RATES AND UNIT RATES

Ratios, Rates and Unit Rates

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Speed Limits in Miles per Hour and Kilometers per Hour

A ratio is a comparison of numbers with the same units.

If we compare the weights of 2 bags of potatoes, one 20 lbs, the other 5 lbs;
the ratio is 20 : 5 or 4 to 1. Both weights are measured in pounds.

A rate is a comparison of numbers with different units.

The speed limit signs above are in miles per hour and kilometers per hour.

These are price rates -- given in cost per pound.

These speed limits and price rates are known as unit rates because they give us the rate per unit -- in the speed limits, the unit is hours -- in the prices, the unit is pounds.

Unit Rates for Every One

Say we know that the average tomato weighs 4 ounces. We could find another unit rate from the one above (99¢/lb) since we know that 1 lb = 16 ounces, so there is an average of 4 tomatoes per pound. The unit price per tomato then is 99¢ ÷ 4 = 24.75¢. If we buy a single tomato, it will cost about 25¢.

The speed limit of 55 miles per hour is the same as the ratio 55 miles to 1 hour.
The unit price of a tomato is 25¢ to 1 tomato or 99¢ to 1 pound.

The second term in a unit rate is always 1!

Everyday at work and play we measure rates. We worry about fuel consumption in miles per gallon or kilometer per liter. On the job, workers are paid in dollars or Euros per hour. The tire pressure on our cars and bicycles is measured in pounds per square inch and good musicians measure beats per minute. Typists are rated by the number of words per minute and runners by the rate of yards or meters per minute.

Finding the Unit Rate

When our data is such that we don't have a unit rate but we know the ratio of one unit to the other, we can find a unit rate -- like we did with the tomatoes. Some things, like packaged cheese or cereal are priced by the box or package. The price tells us the cost of 1 box that weighs a given amount. To find the unit price, we divide the price of a box or package by its weight.

Changing Rates and their Units

In 1971, when Canada "went Metric", car owners had to learn to convert speed limits from miles per hour to kilometers per hour. Their fuel consumption rates changed from miles per gallon to kilometers per liter. The 60 mile per hour speed limit on the highway became 100 kilometers per hour. We simply changed miles to kilometers. Hours stayed the same.

A fuel efficient car with a rating of 30 miles to the gallon had a metric fuel rating of
12.73 kilometers per liter. For this conversion, we had to change both units. Miles became kilometers and gallons changed to liters. Since 30 miles is 48.28 kilometers, and 1 gallon = 3.79 liters, we divide 48.28 km by 3.79 L to get the metric fuel rating of 12.73 kms/ L

Comparing Rates

To compare rates we convert them to unit rates in the same units.

A Rate must include both units of measure.

Example

The 1000 year old cedar tree known as the "Hanging Garden Tree", like all trees, takes water from the soil and returns it to the atmosphere as vapour, at a rate of 300 liters every 10 hours. A large oak tree converts 2100 liters of ground water to vapour every 10 hours.
a) How many liters of water vapour per hour does each tree release?
b) What is the ratio of the two hourly rates?

Solution

a) The cedar tree releases 300 liters every 10 hours: 300 ÷ 10 = 30 L/hr.
The oak tree releases 2100 liters every 10 hours: 2100 ÷ 10 = 210 L/hr.

b) the ratio of oak tree to cedar tree is 7 to 1.

Example: Nathalie Lambert, a world champion speed skater won the 1 kilometer race in a record breaking 1 minute and 45.46 seconds. What was her speed in meters per second?

Solution

Since we want her rate of speed in meters per second, we convert 1 km to 1000 meters and we change 1 min 45.46 seconds to 105.46 seconds (1 minute = 60 seconds). Now, we divide the distance by the time to get her rate of speed in meters per second.

1000 m ÷ 105.46 s = 9.4822 or 9.5 m/s.

Example: A 4 ounce jar of pure strawberry jam costs \$2.80. A 10 ounce jar of the same jam costs \$6.75. Which size jar is the better buy?

Solution

We find the unit price per ounce and then we compare.
The 4 ounce jar sells for 2.80 ÷ 4 = 70¢/ounce, the 10-oz jar sells for 67.5¢/ounce, so the larger jar is the better buy. It costs 2.5¢ less per ounce.

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

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Practice Exercises

1) Find the unit price for these items:

 a) \$4.50 for 5 pens b) milk at \$6.60 for 4 L c) 12 bread rolls at \$2.88 d) 400 g of oats for \$2.79 e) 15 candies for \$0.75 f) 8 pounds of turnips for \$3.87

2) Find the unit rate:

 a) typing 520 words in 10 min b) 180 miles in 3 hours c) \$83.25 pay for 9 hours work

3) Which of these is the better buy?

 a) 8 granola bars for \$1.89 or 12 for \$2.89 b) 10 bus tokens for \$13 or 3 for \$4.50 c) 48 tea bags for \$2.19 or 72 for \$3.39 d) 200 sheets of paper at \$1.98 or 500 at \$4.49

4) a) Jeffrey earned \$105 for 12 hours of work cutting lawns. His friend Brad earned \$126.75 at the local market for 15 hours of work. Which boy had the higher rate of pay and by how much?

b) Nikki won the 800-meter race with a speed of 2 minutes 4.23 seconds. Her friend Angela won the women's 1500-meter race in 4 minutes 15.31 seconds. Which woman ran faster?

c) In 667 basketball games, Michael Jordan scored 21 541 points. In 1560 games, Kareem Abdul-Jabbar scored 38 387 points for his team. Who scored more points per game and by how many?

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Solutions

1) Find the unit price for these items:

 a) \$4.50 for 5 pens\$4.50 ÷ 5 = 90¢/pen b) milk at \$6.60 for 4 L \$6.60 ÷ 4L = \$1.65/L c) 12 bread rolls at \$2.88\$2.88 ÷ 12 = 0.24/roll d) 400 g of oats for \$2.79\$2.79 ÷ 400 g = 0.7¢/g a better unit price is /100 g \$2.79 ÷ 4 = 69.8¢/100 g e) 15 candies for \$0.75\$0.75 ÷ 15 = 5¢/ candy f) 8 pounds for \$3.87\$3.87 ÷ 8 = 48.375¢/lb

2) Find the unit rate:

 a) 520 words in 10 min520 ÷ 10 = 52 words/min b) 180 miles in 3 hours180 ÷ 3 = 60 mi/hr c) \$83.25 for 9 hours work\$83.25 ÷ 9 = \$9.25/hr

3) Which of these is the better buy?

 a) 8 granola bars for \$1.89 or 12 for \$2.89\$1.89 ÷ 8 = 23.63¢/bar \$2.89 ÷ 12 = 24.083 ¢/bar8 granola bars for \$1.89 is the better buy. b) 10 bus tokens for \$13 or 3 for \$4.50\$13 ÷ 10 = \$1.30 / token \$4.50 ÷ 3 = \$1.50 / token10 bus tokens for \$13 is the better buy. c) 48 tea bags for \$2.19 or 72 for \$3.39\$2.19 ÷ 48 = 4.56 ¢/t-bag \$3.39 ÷ 72 = 4.708 ¢/t-bag48 tea bags for \$2.19 is the better buy. d) 200 sheets of paper at \$1.98 or 500 at \$4.49\$1.98 ÷ 200 = 9.9 ¢/sheet \$4.49 ÷ 500 = 8.98 ¢/sheet500 sheets at \$4.49 is the better buy.

4)

 a) \$105 in 12 hrs = \$8.75/hr\$126.75 in 15 hrs = \$8.45/hrJeffrey made \$0.30/hr more. b) 800 m ÷ 124.23 s = 6.44 meters/second1500 m ÷ 255.31 s = 5.88 meters/secondNikki ran faster c) 21 541 points in 667 games = 21 541 ÷ 667 = 32.3 pts/game38 387 points in 1560 games = 38 387 ÷ 1560 = 24.61 pts/game Michael Jordan scored 7.69 more points per game.