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Charts
Example 1
To perform at their best, athletes must be on a strict diet during competitions. The following charts show the recommended proportions of various nutrients:
day before
competition
competition
day of the
competition
competition
day after
competition
competition
protein (%)
fat (%)
carbohydrates (%)
10%
12%
15%
57%
62%
70%
20%
26%
28%
- Based on the charts, which types of nutrients should be increased on the day of the competition and the day after it?
- An athlete weighing \latex{ 75 \;kg } should eat \latex{ 750\; g } of food the day before the competition and \latex{ 1,000\;g } the day after.
Should this athlete consume more carbohydrates before or after the race?
Solution
- Write the information in a table.
protein (%)
fat (%)
carbohydrates (%)
day before
competition
competition
day of the
competition
competition
day after
competition
competition
increased
increased
decreased
decreased
increased
increased
\latex{ 10 }
\latex{ 20 }
\latex{ 70 }
\latex{ 62 }
\latex{ 26 }
\latex{ 12 }
\latex{ 15 }
\latex{ 28 }
\latex{ 57 }
- \latex{ 70 }% of the \latex{ 750 \,g } of food consumed the day before the competition is carbohydrates.
\latex{ 100 }% \latex{ 750\,g}
\latex{ 1 }% \latex{ 7.5\,g}
\latex{ 70 }% \latex{ 70 \times 7.5\,g = 525 \,g }
\latex{ 70 }% \latex{ 70 \times 7.5\,g = 525 \,g }
\latex{ 57 }% of the \latex{ 1,000 \,g } of food consumed after the competition is carbohydrates.
\latex{ 100 }% \latex{ 1,000\,g}
\latex{ 1 }% \latex{ 10\,g}
\latex{ 57 }% \latex{ 57 \times 10\,g = 570 \,g }
The athlete must consume more carbohydrates the day after the competition.
Example 2
Flip \latex{ 3 } coins at the same time. Write down how many (\latex{ 3; 2; 1 } or \latex{ 0 }) heads you got. Repeat this \latex{ 100 } times. Show the frequency of each outcome on a chart.
Solution
The table shows the outcomes of a series of flips.
\latex{ 3 } heads
\latex{ 2 } heads
\latex{ 1 } head
\latex{ 0 } heads
\latex{ 16 }
\latex{ 39 }
\latex{ 36 }
\latex{ 9 }
- The outcomes can be represented using a bar chart, with the \latex{ x }-axis showing the possible outcomes and the \latex{ y }-axis representing their frequencies.
\latex{ frequency }
\latex{ of } \latex{ outcomes }
\latex{ of } \latex{ outcomes }
\latex{ 40 }
\latex{ 30 }
\latex{ 20 }
\latex{ 10 }
\latex{ 0 }
\latex{ 3 } heads
\latex{ 2 } heads
\latex{ 1 } head
\latex{ 0 } heads
\latex{ outcomes }
- The outcomes can be shown in a horizontal bar chart as well.
Draw a \latex{ 10 \;cm = 100 \;mm } long bar, which corresponds to all the flips,
that is, to \latex{ 100 }%. Divide the bar into sections according to the proportion of the outcomes. \latex{ 1 \;mm } should correspond to \latex{ 1 }%.
You flipped the coin \latex{ 100 } times. One flip is \latex{\frac{1}{100}} part,
you got \latex{ 3 } heads \latex{ 9 } times, which is \latex{\frac{9}{100}}, that is, \latex{ 9 }% of all the flips;
you got \latex{ 2 } heads \latex{ 36 } times, which is \latex{\frac{36}{100}}, that is, \latex{ 36 }% of all the flips;
you got \latex{ 1 } head \latex{ 39 } times, which is \latex{\frac{39}{100}}, that is, \latex{ 39 }% of all the flips;
you got \latex{ 0 } heads \latex{ 16 } times, which is \latex{\frac{16}{100}}, that is, \latex{ 16 }% of all the flips.
\latex{9}%
\latex{36}%
\latex{39}%
\latex{16}%
\latex{0} heads
\latex{1} head
\latex{2} heads
\latex{3} heads
- The outcomes can also be represented using a pie chart.
Draw a circle with an arbitrary radius. The complete circle corresponds to \latex{ 100 } flips, that is, to \latex{ 100 }%. Divide the circle according to the frequency of the outcomes.
\latex{ 1 } flip \latex{ 1 }% \latex{ \to} \latex{ 3.6°}
\latex{ 9 } flips \latex{ 9 }% \latex{ \to} \latex{ 9\times3.6°=32.4°}
\latex{ 36 } flips \latex{ 36 }% \latex{ \to} \latex{ 36\times3.6°=129.6°}
\latex{ 39 } flips \latex{ 39 }% \latex{ \to} \latex{ 39\times3.6°=140.4°}
\latex{ 16 } flips \latex{ 16 }% \latex{ \to} \latex{ 16\times3.6°=57.6°}
\latex{ 9 } flips \latex{ 9 }% \latex{ \to} \latex{ 9\times3.6°=32.4°}
\latex{ 36 } flips \latex{ 36 }% \latex{ \to} \latex{ 36\times3.6°=129.6°}
\latex{ 39 } flips \latex{ 39 }% \latex{ \to} \latex{ 39\times3.6°=140.4°}
\latex{ 16 } flips \latex{ 16 }% \latex{ \to} \latex{ 16\times3.6°=57.6°}
\latex{ 36 }%
\latex{ 9 }%
\latex{ 16 }%
\latex{ 39 }%
\latex{ 3 } heads
\latex{ 2 } heads
\latex{ 1 } head
\latex{ 0 } heads
Use a protractor to measure the angles (rounding them to whole numbers).
Example 3
The table shows the results of a midterm maths exam. None of the students got an E or F on their exam.
grade
number of students
A
B
C
D
\latex{ 6 }
\latex{ 8 }
\latex{ 7 }
\latex{ 4 }
- What percentage of the students got an A, B, C or D on their midterm exam?
- Illustrate the data on a chart.
Solution
- Based on the table, there are \latex{ 6 + 8 + 7 + 4 = 25 } students in total.
\latex{\frac{6}{25}} of the students got an A \latex{\to} \latex{\frac{6}{25}=\frac{24}{100}} \latex{\to} \latex{ 24 }%
\latex{\frac{8}{25}} of the students got a B \latex{\to} \latex{\frac{8}{25}=\frac{32}{100}} \latex{\to} \latex{ 32 }%
\latex{\frac{7}{25}} of the students got a C \latex{\to} \latex{\frac{7}{25}=\frac{28}{100}} \latex{\to} \latex{ 28 }%
\latex{\frac{4}{25}} of the students got a D \latex{\to} \latex{\frac{4}{25}=\frac{16}{100}} \latex{\to} \latex{ 16 }%
\latex{ 24 }% of the students got an A, \latex{ 32 }% got a B, \latex{ 28 }% got a C and \latex{ 16 }% got a D on their midterm maths exam.
Illustration on a bar chart
Draw a \latex{ 10\; cm } long bar to represent the total number of students, that is, \latex{ 100 }%. Divide the bar according to the proportion of the marks.
\latex{ 1 \,mm } should correspond to \latex{ 1 }%. This way, you can compare the frequencies of the cases with each other.
\latex{ 24 }%
\latex{ 32 }%
\latex{ 28 }%
\latex{ 16 }%
Using a pie chart
Draw a circle with an arbitrary radius. The complete circle corresponds to \latex{ 100 }%. Divide the circle according to the proportion of the marks.
B
\latex{ 32 }%
\latex{ 32 }%
A
\latex{ 24 }%
\latex{ 24 }%
D
\latex{ 16 }%
\latex{ 16 }%
C
\latex{ 28 }%
\latex{ 28 }%
\latex{ 100 }% \latex{\to} \latex{ 360° }
\latex{ 1 }% \latex{\to} \latex{ 3.6° }
\latex{ 24 }% \latex{\to} \latex{24\times3.6° = 86.4°}
\latex{ 32 }% \latex{\to} \latex{32\times3.6° = 115.2°}
\latex{ 28 }% \latex{\to} \latex{28\times3.6° = 100.8°}
\latex{ 16 }% \latex{\to} \latex{{16\times3.6° = 57.6°}}
\latex{ 1 }% \latex{\to} \latex{ 3.6° }
\latex{ 24 }% \latex{\to} \latex{24\times3.6° = 86.4°}
\latex{ 32 }% \latex{\to} \latex{32\times3.6° = 115.2°}
\latex{ 28 }% \latex{\to} \latex{28\times3.6° = 100.8°}
\latex{ 16 }% \latex{\to} \latex{{16\times3.6° = 57.6°}}
\latex{86.4°+115.2°+100.8°+57.6°=360° }
Use a protractor to measure the angles (rounded to whole numbers).
Bar charts and pie charts are mainly used to compare the proportions with each other and the whole.
Exercises
{{exercise_number}}. Perform the experiment described in Example 2. Make a table and a chart based on the flips.
{{exercise_number}}. Flip two €\latex{ 2 } coins \latex{ 50 } times. Make a table and a chart showing how many times the outcome was two heads, one head and one tail, or two tails.
{{exercise_number}}. Conduct a survey in your class and write the results in a table, then create a pie chart about what percentage of your classmates
- go to school on foot, by bike, by car, by public transport (metro, tram, bus, train);
- eat lunch at school, at home, or somewhere else.
{{exercise_number}}. A newspaper editor received an article that would be published on World Water Day. Make \latex{ 3 } charts that could be used as illustrations in the newspaper.
Is water shortage a real threat?
"We need to take care of the Earth's water supplies. Gone are the days when clean and healthy drinking water was available in unlimited quantities. This has never been true for the entire planet. Seawater constitutes most of the water supplies on Earth; only \latex{3}% is freshwater. Moreover, only one-third of freshwater supplies are suitable for human consumption.
More than \latex{1.2} billion people do not have access to healthy drinking water.
When you open the tap, it is natural that drinking water flows out of it. One-fifth of the water supplies of the European Union are threatened by serious pollution; during the summer months, many member states must introduce restrictions regarding water consumption. The decrease in the groundwater level causes a serious water shortage in certain regions, threatening aquatic habitats (marshes, swamps, lakes) and agricultural production. In Southern Europe, since \latex{1985}, the area of land requiring irrigation has increased by one-fifth."
{{exercise_number}}. Who is responsible for the ozone hole? Could it be me?
At the beginning of the \latex{1990}s, humanity realised that seemingly harmless household substances can have serious environmental impacts.
It turned out that common household chemical substances damage the ozone layer, which protects life on Earth from the harmful effects of solar radiation. When these compounds, called freons, reach the upper atmosphere (an altitude of \latex{18–25} \latex{ km }), they break down the ozone layer, and the depleted layer cannot filter the Sun's rays, which are harmful to the skin, the eyes and the immune system. Today, \latex{750,000} \latex{tonnes} of freons are produced annually. The amount of freons used can be reduced if people buy freon-free refrigerators, deodorants and other products.
Uses of Freons
aerosols,
beauty products
refrigerators
vehicles,
air conditioning
air conditioning
cleaning products
foams,
packaging
packaging
other
\latex{ 2 }%
\latex{ 24 }%
\latex{ 24 }%
\latex{ 20 }%
\latex{ 15 }%
\latex{ 15 }%
Based on the chart, calculate how many \latex{ tonnes } of freons each area uses annually.
Quiz
The speed of downloading is tested on three devices. The download is initiated on all three devices at different moments. At some point, you check the screens of the devices and see the following:
\latex{ 1 }st device:
\latex{ 2 }nd device:
\latex{ 3 }rd device:
\latex{60}%
\latex{82}%
\latex{27}%
remaining time: \latex{ 1\, min \,48\, sec }
remaining time: \latex{ 3\, min \,39\, sec }
remaining time: \latex{45\, sec }
Which device is the fastest?
