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Average
It is often necessary to describe a data set with only one value. For example, instead of listing all the marks on a report card, only their average is indicated.
Example 1
If your checked-in luggage is over \latex{ 20 } \latex{ kg }, you must pay a surcharge. Agatha’s luggage is \latex{ 26 } \latex{ kg }, Carl’s is \latex{ 12 } \latex{ kg }. Agatha argues that the average mass of their luggage does not exceed the limit.
a) Is Agatha right?
b) What is the mass of the items Agatha has to put into Carl's luggage so that their luggage has the same mass?
b) What is the mass of the items Agatha has to put into Carl's luggage so that their luggage has the same mass?
Solution (a)
If the sum of the masses of the luggage is divided equally, then \latex{(26 + 12) \div 2 = 38 \div 2 = 19} (\latex{ kg }).
This is the average of the masses of the luggage:
This is the average of the masses of the luggage:
\latex{(26+12)\div2=19} (\latex{ kg }).
The average mass of the luggage is the sum of their masses divided by two.
The average mass of the luggage does not exceed the \latex{ 20 }-\latex{ kg } limit, so Agatha was right.
Solution (b)
\latex{ 26 }
\latex{ 19 }
\latex{ 12 }
\latex{26-19 = 7} (\latex{ kg }), \latex{19 - 12 = 7} (\latex{ kg }), so a total of \latex{ 7 } \latex{ kg } of items must be moved from Agatha's luggage to Carl's, so that they have the same mass.
Agatha's luggage is \latex{ 7 } \latex{ kg } heavier than the average, which is exactly the same value by which Carl's luggage is lighter than the average.
average
\latex{ 12 }
\latex{ 19 }
\latex{ 26 }
\latex{+7 }
\latex{-7 }
\latex{12+26=19+19}
The average of two numbers is their sum divided by two.
average
of \latex{a}, \latex{b}:
\latex{(a+b)\div2}
of \latex{a}, \latex{b}:
\latex{(a+b)\div2}
The average of two numbers multiplied by two is their sum.
\latex{2\times19=12+26}
Example 2
Kate bought \latex{ 2 } tomatoes on three consecutive days at the market. On the first day, the two tomatoes cost ¢\latex{ 25 }; on the second, ¢\latex{ 20 }; and on the third day, ¢\latex{ 15 }. Judith bought tomatoes for ¢\latex{ 20 } each day. Who paid more for the tomatoes in total?
Solution
Kate paid: \latex{¢ 25+¢20+¢15=60}.
Judith paid: \latex{3\times¢20=60}.
Thus, they both paid the same amount for the tomatoes.
The average of the prices paid by Kate is \latex{(25+20+15)\div3 = 60\div3 = 20}(¢), the same price Judith paid every day.
On average, Kate paid ¢\latex{ 20 } every day, so she spent the same amount as if she had bought tomatoes for ¢\latex{ 20 } each day.
The average of three numbers is their sum divided by three.
average of \latex{ a, b, c } :
\latex{(a+b+c)\div3}
Example 3*
The table below shows the Mathematics and Literature marks of a \latex{ 5 }th-grade class of \latex{ 24 } students. (\latex{ 1 } is the lowest mark, \latex{ 5 } is the highest.)
Marks
Maths
Literature
\latex{ 5 }
\latex{ 4 }
\latex{ 3 }
\latex{ 2 }
\latex{ 1 }
\latex{ 9 } students
\latex{ 8 } students
\latex{ 10 } students
\latex{ 9 } students
\latex{ 2 } students
\latex{ 6 } students
\latex{ 2 } students
\latex{ 1 } students
\latex{ 1 } students
\latex{ 0 } students
Which subject has a higher average of marks in this class?
Solution
Calculate the average of marks in Mathematics first.
\latex{(5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 4 + 4 + 4 + 4 + 4 + 4 + 4 +}
\latex{+ 4 + 4 + 4 + 3 + 3 + 2 + 2 + 1) \div 24 = (5 \times 9+4 \times 10 + 3 \times 2 +}
\latex{+ 2 \times 2 + 1) \div 24 = (45 + 40 + 6 + 4 + 1) \div 24 = 96 \div 24 =} \latex{ 4}
\latex{+ 4 + 4 + 4 + 3 + 3 + 2 + 2 + 1) \div 24 = (5 \times 9+4 \times 10 + 3 \times 2 +}
\latex{+ 2 \times 2 + 1) \div 24 = (45 + 40 + 6 + 4 + 1) \div 24 = 96 \div 24 =} \latex{ 4}
The average of marks in Mathematics in this class is \latex{ 4 }.
Calculate the average of marks in Literature.
\latex{(5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 +}
\latex{+ 4 + 3 + 3 + 3 + 3 + 3 + 3 + 2) \div 24 = (5 \times 8 + 4 \times 9 + 3 \times 6 +}
\latex{+ 2) \div 24 = (40 + 36 + 18 + 2) \div 24 = 96 \div 24 =} \latex{ 4}
\latex{+ 4 + 3 + 3 + 3 + 3 + 3 + 3 + 2) \div 24 = (5 \times 8 + 4 \times 9 + 3 \times 6 +}
\latex{+ 2) \div 24 = (40 + 36 + 18 + 2) \div 24 = 96 \div 24 =} \latex{ 4}
The average (or arithmetic mean) of several numbers can be calculated by dividing the sum of the numbers by the total number of values.
Exercises
{{exercise_number}}. Gabe spent €\latex{ 17 } on the first \latex{ day } of a two-\latex{ day } trip and €\latex{ 13 } on the second. How much did he spend on average each \latex{ day? }
{{exercise_number}}. Pete got €\latex{ 30 } per \latex{ day } for a two-\latex{ day } trip. How much can he spend on the second \latex{ day } if he spent €\latex{ 18 } on the first \latex{ day? }
{{exercise_number}}. David scored \latex{ 17, 26 } and \latex{ 23 } points in three consecutive basketball games. What was his average score for the three games?
{{exercise_number}}. Peter has written two maths tests this month. He scored \latex{ 40 } points on the first and \latex{ 80 } on the second. How many points does he need on the next test if he wants a monthly average of \latex{ 60 } points? (He does not write any more tests this month.)
{{exercise_number}}. Carl’s backpack weighs \latex{ 20 } \latex{ kg }. The backpack itself is \latex{ 2 } \latex{ kg }, and Carl's stuff in it is \latex{ 6 } \latex{ kg }. In addition to this, he is carrying three gift packs for his friends.
a) What is the average mass of the gift packs?
b) How much can each pack weigh knowing that their mass expressed in \latex{ kilograms } is a whole number?
b) How much can each pack weigh knowing that their mass expressed in \latex{ kilograms } is a whole number?
{{exercise_number}}. Two pirates found gold bars in a cave on an island. They calculated that the average number of gold bars found by the two of them was \latex{ 16 }. How many gold bars did each of them find if one of them left the cave with three times as many as the other? Draw a line for each gold bar.
{{exercise_number}}. The chart shows the amount of precipitation fallen in November and December. Colour the column in the middle to show the average rainfall. (→)
Dec.
Nov.
average
\latex{ 100 }
\latex{ 50 }
\latex{ mm }
\latex{ 120 }
\latex{ 100 }
\latex{ 80 }
\latex{ 60 }
\latex{ 40 }
\latex{ 20 }
\latex{ 0 }
{{exercise_number}}. On the chart, the red line shows the change in temperature, and the blue bars represent the amount of precipitation.
- Use the chart to calculate the average rainfall in spring, summer, autumn and winter.
- What is the difference in precipitation between the wettest and driest months?
- Estimate the difference between the average temperature of the hottest and coldest months (annual mean temperature range). (→)
\latex{ °C }
\latex{ 30 }
\latex{ 20 }
\latex{ 10 }
\latex{ 0 }
\latex{ mm }
\latex{ 150 }
\latex{ 100 }
\latex{ 50 }
\latex{ 0 }
Jan.
Mar.
May
Jul.
Sep.
Nov.
Quiz
The average age of five cousins is \latex{ 12 } \latex{years}. The age of each cousin is a whole number. How old is the oldest if none of them has the same age and no one is younger than \latex{ 10 } \latex{years?}
