Grade Level: 6th Skill: Algebra Topic: Rates Goal: Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity. Skill Description: A rate is a ratio that compares quantities with different units of measure. Some examples of rates are, 7 pages per hour, miles per gallon, \$25 per hour, and 2 miles in 20 minutes. In each of these rates two different units of measure are being compares. For example, in the rate 7 pages per hour, the two units of measure are pages and hours. Equivalent rates can be found using common factors or divisors. So, for the rate 7 page/1 hour can be used to find how many pages in 3 hours. To do this, you simply use 3 as a factor and multiply both terms by it. The equivalent rate will be 21 pages in 3 hours. When the rate of the second term is one, you have unit rate. The unit rate is helpful in determining how much for only one. For instance, if you have \$1 for 5 pencils, the unit rate would be 20 ¢ for one pencil. You find the unit rate by dividing the first term by the second term.

### Sample Problems

 (1) Write the rate for 60 miles in 3 hours. (60 miles in 3 hours = 20 mph) (2) Write 3 equivalent rates for 150 words per minute. (300:2, 450:3, 600:4) (3) DVDs are on sale for \$25.00 for 3. What is the unit rate for DVDs? (about \$8.33 each) (4) Danny wants to buy cupcakes for a class party. The sprinkled cupcakes are 6 for \$3.25. The decorated cupcakes are 8 for \$4.85. Which cupcakes should he buy if he’s wants to get the best deal? (sprinkled) (5) If Nina can type 40 words in 20 seconds, how many words can she type in 60 seconds? (120 words)

### Learning Tips

 (1) There are 2 ways to write a ratio (a/b, a:b and a to b). Rates are normally written in fraction form (a/b). (2) It is helpful to begin to write your rate using a word ratio to compare the units of measure. For the rate, 55 miles in 30 minutes, you would write the word ratio ___miles/___minutes. Now, each time you write an equivalent rate or the unit rate you would use this same word ratio and fill in the new numbers. One mistake that children often make is switching the order of the units of measure, and in turn writing in incorrect numbers. (3) The unit rate can be found by dividing the first term by the second term. You can look at this in two ways. We’ll look at both, using the ratio 250 miles in 10 gallons. First, you can simply divide the first term (250 miles) by the second term (10 gallons). When you do this you get a unit rate of 25 miles in one gallon. The reason this works can be seen clearly looking at the terms the second way, as a proportion. 250 miles = 25 miles 10 gallons 1 gallon The goal is to find out how many miles are in one gallon. The way to turn 10 gallons into 1 is to divide by 10. It’s very important that to keep your ratios proportional, that whatever you do to the bottom, you must also do to the top. In conclusion, dividing the first term and second term by 10 made the new ratio. (4) You can either multiply or divide both terms of the ratio by the same number to find equivalent rates. For example, if you run one mile in 10 minutes, you can use this rate to find out how many miles you could run in 40 minutes. To make the computations you need to begin with your word ratio to show the units of measure you’re comparing (mile/minutes). Now, you will fill in the terms, using the rate you know (1 mile/10 minutes). Next, you write another ratio, filling in the term you know and leaving the other blank (make sure you keep your units of measure in the same place) ___miles/40 minutes. If you wanted to write this as a proportion it would look like this: 1 mi = ___mi 10 min. 40 min. Then, find out what was done to the like units. In other words, what was done to change the 10 min. into 40 min.? Since 10 x 4 = 40, we know that 4 was the factor used. Remember, whatever you do to the bottom ratio, you must do to the top. So, 1 must be multiplied by 4 to find the missing top term. This means the equivalent rate is 4 mi in 40 min. (5) There is no need to add or subtract when finding equivalent or unit rates. You will always divide to find unit rate. You can always multiply to find equivalent rates. The important thing to remember is that the numerator and denominator must always be divided or multiplied by the same number.

### Online Resources

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### Extra Help Problems

 (1) Write the rate 3 cm to 5 second as a ratio, three different ways. (3:5, 3/5, 3 to 5) (2) Which ratio represents the rate, 5 apples for \$6? Explain.a) 5/6 b) 6/5 (a, shows the ratio of apples to dollars in the correct order. B is incorrect, because it shows the ration of dollars to apples.) (3) Write a rate that is equivalent to ¾? (6/8) (4) Write 3 ratios that are equivalent to the rate 8 tickets for \$12.00. (Sample answers - 1:\$1.50, 4: \$6, 16:\$24) (5) Write 3 ratios that are equivalent to the rate 9 miles per hour. (Sample answers – 18:2, 27:3, 1:9) (6) Write as a rate and a unit rate. 70 km in 7 hours (70:7, 10:1) (7) Write as a rate and a unit rate. 25 cm in 5 sec. (25:5, 5:1) (8) Write as a rate and a unit rate. 650 miles on 50 gallons (650/50, 13/1) (9) Write as a rate and a unit rate. \$5.60 for 8 lbs. (5.60/8, .70/1) (10) Write as a rate and a unit rate. 14 minutes to run 3 laps (14/3, 4.67/1) (11) Write as a rate and a unit rate. \$25.00 for 6 cds (25/6, 4.17/6) (12) Write as a rate and a unit rate. 72 mm in 5 seconds (72:5, 14/4:1) (13) Write as a rate and a unit rate. 516 miles in 12 hours (516/12, 43/1) (14) Write as a rate and a unit rate. 24 soccer players to 12 baseball players (24:12, 2:1) (15) Write as a rate and a unit rate. 12 calls to 48 text messages (12:48, 1:4) (16) If Stacy gets 2 calls for every 6 texts, how many calls will she get if she has 18 texts. (2:6, 6:18) (17) Jasmin runs at a rate of 4 laps in 12 minutes. How long will it take her to run 6 laps? (4/12, 6/18) (18) Ty can ride his bike 7 miles per hour. How far will he ride in 7 hours? (7/1, 49/7) (19) If Juan visits 4 countries in 6 days, how many countries can he expect to visit in one month (30 days)? (20) (20) If sodas are one sale for \$2.25 for 6, how much would you pay for 18? (\$6.75) (21) Soda 1 is on sale for \$1.25 for 2. Soda 2 costs \$3.35 for 4. Which soda is a better buy? (soda 1) (22) Lacey can buy 6 CDs for \$25.00 or 5 CDs for \$23.25. Which is the better buy? (6 CDs for \$25.00) (23) Tim can run 5 laps in 13. 5 minutes. Jason can run 3 laps in 9.2 minutes. Who is the faster runner? (Tim) (24) Taylor has driven 40 miles in 50 minutes. Sam has driven 30 miles in 30 minutes. Who is driving faster? (Sam) (25) Jill’s racecar can travel 120 mph. What is its maximum rate of speed in miles per minute? (2 miles per minute)