## Problem Set: Ratios, Rates, Probabilities, and Averages

### Calculate the Mean of a Set of Numbers

In the following exercises, find the mean.

1. $3$ , $8$ , $2$ , $2$ , $5$

2. $6$ , $1$ , $9$ , $3$ , $4$ , $7$
3. $65$ , $13$ , $48$ , $32$ , $19$ , $33$

4. $34$ , $45$ , $29$ , $61$ , and $41$
5. $202$ , $241$ , $265$ , $274$

6. $525$ , $532$ , $558$ , $574$
7. $12.45$ , $12.99$ , $10.50$ , $11.25$ , $9.99$ , $12.72$

8. $28.8$ , $32.9$ , $32.5$ , $27.9$ , $30.4$ , $32.5$ , $31.6$ , $32.7$
9. $2,4,1,0,1,\text{and}1$
10. ${270}$ , ${310.50}$ , ${243.75}$ , and ${252.15}$

11. Each workday last week, Yoshie kept track of the number of minutes she had to wait for the bus. She waited $3,0,8,1, and 8$ minutes. Find the mean
12. In the last three months, Raul’s water bills were ${31.45}, {48.76},\text{and} {42.60}$. Find the mean.

### Calculate the Mean of a Set of Numbers in Applications

In the following exercises, find the mean.

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

2. Juan bought $5$ shirts to wear to his new job. The costs of the shirts were ${32.95}$ , ${38.50}$ , ${30.00}$ , ${17.45}$ , and ${24.25}$ . Find the mean cost.
3. 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.

4. Norris bought six books for his classes this semester. The costs of the books were ${74.28}$ , ${120.95}$ , ${52.40}$ , ${10.59}$ , ${35.89}$ , and ${59.24}$ . Find the mean cost.
5. 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.

6. 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.

1. $24$ , $19$ , $18$ , $29$ , $21$

2. $48$ , $51$ , $46$ , $42$ , $50$
3. $65$ , $56$ , $35$ , $34$ , $44$ , $39$ , $55$ , $52$ , $45$

4. $121$ , $115$ , $135$ , $109$ , $136$ , $147$ , $127$ , $119$ , $110$
5. $4$ , $8$ , $1$ , $5$ , $14$ , $3$ , $1$ , $12$

6. $3$ , $9$ , $2$ , $6$ , $20$ , $3$ , $3$ , $10$
7. $99.2$ , $101.9$ , $98.6$ , $99.5$ , $100.8$ , $99.8$

8. $28.8$ , $32.9$ , $32.5$ , $27.9$ , $30.4$ , $32.5$ , $31.6$ , $32.7$
9. $41$ , $45$ , $32$ , $60$ , $58$
10. $25$ , $23$ , $24$ , $26$ , $29$ , $19$ , $18$ , $32$

### Find the Median of a Set of Numbers in Applications

In the following exercises, find the median.

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

2. 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.
3. 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.

4. Sal recorded the amount he spent for gas each week for the past $8$ weeks. The amounts were ${38.65}$, ${32.18}$, ${40.23}$, ${51.50}$, ${43.68}$, ${30.96}$, ${41.37}$, and ${44.72}$. Find the median amount.
5. The ages of the eight men in Jerry’s model train club are $52,63,45,51,55,75,60,\text{and}59$. Find the median age.
6. The number of clients at Miranda’s beauty salon each weekday last week were $18,7,12,16,\text{and}20$. Find the median number of clients.

### Identify the Mode of a Set of Numbers

In the following exercises, identify the mode.

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

2. $8$ , $5$ , $1$ , $3$ , $7$ , $1$ , $1$ , $7$ , $1$ , $8$ , $7$
3. $18$ , $22$ , $17$ , $20$ , $19$ , $20$ , $22$ , $19$ , $29$ , $18$ , $23$ , $25$ , $22$ , $24$ , $23$ , $22$ , $18$ , $20$ , $22$ , $20$

4. $42$ , $28$ , $32$ , $35$ , $24$ , $32$ , $48$ , $32$ , $32$ , $24$ , $35$ , $28$ , $30$ , $35$ , $45$ , $32$ , $28$ , $32$ , $42$ , $42$ , $30$
5. The number of children per house on one block: $1$ , $4$ , $2$ , $3$ , $3$ , $2$ , $6$ , $2$ , $4$ , $2$ , $0$ , $3$ , $0$.

6. The number of movies watched each month last year: $2$ , $0$ , $3$ , $0$ , $0$ , $8$ , $6$ , $5$ , $0$ , $1$ , $2$ , $3$.
7. The number of units being taken by students in one class: $12$ , $5$ , $11$ , $10$ , $10$ , $11$ , $5$ , $11$ , $11$ , $11$ , $10$ , $12$ .

8. The number of hours of sleep per night for the past two weeks: $8$ , $5$ , $7$ , $8$ , $8$ ,
9. $6$ , $6$ , $6$ , $6$ , $9$ , $7$ , $8$ , $8$ , $8$ .
10. $6$ , $4$ , $4,5$ , $6,6$ , $4$ , $4$ , $4$ , $3$ , $5$
11. The number of siblings of a group of students: $2$ , $0$ , $3$ , $2$ , $4$ , $1$ , $6$ , $5$ , $4$ , $1$ , $2$ , $3$

### 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.)

1. 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.

2. 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.
3. 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.

4. 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.
5. 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.

6. 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.
7. 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.

8. 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.
9. The Sustainability Club sells $200$ tickets to a raffle, and Albert buys one ticket. One ticket will be selected at random to win the grand prize. Find the probability Albert will win the grand prize. Express your answer as a fraction and as a decimal.
10. Luc has to read $3$ novels and $12$ short stories for his literature class. The professor will choose one reading at random for the final exam. Find the probability that the professor will choose a novel for the final exam. Express your answer as a fraction and as a decimal.

### Everyday Math

1. Joaquin gets paid every Friday. His paychecks for the past $8$ Fridays were ${315}$, ${236.25}$, ${236.25}$, ${236.25}$, ${315}$, ${315}$, ${236.25}$, ${393.75}$. Find the ⓐ mean, ⓑ median, and ⓒ mode.

2. The cash register receipts each day last week at a coffee shop were ${1,845}$, ${1,520}$, ${1,438}$, ${1,682}$, ${1,850}$, ${2,721}$, ${2,539}$. Find the ⓐ mean, ⓑ median, and ⓒ mode.

### Writing Exercises

Explain in your own words the difference between the mean, median, and mode of a set of numbers.

Make an example of probability that relates to your life. Write your answer as a fraction and explain what the numerator and denominator represent.

### Write a Ratio as a Fraction

In the following exercises, write each ratio as a fraction.

1. $20$ to $36$

2. $20$ to $32$
3. $42$ to $48$

4. $45$ to $54$
5. $49$ to $21$

6. $56$ to $16$
7. $84$ to $36$

8. $6.4$ to $0.8$
9. $0.56$ to $2.8$

10. $1.26$ to $4.2$
11. $1\Large\frac{2}{3}$ to $2\Large\frac{5}{6}$

12. $1\Large\frac{3}{4}$ to $2\Large\frac{5}{8}$
13. $4\Large\frac{1}{6}$ to $3\Large\frac{1}{3}$

14. $5\Large\frac{3}{5}$ to $3\Large\frac{3}{5}$
15. ${18}$ to ${63}$

16. ${16}$ to ${72}$
17. ${1.21}$ to ${0.44}$

18. ${1.38}$ to ${0.69}$
19. $28$ ounces to $84$ ounces

20. $32$ ounces to $128$ ounces
21. $12$ feet to $46$ feet

22. $15$ feet to $57$ feet
23. $246$ milligrams to $45$ milligrams

24. $304$ milligrams to $48$ milligrams
25. total cholesterol of $175$ to HDL cholesterol of $45$

26. total cholesterol of $215$ to HDL cholesterol of $55$
27. $27$ inches to $1$ foot

28. $28$ inches to $1$ foot
29. $28$ to $40$$56$ to $32$

30. $3.5$ to $0.5$
31. $1.2$ to $1.8$

32. $1\Large\frac{3}{4}\normalsize\text{to}1\Large\frac{5}{8}$
33. $2\Large\frac{1}{3}\normalsize\text{to}5\Large\frac{1}{4}$

34. $64$ ounces to $30$ ounces
35. $28$ inches to $3$ feet

### Write a Rate as a Fraction

In the following exercises, write each rate as a fraction.

1. $140$ calories per $12$ ounces

2. $180$ calories per $16$ ounces
3. $8.2$ pounds per $3$ square inches

4. $9.5$ pounds per $4$ square inches
5. $488$ miles in $7$ hours

6. $527$ miles in $9$ hours
7. ${595}$ for $40$ hours

8. ${798}$ for $40$ hours
9. $180$ calories per $8$ ounces
10. $90$ pounds per $7.5$ square inches

11. $126$ miles in $4$ hours
12. ${612.50}$ for $35$ hours

### Find Unit Rates

#### Exercise 1

In the following exercises, find the unit rate. Round to two decimal places, if necessary.

1. $140$ calories per $12$ ounces

2. $180$ calories per $16$ ounces
3. $8.2$ pounds per $3$ square inches

4. $9.5$ pounds per $4$ square inches
5. $488$ miles in $7$ hours

6. $527$ miles in $9$ hours
7. ${595}$ for $40$ hours

8. ${798}$ for $40$ hours
9. $576$ miles on $18$ gallons of gas

10. $435$ miles on $15$ gallons of gas
11. $43$ pounds in $16$ weeks

12. $57$ pounds in $24$ weeks
13. $46$ beats in $0.5$ minute

14. $54$ beats in $0.5$ minute
15. $180$ calories per $8$ ounces
16. $90$ pounds per $7.5$ square inches

17. $126$ miles in $4$ hours
18. ${612.50}$ for $35$ hours

#### Exercise 2

1. The bindery at a printing plant assembles $96,000$ magazines in $12$ hours. How many magazines are assembled in one hour?

2. The pressroom at a printing plant prints $540,000$ sections in $12$ hours. How many sections are printed per hour?

### Find Unit Price

#### Exercise 1

In the following exercises, find the unit price. Round to the nearest cent.

1. Soap bars at $8$ for ${8.69}$

2. Soap bars at $4$ for ${3.39}$
3. Women’s sports socks at $6$ pairs for ${7.99}$

4. Men’s dress socks at $3$ pairs for ${8.49}$
5. Snack packs of cookies at $12$ for ${5.79}$

6. Granola bars at $5$ for ${3.69}$
7. CD-RW discs at $25$ for ${14.99}$

8. CDs at $50$ for ${4.49}$
9. t-shirts: $3$ for ${8.97}$
10. Highlighters: $6$ for ${2.52}$

11. An office supply store sells a box of pens for ${11}$. The box contains $12$ pens. How much does each pen cost?
12. Anna bought a pack of $8$ kitchen towels for ${13.20}$. How much did each towel cost? Round to the nearest cent if necessary.

#### Exercise 2

1. The grocery store has a special on macaroni and cheese. The price is ${3.87}$ for $3$ boxes. How much does each box cost?

2. The pet store has a special on cat food. The price is ${4.32}$ for $12$ cans. How much does each can cost?

#### Exercise 3

In the following exercises, find each unit price and then identify the better buy. Round to three decimal places.

1. Mouthwash, ${50.7-ounce}$ size for ${6.99}$ or ${33.8-ounce}$ size for ${4.79}$

2. Toothpaste, $6$ ounce size for ${3.19}$ or $7.8-ounce$ size for ${5.19}$
3. Breakfast cereal, $18$ ounces for ${3.99}$ or $14$ ounces for ${3.29}$

4. Breakfast Cereal, $10.7$ ounces for ${2.69}$ or $14.8$ ounces for ${3.69}$
5. Ketchup, ${40-ounce}$ regular bottle for ${2.99}$ or ${64-ounce}$ squeeze bottle for ${4.39}$

6. Mayonnaise ${15-ounce}$ regular bottle for ${3.49}$ or ${22-ounce}$ squeeze bottle for ${4.99}$
7. Cheese ${6.49}$ for $1$ lb. block or ${3.39}$ for $\Large\frac{1}{2}$ lb. block

8. Candy ${10.99}$ for a $1$ lb. bag or ${2.89}$ for $\Large\frac{1}{4}$ lb. of loose candy
9. Shampoo: $12$ ounces for ${4.29}$ or $22$ ounces for ${7.29}?$
10. Vitamins: $60$ tablets for ${6.49}$ or $100$ for ${11.99}?$

### Translate Phrases to Expressions with Fractions

In the following exercises, translate the English phrase into an algebraic expression.

1. $793$ miles per $p$ hours

2. $78$ feet per $r$ seconds
3. ${3}$ for $0.5$ lbs.

4. $j$ beats in $0.5$ minutes
5. $105$ calories in $x$ ounces

6. $400$ minutes for $m$ dollars
7. the ratio of $y$ and $5x$

8. the ratio of $12x$ and $y$
9. $535$ miles per $h\text{hours}$
10. $a$ adults to $45$ children

11. the ratio of $4y$ and the difference of $x$ and $10$
12. the ratio of $19$ and the sum of $3$ and $n$

### Everyday math

1. One elementary school in Ohio has $684$ students and $45$ teachers. Write the student-to-teacher ratio as a unit rate.

2. The average American produces about $1,600$ pounds of paper trash per year (365 days). How many pounds of paper trash does the average American produce each day? (Round to the nearest tenth of a pound.)
3. A popular fast food burger weighs $7.5$ ounces and contains $540$ calories, $29$ grams of fat, $43$ grams of carbohydrates, and $25$ grams of protein. Find the unit rate of ⓐ calories per ounce ⓑ grams of fat per ounce ⓒ grams of carbohydrates per ounce ⓓ grams of protein per ounce. Round to two decimal places.

4. A $16-ounce$ chocolate mocha coffee with whipped cream contains $470$ calories, $18$ grams of fat, $63$ grams of carbohydrates, and $15$ grams of protein. Find the unit rate of ⓐ calories per ounce ⓑ grams of fat per ounce ⓒ grams of carbohydrates per ounce ⓓ grams of protein per ounce.

### Writing Exercises

1. Would you prefer the ratio of your income to your friend’s income to be $\text{3/1}$ or $1/3?$ Explain your reasoning.
2. The parking lot at the airport charges ${0.75}$ for every $15$ minutes. ⓐ How much does it cost to park for $1$ hour? ⓑ Explain how you got your answer to part ⓐ. Was your reasoning based on the unit cost or did you use another method?
3. Kathryn ate a $4-ounce$ cup of frozen yogurt and then went for a swim. The frozen yogurt had $115$ calories. Swimming burns $422$ calories per hour. For how many minutes should Kathryn swim to burn off the calories in the frozen yogurt? Explain your reasoning.
4. Mollie had a $16-ounce$ cappuccino at her neighborhood coffee shop. The cappuccino had $110$ calories. If Mollie walks for one hour, she burns $246$ calories. For how many minutes must Mollie walk to burn off the calories in the cappuccino? Explain your reasoning.

Simplify Expressions with Square Roots

In the following exercises, simplify.

1. $\sqrt{36}$

2. $\sqrt{4}$
3. $\sqrt{64}$

4. $\sqrt{144}$
5. $-\sqrt{4}$

6. $-\sqrt{100}$
7. $-\sqrt{1}$

8. $-\sqrt{121}$
9. $\sqrt{-121}$

10. $\sqrt{-36}$
11. $\sqrt{-9}$

12. $\sqrt{-49}$
13. $\sqrt{9+16}$

14. $\sqrt{25+144}$
15. $\sqrt{9}+\sqrt{16}$

16. $\sqrt{25}+\sqrt{144}$
17. $\sqrt{64}$
18. $\sqrt{144}$

19. $-\sqrt{25}$
20. $-\sqrt{81}$

21. $\sqrt{-9}$
22. $\sqrt{-36}$

23. $\sqrt{64}+\sqrt{225}$
24. $\sqrt{64+225}$

### Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

1. $\sqrt{70}$

2. $\sqrt{55}$
3. $\sqrt{200}$

4. $\sqrt{172}$
5. $\sqrt{28}$
6. $\sqrt{155}$

### Approximate Square Roots with a Calculator

In the following exercises, use a calculator to approximate each square root and round to two decimal places.

1. $\sqrt{19}$

2. $\sqrt{21}$
3. $\sqrt{53}$

4. $\sqrt{47}$
5. $\sqrt{15}$
6. $\sqrt{57}$

### Simplify Variable Expressions with Square Roots

In the following exercises, simplify. (Assume all variables are greater than or equal to zero.)

1. $\sqrt{{y}^{2}}$

2. $\sqrt{{b}^{2}}$
3. $\sqrt{49{x}^{2}}$

4. $\sqrt{100{y}^{2}}$
5. $-\sqrt{64{a}^{2}}$

6. $-\sqrt{25{x}^{2}}$
7. $\sqrt{144{x}^{2}{y}^{2}}$

8. $\sqrt{196{a}^{2}{b}^{2}}$
9. $\sqrt{{q}^{2}}$
10. $\sqrt{64{b}^{2}}$

11. $-\sqrt{121{a}^{2}}$
12. $\sqrt{225{m}^{2}{n}^{2}}$

13. $-\sqrt{100{q}^{2}}$
14. $\sqrt{49{y}^{2}}$

15. $\sqrt{4{a}^{2}{b}^{2}}$
16. $\sqrt{121{c}^{2}{d}^{2}}$

### Use Square Roots in Applications

In the following exercises, solve. Round to one decimal place.

Landscaping Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of $75$ square feet. How long can a side of his garden be?

Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of $130$ square feet. How long can a side of his patio be?

Gravity An airplane dropped a flare from a height of $1,024$ feet above a lake. How many seconds did it take for the flare to reach the water?

Gravity A hiker dropped a granola bar from a lookout spot $576$ feet above a valley. How long did it take the granola bar to reach the valley floor?

Gravity A hang glider dropped his cell phone from a height of $350$ feet. How many seconds did it take for the cell phone to reach the ground?

Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, $4,000$ feet above the Colorado River. How many seconds did it take for the hammer to reach the river?