Notes on Ratios and Unit Rates

Ratios

  • Ratios compare two quantities to show how they relate to each other, rather than looking at numbers alone.

  • They help us understand relationships, make comparisons, and inform decisions in real-world contexts.

  • Notation:

    • A ratio is written as a : b (e.g., a : b).

    • The same relationship is expressed as a fraction a/b.

  • Applications:

    • Budgeting and finances: compare housing costs to income or food costs to total expenses.

    • Financial planning: advisors use ratios to identify overspending in a category.

  • Example emphasis: ratios are the first step to understanding how quantities relate; unit rates build on ratios by showing the amount per one unit.

Practice: Write each as a ratio a : b and as a simplest form fraction

1) Food costs to total expenses: Food = $900, Total = $4,800.

  • Ratio: 900:4800=3:16900:4800 = 3:16

  • Fraction: rac316rac{3}{16}

2) Book cost to class cost: Book = $120, Class = $1,200.

  • Ratio: 120:1200=1:10120:1200 = 1:10

  • Fraction: rac110rac{1}{10}

3) Restaurant drink cost to meal cost: Drink = $3, Meal = $12.

  • Ratio: 3:12=1:43:12 = 1:4

  • Fraction: rac14rac{1}{4}

4) Fast food cost to meal prep cost: Fast food = $8, Meal prep = $5.

  • Ratio: 8:58:5

  • Fraction: rac85rac{8}{5}

Shopping and Unit Rates

  • For cost problems, express the answer as a unit rate (cost per item) rather than items per cost.

  • A unit rate is a ratio where the second quantity is 1, answering: “How much for one?” or “How many per one?”

  • Unit rates are especially useful for comparing shopping options on a per-item basis.

Practice: Unit rates (cost per one)

5) 2 notebooks for $5.

  • Unit rate: rac52=2.50rac{5}{2} = 2.50 dollars per notebook.

6) A grocery store sells 3 pounds of apples for $6.

  • Unit rate: rac63=2.00rac{6}{3} = 2.00 dollars per pound.

7) A 12-pack of soda costs $4.80.

  • Unit rate: rac4.8012=0.40rac{4.80}{12} = 0.40 dollars per can.

8) You buy 5 shirts for $60.

  • Unit rate: rac605=12rac{60}{5} = 12 dollars per shirt.

Travel and Efficiency

  • Miles per gallon (mpg) compares distance to fuel; higher mpg means more miles per gallon of fuel.

Practice: Travel and efficiency

9) A car travels 240 miles on 8 gallons of gas.

  • mpg: rac2408=30rac{240}{8} = 30 mpg

10) A bus travels 150 miles on 25 gallons of gas.

  • mpg: rac15025=6rac{150}{25} = 6 mpg

11) A runner completes 10 laps in 25 minutes.

  • laps per minute: rac1025=0.4rac{10}{25} = 0.4 laps/min

12) A bicycle covers 30 miles in 2 hours.

  • mph: rac302=15rac{30}{2} = 15 mph

Population and Statistics

  • Ratios describe student-to-faculty ratios or population density.

Practice: Population and statistics

13) A college has 2,000 students and 100 faculty.

  • Ratio: 2000:100=20:12000:100 = 20:1 (20 students per faculty)

14) A city has 120,000 people and 30,000 houses.

  • People per house: rac120,00030,000=4rac{120{,}000}{30{,}000} = 4 people per house

15) A school has 450 boys and 550 girls.

  • Ratio: 450:550=9:11450:550 = 9:11

16) A country has 50 million people in 5 million square miles.

  • People per square mile: rac505=10rac{50}{5} = 10 people per square mile

Comparing Options: Price-to-Rent and Related Ratios

  • Price-to-rent ratio helps decide whether to buy or rent a home; ratios appear in various domains like sports, health care, and cooking.

  • When comparing rent vs. price, think in terms of a ratio of cost to cost to assess affordability.

Practice: Additional ratios

17) A basketball player scores 28 points in 7 games.

  • Points per game: rac287=4rac{28}{7} = 4 points per game

18) A recipe uses 3 cups of flour and 2 cups of sugar.

  • Ratio: 3:23:2; Fraction: rac32rac{3}{2}

19) A house costs $240{,}000 while yearly rent is $12{,}000.

  • Price-to-rent ratio: rac240,00012,000=20:1rac{240{,}000}{12{,}000} = 20:1

20) A hospital has 300 nurses for 600 patients.

  • Patients per nurse: rac600300=2rac{600}{300} = 2 patients per nurse (expressed as 2:12:1)

Unit Rates (revisited)

  • A unit rate is a ratio with the second quantity equal to 1; always express shopping answers as cost per one.

Practice: Additional unit rates

21) A 16-oz box of cereal costs $4.50.

  • Unit rate: rac{4.50}{16} = 0.28125 \

Ratios

  • Ratios compare two quantities to show how they relate to each other, rather than looking at numbers alone.

  • They help us understand relationships, make comparisons, and inform decisions in real-world contexts.

  • Notation:- A ratio is written as a : b (e.g., a : b).

    • The same relationship is expressed as a fraction a/b.

  • Applications:- Budgeting and finances: compare housing costs to income or food costs to total expenses.

    • Financial planning: advisors use ratios to identify overspending in a category.

  • Example emphasis: ratios are the first step to understanding how quantities relate; unit rates build on ratios by showing the amount per one unit.

Practice: Write each as a ratio a : b and as a simplest form fraction

1) Food costs to total expenses: Food = $900, Total = $4,800.

  • Ratio: 900:4800 = 3:16</p></li><li><p>Fraction:</p></li><li><p>Fraction:\frac{3}{16}

2) Book cost to class cost: Book = $120, Class = $1,200.

  • Ratio: 120:1200 = 1:10</p></li><li><p>Fraction:</p></li><li><p>Fraction:\frac{1}{10}

3) Restaurant drink cost to meal cost: Drink = $3, Meal = $12.

  • Ratio: 3:12 = 1:4</p></li><li><p>Fraction:</p></li><li><p>Fraction:\frac{1}{4}

4) Fast food cost to meal prep cost: Fast food = $8, Meal prep = $5.

  • Ratio: 8:5</p></li><li><p>Fraction:</p></li><li><p>Fraction:\frac{8}{5}

Shopping and Unit Rates

  • For cost problems, express the answer as a unit rate (cost per item) rather than items per cost.

  • A unit rate is a ratio where the second quantity is 1, answering: “How much for one?” or “How many per one?”

  • Unit rates are especially useful for comparing shopping options on a per-item basis.

Practice: Unit rates (cost per one)

5) 2 notebooks for $5.

  • Unit rate: \frac{5}{2} = 2.50 dollars per notebook.

6) A grocery store sells 3 pounds of apples for $6.

  • Unit rate: \frac{6}{3} = 2.00 dollars per pound.

7) A 12-pack of soda costs $4.80.

  • Unit rate: \frac{4.80}{12} = 0.40 dollars per can.

8) You buy 5 shirts for $60.

  • Unit rate: \frac{60}{5} = 12dollarspershirt.</p></li></ul><h4id="d9bf8eec−365d−4dce−8dd6−ea89cb8f379c"data−toc−id="d9bf8eec−365d−4dce−8dd6−ea89cb8f379c"collapsed="false"seolevelmigrated="true">TravelandEfficiency</h4><ul><li><p>Milespergallon(mpg)comparesdistancetofuel;highermpgmeansmoremilespergallonoffuel.</p></li></ul><h6id="4f8b1c9e−762b−49ac−bd0d−134924dfdbc6"data−toc−id="4f8b1c9e−762b−49ac−bd0d−134924dfdbc6"collapsed="false"seolevelmigrated="true">Practice:Travelandefficiency</h6><p>9)Acartravels240mileson8gallonsofgas.</p><ul><li><p>mpg:dollars per shirt.</p></li></ul><h4 id="d9bf8eec-365d-4dce-8dd6-ea89cb8f379c" data-toc-id="d9bf8eec-365d-4dce-8dd6-ea89cb8f379c" collapsed="false" seolevelmigrated="true">Travel and Efficiency</h4><ul><li><p>Miles per gallon (mpg) compares distance to fuel; higher mpg means more miles per gallon of fuel.</p></li></ul><h6 id="4f8b1c9e-762b-49ac-bd0d-134924dfdbc6" data-toc-id="4f8b1c9e-762b-49ac-bd0d-134924dfdbc6" collapsed="false" seolevelmigrated="true">Practice: Travel and efficiency</h6><p>9) A car travels 240 miles on 8 gallons of gas.</p><ul><li><p>mpg:\frac{240}{8} = 30mpg</p></li></ul><p>10)Abustravels150mileson25gallonsofgas.</p><ul><li><p>mpg:mpg</p></li></ul><p>10) A bus travels 150 miles on 25 gallons of gas.</p><ul><li><p>mpg:\frac{150}{25} = 6mpg</p></li></ul><p>11)Arunnercompletes10lapsin25minutes.</p><ul><li><p>lapsperminute:mpg</p></li></ul><p>11) A runner completes 10 laps in 25 minutes.</p><ul><li><p>laps per minute:\frac{10}{25} = 0.4laps/min</p></li></ul><p>12)Abicyclecovers30milesin2hours.</p><ul><li><p>mph:laps/min</p></li></ul><p>12) A bicycle covers 30 miles in 2 hours.</p><ul><li><p>mph:\frac{30}{2} = 15mph</p></li></ul><h4id="a52e992d−d0a6−42fa−bb50−6f34082f94c6"data−toc−id="a52e992d−d0a6−42fa−bb50−6f34082f94c6"collapsed="false"seolevelmigrated="true">PopulationandStatistics</h4><ul><li><p>Ratiosdescribestudent−to−facultyratiosorpopulationdensity.</p></li></ul><h6id="994df52c−c72c−4c83−80dd−802fdc1f794e"data−toc−id="994df52c−c72c−4c83−80dd−802fdc1f794e"collapsed="false"seolevelmigrated="true">Practice:Populationandstatistics</h6><p>13)Acollegehas2,000studentsand100faculty.</p><ul><li><p>Ratio:mph</p></li></ul><h4 id="a52e992d-d0a6-42fa-bb50-6f34082f94c6" data-toc-id="a52e992d-d0a6-42fa-bb50-6f34082f94c6" collapsed="false" seolevelmigrated="true">Population and Statistics</h4><ul><li><p>Ratios describe student-to-faculty ratios or population density.</p></li></ul><h6 id="994df52c-c72c-4c83-80dd-802fdc1f794e" data-toc-id="994df52c-c72c-4c83-80dd-802fdc1f794e" collapsed="false" seolevelmigrated="true">Practice: Population and statistics</h6><p>13) A college has 2,000 students and 100 faculty.</p><ul><li><p>Ratio:2000:100 = 20:1(20studentsperfaculty)</p></li></ul><p>14)Acityhas120,000peopleand30,000houses.</p><ul><li><p>Peopleperhouse:(20 students per faculty)</p></li></ul><p>14) A city has 120,000 people and 30,000 houses.</p><ul><li><p>People per house:\frac{120,000}{30,000} = 4peopleperhouse</p></li></ul><p>15)Aschoolhas450boysand550girls.</p><ul><li><p>Ratio:people per house</p></li></ul><p>15) A school has 450 boys and 550 girls.</p><ul><li><p>Ratio:450:550 = 9:11</p></li></ul><p>16)Acountryhas50millionpeoplein5millionsquaremiles.</p><ul><li><p>Peoplepersquaremile:</p></li></ul><p>16) A country has 50 million people in 5 million square miles.</p><ul><li><p>People per square mile:\frac{50}{5} = 10peoplepersquaremile</p></li></ul><h4id="599b0777−f66b−4032−89a7−4bad102c290b"data−toc−id="599b0777−f66b−4032−89a7−4bad102c290b"collapsed="false"seolevelmigrated="true">ComparingOptions:Price−to−RentandRelatedRatios</h4><ul><li><p>Price−to−rentratiohelpsdecidewhethertobuyorrentahome;ratiosappearinvariousdomainslikesports,healthcare,andcooking.</p></li><li><p>Whencomparingrentvs.price,thinkintermsofaratioofcosttocosttoassessaffordability.</p></li></ul><h6id="2642873d−d12b−4104−a0bd−584c02bde270"data−toc−id="2642873d−d12b−4104−a0bd−584c02bde270"collapsed="false"seolevelmigrated="true">Practice:Additionalratios</h6><p>17)Abasketballplayerscores28pointsin7games.</p><ul><li><p>Pointspergame:people per square mile</p></li></ul><h4 id="599b0777-f66b-4032-89a7-4bad102c290b" data-toc-id="599b0777-f66b-4032-89a7-4bad102c290b" collapsed="false" seolevelmigrated="true">Comparing Options: Price-to-Rent and Related Ratios</h4><ul><li><p>Price-to-rent ratio helps decide whether to buy or rent a home; ratios appear in various domains like sports, health care, and cooking.</p></li><li><p>When comparing rent vs. price, think in terms of a ratio of cost to cost to assess affordability.</p></li></ul><h6 id="2642873d-d12b-4104-a0bd-584c02bde270" data-toc-id="2642873d-d12b-4104-a0bd-584c02bde270" collapsed="false" seolevelmigrated="true">Practice: Additional ratios</h6><p>17) A basketball player scores 28 points in 7 games.</p><ul><li><p>Points per game:\frac{28}{7} = 4pointspergame</p></li></ul><p>18)Arecipeuses3cupsofflourand2cupsofsugar.</p><ul><li><p>Ratio:points per game</p></li></ul><p>18) A recipe uses 3 cups of flour and 2 cups of sugar.</p><ul><li><p>Ratio:3:2;Fraction:; Fraction:\frac{3}{2}

19) A house costs $240,000 while yearly rent is $12,000.

  • Price-to-rent ratio: \frac{240,000}{12,000} = 20:1</p></li></ul><p>20)Ahospitalhas300nursesfor600patients.</p><ul><li><p>Patientspernurse:</p></li></ul><p>20) A hospital has 300 nurses for 600 patients.</p><ul><li><p>Patients per nurse:\frac{600}{300} = 2patientspernurse(expressedaspatients per nurse (expressed as2:1)

Unit Rates (revisited)

  • A unit rate is a ratio with the second quantity equal to 1; always express shopping answers as cost per one.

Practice: Additional unit rates

21) A 16-oz box of cereal costs $4.50.

  • Unit rate: \frac{4.50}{16} = 0.28125 dollars per ounce.

22) 4 liters of juice cost $8.00.

  • Unit rate: \frac{8.00}{4} = 2.00dollarsperliter.</p></li></ul><p>23)Abakermakes72cookiesin3hours.</p><ul><li><p>Unitrate:dollars per liter.</p></li></ul><p>23) A baker makes 72 cookies in 3 hours.</p><ul><li><p>Unit rate:\frac{72}{3} = 24cookiesperhour.</p></li></ul><p>24)Aprinterprints300pagesin15minutes.</p><ul><li><p>Unitrate:cookies per hour.</p></li></ul><p>24) A printer prints 300 pages in 15 minutes.</p><ul><li><p>Unit rate:\frac{300}{15} = 20 pages per minute.

25) A 6-pack of yogurt costs $3.90.

  • Unit rate: \frac{3.90}{6} = 0.65dollarsperyogurtcup.</p></li></ul><p>26)Arunnerruns10kilometersin50minutes.</p><ul><li><p>Unitrate:dollars per yogurt cup.</p></li></ul><p>26) A runner runs 10 kilometers in 50 minutes.</p><ul><li><p>Unit rate:\frac{10}{50} = 0.2kilometersperminute.</p></li></ul><p>27)Acompanyproduces150widgetsin5days.</p><ul><li><p>Unitrate:kilometers per minute.</p></li></ul><p>27) A company produces 150 widgets in 5 days.</p><ul><li><p>Unit rate:\frac{150}{5} = 30 widgets per day.

28) A 20-pound bag of dog food costs $24.00.

  • Unit rate: \frac{24.00}{20} = 1.20$$ dollars per pound.