MATH 20 - Chapter 5 - Rate, Ratio and Proportion

1. Write the ratio in lowest terms, then write it in the other two forms.

   Ratio	Lowest Terms	Other Forms
   12 16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGOmaaqaaiaaigdacaaI2aaaaaaa@38F9@
   10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic daaaa@376C@
   26:12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaiA dacaGG6aGaaGymaiaaikdaaaa@39A8@

2. Write the rate as a unit rate.

   1. 450 miles 18 gallons = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI0aGaaGynaiaaicdacaqGGaGaaeyBaiaabMgacaqGSbGaaeyzaiaa bohaaeaacaaIXaGaaGioaiaabccacaqGNbGaaeyyaiaabYgacaqGSb Gaae4Baiaab6gacaqGZbaaaiabg2da9aaa@4735@
   2. 5 feet 2 seconds = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aGaaeiiaiaabAgacaqGLbGaaeyzaiaabshaaeaacaaIYaGaaeii aiaabohacaqGLbGaae4yaiaab+gacaqGUbGaaeizaiaabohaaaGaey ypa0daaa@4402@
   3. 228 calories 6 ounces = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaaGOmaiaaiIdacaqGGaGaae4yaiaabggacaqGSbGaae4Baiaa bkhacaqGPbGaaeyzaiaabohaaeaacaaI2aGaaeiiaiaab+gacaqG1b GaaeOBaiaabogacaqGLbGaae4CaaaacqGH9aqpaaa@4856@
   4. 260 miles 4 hours = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaaGOnaiaaicdacaqGGaGaaeyBaiaabMgacaqGSbGaaeyzaiaa bohaaeaacaaI0aGaaeiiaiaabIgacaqGVbGaaeyDaiaabkhacaqGZb aaaiabg2da9aaa@44B0@
   5. $14.95 5 pounds = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca GGKaGaaGymaiaaisdacaGGUaGaaGyoaiaaiwdaaeaacaaI1aGaaeii aiaabchacaqGVbGaaeyDaiaab6gacaqGKbGaae4CaaaacqGH9aqpaa a@426F@
   6. $390 40 hours = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca GGKaGaaG4maiaaiMdacaaIWaaabaGaaGinaiaaicdacaqGGaGaaeiA aiaab+gacaqG1bGaaeOCaiaabohaaaGaeyypa0daaa@40CA@

3. There are 45 members in a marching band. Fifteen play the trumpet, 12 play the drums, 6 play the French horn, 3 play the saxophone, 2 play the tuba, and 1 plays the cymbals.

   1. What is the ratio of trumpet players to saxophone players? Write your answer in lowest terms.
   2. What is the ratio of trumpet players to band members? Write your answer in lowest terms.
   3. What is the ratio of tuba players to drum players? Write your answer in lowest terms.

Use the internet to find two comparable products online. You can use the same brand in two different sizes or from two different stores, or two similar products from different brands. Record the information from each product and calculate the unit price. Repeat for another pair of comparable products.

• Product, size and price:

  Unit Price:

• Comparable product, size and price:

  Unit Price:

Which one would you buy? Why?

• Product, size and price:

  Unit Price:

• Comparable product, size and price:

  Unit Price:

Which one would you buy? Why?

1. Decide whether the proportion is true or false and then compare the cross products to verify your answer.

   1. 1 2 = 2 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaaaacqGH9aqpdaWcaaqaaiaaikdaaeaacaaI0aaa aaaa@3A0D@
   2. 7 3 = 9 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aaabaGaaG4maaaacqGH9aqpdaWcaaqaaiaaiMdaaeaacaaI0aaa aaaa@3A1C@
   3. 15 18 = 10 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaGaaGynaaqaaiaaigdacaaI4aaaaiabg2da9maalaaabaGaaGym aiaaicdaaeaacaaIXaGaaGOmaaaaaaa@3D00@

2. Solve each proportion:

   1. 7 14 = 2 x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI3aaabaGaaGymaiaaisdaaaGaeyypa0ZaaSaaaeaacaaIYaaabaGa amiEaaaaaaa@3B0F@
   2. 8 12 = x 15 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI4aaabaGaaGymaiaaikdaaaGaeyypa0ZaaSaaaeaacaWG4baabaGa aGymaiaaiwdaaaaaaa@3BCD@
   3. x 20 = 5 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WG4baabaGaaGOmaiaaicdaaaGaeyypa0ZaaSaaaeaacaaI1aaabaGa aGioaaaaaaa@3B11@

Set up a proportion to solve the following rate and proportion problems. Write your answer using a complete sentence.

3. Eduardo spent 6 hours completing 21 math problems last week. How long might he expect to spend completing 28 math problems this week?

       =     MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaWaau IhaeaacaaMf8oaaaqaamaaL4babaGaaGzbVdaaaaGaeyypa0ZaaSaa aeaadaqjEaqaaiaaywW7aaaabaWaauIhaeaacaaMf8oaaaaaaaa@3E66@
4. The sales tax on an item that costs $35 is $2.10. What is the sales tax on an item that costs $68?
5. On a map, 6 centimeters represents 2 miles. If Cowpoke Junction is 9 centimeters from Wild Bill Butte on the map, how many miles apart are these two locations?
6. Horse A can travel 40 kilometers in 4 hours. Horse B can travel 63 kilometers in 7 hours. Which horse travelled faster?
7. Which is steeper a ramp that is 5 feet high and 10 feet long or a ramp that is 8 feet high and 15 feet long? How do you know?
8. Heart disease is the leading cause of death in the U.S. for both men and women. If 1 in 4 deaths is due to heart disease, then how many people in a 32 person class may die of heart disease?
9. Three scoops of coffee are used with 4 cups of water in a coffee machine. Four scoops of coffee are used with 16 cups of water in another coffee machine. Which brew will be stronger? Show your reasoning steps. A diagram may help.

10. A recipe calls for 2 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaala aabaGaaGymaaqaaiaaikdaaaaaaa@383A@ cups of flour and 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaaabaGaaG4maaaaaaa@3780@ cup of sugar.

    1. If you use 1 cup of sugar, how much flour will you need? Show your reasoning.
    2. If you use 4 cups of flour, how much sugar will you need? Show your reasoning.

Conversion Factors

• 1 foot = 12 inches
• 1 yard = 3 feet
• 1 mile = 5280 feet
• 1 minute = 60 seconds
• 1 hour = 60 minutes
• 1 day = 24 hours
• 1 pound = 16 ounces
• 1 ton = 2000 pounds
• 1 cup = 8 fluid ounces
• 1 pint = 2 cups
• 1 quart = 2 pints
• 1 gallon = 4 quarts

Example 1: Convert 3 yards to feet.

Start with your given units and place them over one to make it a fraction. Then multiply by unit fractions. Set up your units to cancel out. For example, if you have yards in the numerator, place yards in the denominator. 3 yards =  3 yards 1 ⋅ 3 feet 1 yard =9 feet MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaabc cacaqG5bGaaeyyaiaabkhacaqGKbGaae4CaiaabccacaqG9aGaaeii amaalaaabaGaaG4maiaabccacaqG5bGaaeyyaiaabkhacaqGKbGaae 4CaaqaaiaaigdaaaGaeyyXIC9aaSaaaeaacaaIZaGaaeiiaiaabAga caqGLbGaaeyzaiaabshaaeaacaaIXaGaaeiiaiaabMhacaqGHbGaae OCaiaabsgaaaGaeyypa0JaaGyoaiaabccacaqGMbGaaeyzaiaabwga caqG0baaaa@578B@

Example 2: Convert 5 days into minutes.

5 days =  5 days 1 ⋅ 24 hours 1 day ⋅ 60 minutes 1 hour =7200 minutes MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynaiaabc cacaqGKbGaaeyyaiaabMhacaqGZbGaaeiiaiaab2dacaqGGaWaaSaa aeaacaqG1aGaaeiiaiaabsgacaqGHbGaaeyEaiaabohaaeaacaaIXa aaaiabgwSixpaalaaabaGaaeOmaiaabsdacaqGGaGaaeiAaiaab+ga caqG1bGaaeOCaiaabohaaeaacaaIXaGaaeiiaiaabsgacaqGHbGaae yEaaaacqGHflY1daWcaaqaaiaabAdacaqGWaGaaeiiaiaab2gacaqG PbGaaeOBaiaabwhacaqG0bGaaeyzaiaabohaaeaacaaIXaGaaeiiai aabIgacaqGVbGaaeyDaiaabkhaaaGaeyypa0JaaG4naiaaikdacaaI WaGaaGimaiaabccacaqGTbGaaeyAaiaab6gacaqG1bGaaeiDaiaabw gacaqGZbaaaa@6BAB@

Try it! Fill in the missing parts to convert 2 quarts to cups.

2 quarts =  2 quarts 1 ⋅ 2                    quarts ⋅         cups =          cups MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOmaiaabc cacaqGXbGaaeyDaiaabggacaqGYbGaaeiDaiaabohacaqGGaGaaeyp aiaabccadaWcaaqaaiaabkdacaqGGaGaaeyCaiaabwhacaqGHbGaae OCaiaabshacaqGZbaabaGaaGymaaaacqGHflY1daWcaaqaaiaabkda caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaaabaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaeyCaiaabwhacaqGHbGaaeOCaiaabs hacaqGZbaaaiabgwSixpaalaaabaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGJbGaaeyDaiaabchacaqGZb aabaaaaiabg2da9iaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGJbGaaeyDaiaabchacaqGZb aaaa@7219@

Practice:

1. 33 yards to feet
2. 80 ounces to pounds
3. 10,000 feet to miles
4. 128 fluid ounces to gallons
5. 2.5 tons to ounces
6. How many seconds are there in one year?

Conversion Factors:

• 1 kilometer = 1000 meters
• 1 hectometer (hm) = 100 meters
• 1 dekameter (dam) = 10 meters
• 1 meter = 10 decimeters (dm)
• 1 meter = 100 centimeters (cm)
• 1 meter = 1000 millimeters (mm)

In the metric system you can use unit fractions or just move the decimal point. The chart below shows how many places to move the decimal point and which direction to move it.

Example 1: Convert 10 kilometers to centimeters.

To get from kilometers to centimeters in the chart, move 5 decimal places to the right.

10 kilometers = 1,000,000 centimeters

Example 2: Convert 1000 milligrams to grams

To get from milligrams to grams in the chart, move 3 decimal places to the left.

1000 milligrams = 1 gram

Practice:

1. 400 grams to kilograms
2. 400 liters to milliliters
3. 500 milligrams to centigrams
4. 1 centimeter to meters
5. 45.5 centimeters to millimeters
6. 0.0034 milliliters to liters
7. Your child’s doctor prescribed 0.0015 grams of medication once per day. Your measuring spoon is labeled with milligrams. How many milligrams of medication should you give your child?