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Averages and Probability

Calculate the mean of a set of numbers

Calculate the Mean of a Set of Numbers

The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size.

Suppose Ethan’s first three test scores were $85,88,\text{and}\ 94.$ To find the mean score, he would add them and divide by $3.$

\[\begin{array}{l} \\ \frac{85+88+94}{3} \\ \frac{267}{3} \\ 89\end{array}\]

His mean test score is $89$ points.

Example

Try it.

Find the mean of the numbers $8,12,15,9,\text{and}\ 6.$

Solution

Write the formula for the mean:	$\text{mean}=\frac{\text{sum of all the numbers}}{n}$
Write the sum of the numbers in the numerator.	$\text{mean}=\frac{8+12+15+9+6}{n}$
Count how many numbers are in the set. There are 5 numbers in the set, so $n=5$.	$\text{mean}=\frac{8+12+15+9+6}{5}$
Add the numbers in the numerator.	$\text{mean}=\frac{50}{5}$
Then divide.	$\text{mean}=10$
Check to see that the mean is 'typical': 10 is neither less than 6 nor greater than 15.	The mean is 10.

Example

Try it.

The ages of the members of a family who got together for a birthday celebration were $16,26,53,56,65,70,93,\text{and}\ 97$ years. Find the mean age.

Solution

Write the formula for the mean:	$\text{mean}=\frac{\text{sum of all the numbers}}{n}$
Write the sum of the numbers in the numerator.	$\text{mean}=\frac{16+26+53+56+65+70+93+97}{n}$
Count how many numbers are in the set. Call this $n$ and write it in the denominator.	$\text{mean}=\frac{16+26+53+56+65+70+93+97}{8}$
Simplify the fraction.	$\text{mean}=\frac{476}{8}$
	$\text{mean}=59.5$

Is $59.5$ ‘typical’? Yes, it is neither less than $16$ nor greater than $97.$ The mean age is $59.5$ years.

Condensed — the full section is in OpenStax Prealgebra 2e.

Find the Median of a Set of Numbers

When Ann, Bianca, Dora, Eve, and Francine sing together on stage, they line up in order of their heights. Their heights, in inches, are shown in .

Ann	Bianca	Dora	Eve	Francine
$59$	$60$	$65$	$68$	$70$

Dora is in the middle of the group. Her height, $65\text{″},$ is the median of the girls’ heights. Half of the heights are less than or equal to Dora’s height, and half are greater than or equal. The median is the middle value.

What if Carmen, the pianist, joins the singing group on stage? Carmen is $62$ inches tall, so she fits in the height order between Bianca and Dora. Now the data set looks like this:

\[59,60,62,65,68,70\]

There is no single middle value. The heights of the six girls can be divided into two equal parts.

Statisticians have agreed that in cases like this the median is the mean of the two values closest to the middle. So the median is the mean of $62\ \text{and}\ 65,\ \frac{62+65}{2}.$ The median height is $63.5$ inches.

Notice that when the number of girls was $5,$ the median was the third height, but when the number of girls was $6,$ the median was the mean of the third and fourth heights. In general, when the number of values is odd, the median will be the one value in the middle, but when the number is even, the median is the mean of the two middle values.

Example

Try it.

Find the median of $12,13,19,9,11,15,\text{and}\ 18.$

Solution

List the numbers in order from smallest to largest.	9, 11, 12, 13, 15, 18, 19
Count how many numbers are in the set. Call this $n$.	$n=7$
Is $n$ odd or even?	odd
The median is the middle value.
The middle is the number in the 4th position.	So the median of the data is 13.

Condensed — the full section is in OpenStax Prealgebra 2e.

Identify the Mode of a Set of Numbers

The average is one number in a set of numbers that is somehow typical of the whole set of numbers. The mean and median are both often called the average. Yes, it can be confusing when the word average refers to two different numbers, the mean and the median! In fact, there is a third number that is also an average. This average is the mode. The mode of a set of numbers is the number that occurs the most. The frequency, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.

Suppose Jolene kept track of the number of miles she ran since the start of the month, as shown in .

If we list the numbers in order it is easier to identify the one with the highest frequency.

\[2,3,5,8,8,8,15\]

Jolene ran $8$ miles three times, and every other distance is listed only once. So the mode of the data is $8$ miles.

Example

Try it.

The ages of students in a college math class are listed below. Identify the mode.

$18,18,18,18,19,19,19,20,20,20,20,20,20,20,21,21,22,22,22,22,22,23,24,24,25,29,30,40,44$

Solution

The ages are already listed in order. We will make a table of frequencies to help identify the age with the highest frequency.

Now look for the highest frequency. The highest frequency is $7,$ which corresponds to the age $20.$ So the mode of the ages in this class is $20$ years.

Example

Try it.

The data lists the heights (in inches) of students in a statistics class. Identify the mode.

$56$	$61$	$63$	$64$	$65$	$66$	$67$	$67$
$60$	$62$	$63$	$64$	$65$	$66$	$67$	$70$
$60$	$63$	$63$	$64$	$66$	$66$	$67$	$74$
$61$	$63$	$64$	$65$	$66$	$67$	$67$

Solution

List each number with its frequency.

Now look for the highest frequency. The highest frequency is $6,$ which corresponds to the height $67$ inches. So the mode of this set of heights is $67$ inches.

Condensed — the full section is in OpenStax Prealgebra 2e.

Use the Basic Definition of Probability

The probability of an event tells us how likely that event is to occur. We usually write probabilities as fractions or decimals.

For example, picture a fruit bowl that contains five pieces of fruit - three bananas and two apples.

If you want to choose one piece of fruit to eat for a snack and don’t care what it is, there is a $\frac{3}{5}$ probability you will choose a banana, because there are three bananas out of the total of five pieces of fruit. The probability of an event is the number of favorable outcomes divided by the total number of outcomes.

Converting the fraction $\frac{3}{5}$ to a decimal, we would say there is a $0.6$ probability of choosing a banana.

\[\begin{array}{l} \\ \text{Probability of choosing a banana}=\frac{3}{5} \\ \text{Probability of choosing a banana}=0.6\end{array}\]

This basic definition of probability assumes that all the outcomes are equally likely to occur. If you study probabilities in a later math class, you’ll learn about several other ways to calculate probabilities.

Example

Try it.

The ski club is holding a raffle to raise money. They sold $100$ tickets. All of the tickets are placed in a jar. One ticket will be pulled out of the jar at random, and the winner will receive a prize. Cherie bought one raffle ticket.

ⓐ Find the probability she will win the prize.

ⓑ Convert the fraction to a decimal.

Solution

ⓐ
What are you asked to find?	The probability Cherie wins the prize.
What is the number of favorable outcomes?	1, because Cherie has 1 ticket.
Use the definition of probability.	$\text{Probability of an event}=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$
Substitute into the numerator and denominator.	$\text{Probability Cherie wins}=\frac{1}{100}$

ⓑ
Convert the fraction to a decimal.
Write the probability as a fraction.	$\text{Probability}=\frac{1}{100}$
Convert the fraction to a decimal.	$\text{Probability}=0.01$

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Calculate the mean of a set of numbers.
1. Write the formula for the mean $\text{mean}=\frac{\text{sum of values in data set}}{n}$
2. Find the sum of all the values in the set. Write the sum in the numerator.
3. Count the number, n, of values in the set. Write this number in the denominator.
4. Simplify the fraction.
5. Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.
Find the median of a set of numbers.
1. List the numbers from least to greatest.
2. Count how many numbers are in the set. Call this $n$.
3. Is $n$ odd or even?
  If $n$ is an odd number, the median is the middle value.
  If $n$ is an even number, the median is the mean of the two middle values
Identify the mode of a set of numbers.
1. List the data values in numerical order.
2. Count the number of times each value appears.
3. The mode is the value with the highest frequency.

Averages and Probability

Calculate the Mean of a Set of Numbers

In the following exercises, find the mean.

Try it.

$3$, $8$, $2$, $2$, $5$

Solution

Try it.

$6$, $1$, $9$, $3$, $4$, $7$

Try it.

$65$, $13$, $48$, $32$, $19$, $33$

Solution

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$34$, $45$, $29$, $61$, and $41$

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$202$, $241$, $265$, $274$

Solution

245.5

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$525$, $532$, $558$, $574$

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$12.45$, $12.99$, $10.50$, $11.25$, $9.99$, $12.72$

Solution

11.65

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$28.8$, $32.9$, $32.5$, $27.9$, $30.4$, $32.5$, $31.6$, $32.7$

Try it.

Four girls leaving a mall were asked how much money they had just spent. The amounts were $\text{\$0}$, $\text{\$14.95}$, $\text{\$35.25}$, and $\ \text{\$25.16}$. Find the mean amount of money spent.

Solution

$18.84

Try it.

Juan bought $5$ shirts to wear to his new job. The costs of the shirts were $\text{\$32.95}$, $\text{\$38.50}$, $\text{\$30.00}$, $\text{\$17.45}$, and $\text{\$24.25}$. Find the mean cost.

Try it.

The number of minutes it took Jim to ride his bike to school for each of the past six days was $21$, $18$, $16$, $19$, $24$, and $19$. Find the mean number of minutes.

Solution

19.5 minutes

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Norris bought six books for his classes this semester. The costs of the books were $\text{\$74.28}$, $\text{\$120.95}$, $\text{\$52.40}$, $\text{\$10.59}$, $\text{\$35.89}$, and $\text{\$59.24}$. Find the mean cost.

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The top eight hitters in a softball league have batting averages of $.373$, $.360$, $.321$, $.321$, $.320$, $.312$, $.311$, and $.311$. Find the mean of the batting averages. Round your answer to the nearest thousandth.

Solution

0.329

Try it.

The monthly snowfall at a ski resort over a six-month period was $60.3,$ $79.7,$ $50.9,$ $28.0,$ $47.4,$ and $46.1$ inches. Find the mean snowfall.

Find the Median of a Set of Numbers

In the following exercises, find the median.

Try it.

$24$, $19$, $18$, $29$, $21$

Solution

Try it.

$48$, $51$, $46$, $42$, $50$

Try it.

$65$, $56$, $35$, $34$, $44$, $39$, $55$, $52$, $45$

Solution

Try it.

$121$, $115$, $135$, $109$, $136$, $147$, $127$, $119$, $110$

Try it.

$4$, $8$, $1$, $5$, $14$, $3$, $1$, $12$

Solution

4.5

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$3$, $9$, $2$, $6$, $20$, $3$, $3$, $10$

Try it.

$99.2$, $101.9$, $98.6$, $99.5$, $100.8$, $99.8$

Solution

99.65

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$28.8$, $32.9$, $32.5$, $27.9$, $30.4$, $32.5$, $31.6$, $32.7$

Try it.

Last week Ray recorded how much he spent for lunch each workday. He spent $\text{\$6.50}$, $\text{\$7.25}$, $\text{\$4.90}$, $\text{\$5.30}$, and $\text{\$12.00}$. Find the median.

Solution

$6.50

Try it.

Michaela is in charge of 6 two-year olds at a daycare center. Their ages, in months, are $25$, $24$, $28$, $32$, $29$, and $31$. Find the median age.

Try it.

Brian is teaching a swim class for $6$ three-year olds. Their ages, in months, are $38,41,45,36,40,\text{and}\ 42.$ Find the median age.

Solution

40.5 months

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Sal recorded the amount he spent for gas each week for the past $8$ weeks. The amounts were $\text{\$38.65},$ $\text{\$32.18},$ $\text{\$40.23},$ $\text{\$51.50},$ $\text{\$43.68},$ $\text{\$30.96},$ $\text{\$41.37},$ and $\text{\$44.72}.$ Find the median amount.

Identify the Mode of a Set of Numbers

In the following exercises, identify the mode.

Try it.

$2$, $5$, $1$, $5$, $2$, $1$, $2$, $3$, $2$, $3$, $1$

Solution

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$8$, $5$, $1$, $3$, $7$, $1$, $1$, $7$, $1$, $8$, $7$

Try it.

$18$, $22$, $17$, $20$, $19$, $20$, $22$, $19$, $29$, $18$, $23$, $25$, $22$, $24$, $23$, $22$, $18$, $20$, $22$, $20$

Solution

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$42$, $28$, $32$, $35$, $24$, $32$, $48$, $32$, $32$, $24$, $35$, $28$, $30$, $35$, $45$, $32$, $28$, $32$, $42$, $42$, $30$

Try it.

The number of children per house on one block: $1$, $4$, $2$, $3$, $3$, $2$, $6$, $2$, $4$, $2$, $0$, $3$, $0.$

Solution

2 children

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The number of movies watched each month last year: $2$, $0$, $3$, $0$, $0$, $8$, $6$, $5$, $0$, $1$, $2$, $3.$

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The number of units being taken by students in one class: $12$, $5$, $11$, $10$, $10$, $11$, $5$, $11$, $11$, $11$, $10$, $12$.

Solution

11 units

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The number of hours of sleep per night for the past two weeks: $8$, $5$, $7$, $8$, $8$,  $6$, $6$, $6$, $6$, $9$, $7$, $8$, $8$, $8$.

Use the Basic Definition of Probability

In the following exercises, express the probability as both a fraction and a decimal. (Round to three decimal places, if necessary.)

Try it.

Josue is in a book club with $20$ members. One member is chosen at random each month to select the next month’s book. Find the probability that Josue will be chosen next month.

Solution

$\frac{1}{20},0.05$

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Jessica is one of eight kindergarten teachers at Mandela Elementary School. One of the kindergarten teachers will be selected at random to attend a summer workshop. Find the probability that Jessica will be selected.

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There are $24$ people who work in Dane’s department. Next week, one person will be selected at random to bring in doughnuts. Find the probability that Dane will be selected. Round your answer to the nearest thousandth.

Solution

$\frac{1}{24},0.041\overset{-}{6}\approx 0.042$

Try it.

Monica has two strawberry yogurts and six banana yogurts in her refrigerator. She will choose one yogurt at random to take to work. Find the probability Monica will choose a strawberry yogurt.

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Michel has four rock CDs and six country CDs in his car. He will pick one CD to play on his way to work. Find the probability Michel will pick a rock CD.

Solution

$\frac{2}{5},0.4$

Try it.

Noah is planning his summer camping trip. He can’t decide among six campgrounds at the beach and twelve campgrounds in the mountains, so he will choose one campground at random. Find the probability that Noah will choose a campground at the beach.

Try it.

Donovan is considering transferring to a $\text{4-year college}.$ He is considering $10$ out-of state colleges and $4$ colleges in his state. He will choose one college at random to visit during spring break. Find the probability that Donovan will choose an out-of-state college.

Solution

$\frac{5}{7},0.\overset{\text{———}}{714285}\approx 0.714$

Try it.

There are $258,890,850$ number combinations possible in the Mega Millions lottery. One winning jackpot ticket will be chosen at random. Brent chooses his favorite number combination and buys one ticket. Find the probability Brent will win the jackpot. Round the decimal to the first digit that is not zero, then write the name of the decimal.

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: $\frac{4+9+2}{3}.$
If you missed this problem, review .

Lafunua jibu

$5$
Simplify: $4(8)+6(3).$
If you missed this problem, review .

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$50$
Convert $\frac{5}{2}$ to a decimal.
If you missed this problem, review .

Lafunua jibu

$2.5$

Find the mean of the numbers $8,12,15,9,\text{and}\ 6.$

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Write the formula for the mean:	$\text{mean}=\frac{\text{sum of all the numbers}}{n}$
Write the sum of the numbers in the numerator.	$\text{mean}=\frac{8+12+15+9+6}{n}$
Count how many numbers are in the set. There are 5 numbers in the set, so $n=5$.	$\text{mean}=\frac{8+12+15+9+6}{5}$
Add the numbers in the numerator.	$\text{mean}=\frac{50}{5}$
Then divide.	$\text{mean}=10$
Check to see that the mean is 'typical': 10 is neither less than 6 nor greater than 15.	The mean is 10.

Find the mean of the numbers: $8,9,7,12,10,5.$

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8.5
Find the mean of the numbers: $9,13,11,7,5.$

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9

The ages of the members of a family who got together for a birthday celebration were $16,26,53,56,65,70,93,\text{and}\ 97$ years. Find the mean age.

Lafunua jibu

Write the formula for the mean:	$\text{mean}=\frac{\text{sum of all the numbers}}{n}$
Write the sum of the numbers in the numerator.	$\text{mean}=\frac{16+26+53+56+65+70+93+97}{n}$
Count how many numbers are in the set. Call this $n$ and write it in the denominator.	$\text{mean}=\frac{16+26+53+56+65+70+93+97}{8}$
Simplify the fraction.	$\text{mean}=\frac{476}{8}$
	$\text{mean}=59.5$

Is $59.5$ ‘typical’? Yes, it is neither less than $16$ nor greater than $97.$ The mean age is $59.5$ years.

The ages of the four students in Ben’s carpool are $25,18,21,\text{and}\ 22.$ Find the mean age of the students.

Lafunua jibu

21.5 years
Yen counted the number of emails she received last week. The numbers were $4,9,15,12,10,12,\text{and}\ 8.$ Find the mean number of emails.

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10

For the past four months, Daisy’s cell phone bills were $\text{\$42.75},\text{\$50.12},\text{\$41.54},\text{\$48.15}.$ Find the mean cost of Daisy’s cell phone bills.

Lafunua jibu

Write the formula for the mean.	$\text{mean}=\frac{\text{sum of all the numbers}}{n}$
Count how many numbers are in the set. Call this $n$ and write it in the denominator.	$\text{mean}=\frac{\text{sum of all the numbers}}{4}$
Write the sum of all the numbers in the numerator.	$\text{mean}=\frac{42.75+50.12+41.54+48.15}{4}$
Simplify the fraction.	$\text{mean}=\frac{182.56}{4}$
	$\text{mean}=45.64$

Does $\text{\$45.64}$ seem ‘typical’ of this set of numbers? Yes, it is neither less than $\text{\$41.54}$ nor greater than $\text{\$50.12}.$

The mean cost of her cell phone bill was $\text{\$45.64}$

Last week Ray recorded how much he spent for lunch each workday. He spent $\text{\$6.50},\text{\$7.25},\text{\$4.90},\text{\$5.30},\text{and}\ \text{\$12.00}.$ Find the mean of how much he spent each day.

Lafunua jibu

$7.19
Lisa has kept the receipts from the past four trips to the gas station. The receipts show the following amounts: $\text{\$34.87},\text{\$42.31},\text{\$38.04},\text{and}\ \text{\$43.26}.$ Find the mean.

Lafunua jibu

$39.62

Find the median of $12,13,19,9,11,15,\text{and}\ 18.$

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List the numbers in order from smallest to largest.	9, 11, 12, 13, 15, 18, 19
Count how many numbers are in the set. Call this $n$.	$n=7$
Is $n$ odd or even?	odd
The median is the middle value.
The middle is the number in the 4th position.	So the median of the data is 13.

Find the median of the data set: $43,38,51,40,46.$

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43
Find the median of the data set: $15,35,20,45,50,25,30.$

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30

Kristen received the following scores on her weekly math quizzes:

$83,79,85,86,92,100,76,90,88,\text{and}\ 64.$ Find her median score.

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Find the median of 83, 79, 85, 86, 92, 100, 76, 90, 88, and 64.
List the numbers in order from smallest to largest.	64, 76, 79, 83, 85, 86, 88, 90, 92, 100
Count the number of data values in the set. Call this $n.$	$n=10$
Is $n$ odd or even?	even
The median is the mean of the two middle values, the 5th and 6th numbers.
Find the mean of 85 and 86.	$\text{mean}=\frac{85+86}{2}$
	$\text{mean}=85.5$
	Kristen's median score is 85.5.

Find the median of the data set: $8,7,5,10,9,12.$

Lafunua jibu

8.5
Find the median of the data set: $21,25,19,17,22,18,20,24.$

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20.5
The ages of students in a college math class are listed below. Identify the mode.
$18,18,18,18,19,19,19,20,20,20,20,20,20,20,21,21,22,22,22,22,22,23,24,24,25,29,30,40,44$

Lafunua jibu

The ages are already listed in order. We will make a table of frequencies to help identify the age with the highest frequency.
Now look for the highest frequency. The highest frequency is $7,$ which corresponds to the age $20.$ So the mode of the ages in this class is $20$ years.
The number of sick days employees used last year: $3,6,2,3,7,5,6,2,4,2.$ Identify the mode.

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2
The number of handbags owned by women in a book club: $5,6,3,1,5,8,1,5,8,5.$ Identify the mode.

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5
The data lists the heights (in inches) of students in a statistics class. Identify the mode.
$56$ $61$ $63$ $64$ $65$ $66$ $67$ $67$
$60$ $62$ $63$ $64$ $65$ $66$ $67$ $70$
$60$ $63$ $63$ $64$ $66$ $66$ $67$ $74$
$61$ $63$ $64$ $65$ $66$ $67$ $67$

Lafunua jibu

List each number with its frequency.
Now look for the highest frequency. The highest frequency is $6,$ which corresponds to the height $67$ inches. So the mode of this set of heights is $67$ inches.
The ages of the students in a statistics class are listed here: $19$, $20$, $23$, $23$, $38$, $21$, $19$, $21$, $19$, $21$, $20$, $43$, $20$, $23$, $17$, $21$, $21$, $20$, $29$, $18$, $28$. What is the mode?

Lafunua jibu

21
Students listed the number of members in their household as follows: $6$, $2$, $5$, $6$, $3$, $7$, $5$, $6$, $5$, $3$, $4$, $4$, $5$, $7$, $6$, $4$, $5$, $2$, $1$, $5$. What is the mode?

Lafunua jibu

5

ⓐ Find the probability she will win the prize.

ⓑ Convert the fraction to a decimal.

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ⓐ
What are you asked to find?	The probability Cherie wins the prize.
What is the number of favorable outcomes?	1, because Cherie has 1 ticket.
Use the definition of probability.	$\text{Probability of an event}=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$
Substitute into the numerator and denominator.	$\text{Probability Cherie wins}=\frac{1}{100}$

ⓑ
Convert the fraction to a decimal.
Write the probability as a fraction.	$\text{Probability}=\frac{1}{100}$
Convert the fraction to a decimal.	$\text{Probability}=0.01$

Ignaly is attending a fashion show where the guests are seated at tables of ten. One guest from each table will be selected at random to receive a door prize. ⓐ Find the probability Ignaly will win the door prize for her table. ⓑ Convert the fraction to a decimal.
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1. ⓐ $\ \frac{1}{10}\$
2. ⓑ $\ 0.1$
Hoang is among $20$ people available to sit on a jury. One person will be chosen at random from the $20.$ ⓐ Find the probability Hoang will be chosen. ⓑ Convert the fraction to a decimal.
Lafunua jibu
1. ⓐ $\ \frac{1}{20}\$
2. ⓑ $\ 0.05$

Three women and five men interviewed for a job. One of the candidates will be offered the job.

ⓐ Find the probability the job is offered to a woman.

ⓑ Convert the fraction to a decimal.

Lafunua jibu

ⓐ
What are you asked to find?	The probability the job is offered to a woman.
What is the number of favorable outcomes?	3, because there are three women.
What are the total number of outcomes?	8, because 8 people interviewed.
Use the definition of probability.	$\text{Probability of an event}=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$
Substitute into the numerator and denominator.	$\text{Probability}=\frac{3}{8}$

ⓑ
Convert the fraction to a decimal.
Write the probability as a fraction.	$\text{Probability}=\frac{3}{8}$
Convert the fraction to a decimal.	$\text{Probability}=0.375$

A bowl of Halloween candy contains $5$ chocolate candies and $3$ lemon candies. Tanya will choose one piece of candy at random. ⓐ Find the probability Tanya will choose a chocolate candy. ⓑ Convert the fraction to a decimal.
Lafunua jibu
1. ⓐ $\frac{5}{8}\$
2. ⓑ $\ 0.625$
Dan has $2$ pairs of black socks and $6$ pairs of blue socks. He will choose one pair at random to wear tomorrow. ⓐ Find the probability Dan will choose a pair of black socks ⓑ Convert the fraction to a decimal.
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1. ⓐ $\ \frac{2}{8}\$
2. ⓑ $\ 0.25$
$3$, $8$, $2$, $2$, $5$

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4
$6$, $1$, $9$, $3$, $4$, $7$
$65$, $13$, $48$, $32$, $19$, $33$

Lafunua jibu

35
$34$, $45$, $29$, $61$, and $41$
$202$, $241$, $265$, $274$

Lafunua jibu

245.5
$525$, $532$, $558$, $574$
$12.45$, $12.99$, $10.50$, $11.25$, $9.99$, $12.72$

Lafunua jibu

11.65
$28.8$, $32.9$, $32.5$, $27.9$, $30.4$, $32.5$, $31.6$, $32.7$
Four girls leaving a mall were asked how much money they had just spent. The amounts were $\text{\$0}$, $\text{\$14.95}$, $\text{\$35.25}$, and $\ \text{\$25.16}$. Find the mean amount of money spent.

Lafunua jibu

$18.84
Juan bought $5$ shirts to wear to his new job. The costs of the shirts were $\text{\$32.95}$, $\text{\$38.50}$, $\text{\$30.00}$, $\text{\$17.45}$, and $\text{\$24.25}$. Find the mean cost.

Symbols used here

\approx

approximately equal
Equal to the precision shown, not exactly.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Averages and Probability

Calculate the mean of a set of numbers
Find the median of a set of numbers
Find the mode of a set of numbers
Apply the basic definition of probability
Write the formula for the mean
Find the sum of all the values in the set. Write the sum in the numerator.
Count the number,
Simplify the fraction.

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Jaribu kufanya mambo yako mwenyewe

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Mengi zaidi katika Arithmetic

Order of operations Fractions Percentages Powers and roots

Write the formula for the mean:	\(\text{mean}=\frac{\text{sum of all the numbers}}{n}\)
Write the sum of the numbers in the numerator.	\(\text{mean}=\frac{16+26+53+56+65+70+93+97}{n}\)
Count how many numbers are in the set. Call this \(n\) and write it in the denominator.	\(\text{mean}=\frac{16+26+53+56+65+70+93+97}{8}\)
Simplify the fraction.	\(\text{mean}=\frac{476}{8}\)
	\(\text{mean}=59.5\)

\(56\)	\(61\)	\(63\)	\(64\)	\(65\)	\(66\)	\(67\)	\(67\)
\(60\)	\(62\)	\(63\)	\(64\)	\(65\)	\(66\)	\(67\)	\(70\)
\(60\)	\(63\)	\(63\)	\(64\)	\(66\)	\(66\)	\(67\)	\(74\)
\(61\)	\(63\)	\(64\)	\(65\)	\(66\)	\(67\)	\(67\)

Write the formula for the mean:	\(\text{mean}=\frac{\text{sum of all the numbers}}{n}\)
Write the sum of the numbers in the numerator.	\(\text{mean}=\frac{16+26+53+56+65+70+93+97}{n}\)
Count how many numbers are in the set. Call this \(n\) and write it in the denominator.	\(\text{mean}=\frac{16+26+53+56+65+70+93+97}{8}\)
Simplify the fraction.	\(\text{mean}=\frac{476}{8}\)
	\(\text{mean}=59.5\)

Write the formula for the mean.	\(\text{mean}=\frac{\text{sum of all the numbers}}{n}\)
Count how many numbers are in the set. Call this \(n\) and write it in the denominator.	\(\text{mean}=\frac{\text{sum of all the numbers}}{4}\)
Write the sum of all the numbers in the numerator.	\(\text{mean}=\frac{42.75+50.12+41.54+48.15}{4}\)
Simplify the fraction.	\(\text{mean}=\frac{182.56}{4}\)
	\(\text{mean}=45.64\)

\(56\)	\(61\)	\(63\)	\(64\)	\(65\)	\(66\)	\(67\)	\(67\)
\(60\)	\(62\)	\(63\)	\(64\)	\(65\)	\(66\)	\(67\)	\(70\)
\(60\)	\(63\)	\(63\)	\(64\)	\(66\)	\(66\)	\(67\)	\(74\)
\(61\)	\(63\)	\(64\)	\(65\)	\(66\)	\(67\)	\(67\)

ⓑ
Convert the fraction to a decimal.
Write the probability as a fraction.	\(\text{Probability}=\frac{1}{100}\)
Convert the fraction to a decimal.	\(\text{Probability}=0.01\)

ⓑ
Convert the fraction to a decimal.
Write the probability as a fraction.	\(\text{Probability}=\frac{3}{8}\)
Convert the fraction to a decimal.	\(\text{Probability}=0.375\)