the number of students that did not work decreased with age.

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