1. Ratios
Fall 2014

2. Ratios
A to B, A:B, A/B
We describe ratio relationships with words, such as to, for each, for every.
No zeros! Or Negatives.
Order Matters! A:B ≠ B:A unless A and B are equal
Similar to fractions but not a fraction. Ratios are a comparison of two quantities
No units needed

3. Modeling
Table
Tape diagram
Double line diagram

4. Equivalent Ratios
If you can multiply BOTH quantities in a ratio by the same number, then they are equivalent
2:3 = 12 : 18 because 2 x 6 : 3 x6
Or 2/3 = 12/18

5. In a bag of mixed walnuts and cashews, the ratio of the number of walnuts to the number of cashews is 5:6. Determine the amount of walnuts that are in the bag if there are 54 cashews. Use a tape diagram to support your work. Justify your answer by showing that the new ratio you created of the number of walnuts to the number of cashews is equivalent to 5:6.

6. RATIOS vs FRACTIONS Fractions are part/whole
Ratios can be part to part; part to whole, whole to part, etc

7. Multiple Ratios from 2 quantities
In our class there are 6 girls and 7 boys. Write the following ratios:
girls to boys
boys to girls
girls to total
Total to girls
Boys to total
Total to boys
6:7
7:6
6:13
13:6
7:13
13:7

8. In the month of August, a total of 192 registrations were purchased for passenger cars and pickup trucks at the local Department of Motor Vehicles (DMV). The DMV reported that in the month of August, for every 5 passenger cars registered, there were 7 pickup trucks registered.
How many of each type of vehicle were registered in the county in the month of August?
Make a table and a diagram to support your answer.

9. Differences and Total
There is more information in a ratio than just the two quantities.
Add third line to your table. Label it total and enter the totals in your table. What do you notice?
cars 5 10 15 50
trucks 7 14 21 70
total 12 24 36
120

10. Solve the following. Use a table or a diagram to support your answer:
The ratio of the number of people who own a smartphone to the number of people who own a flip phone is 4:3. If 500 more people own a smartphone than a flip phone, how many people own each type of phone?
Sammy and David were selling water bottles to raise money for new football uniforms. Sammy sold 5 water bottles for every 3 water bottles David sold. Together they sold 160 water bottles. How many did each boy sell
At a country concert, the ratio of the number of boys to the number of girls is 2:7. If there are 250 more girls than boys, how many boys are at the concert?

11. The Business Direct Hotel caters to people who travel for different types of business trips. On Saturday night there is not a lot of business travel, so the ratio of the number of occupied rooms to the number of unoccupied rooms is 2:5. However, on Sunday night the ratio of the number of occupied rooms to the number of unoccupied rooms is 6:1 due to the number of business people attending a large conference in the area. If the Business Direct Hotel has 432 occupied rooms on Sunday night, how many unoccupied rooms does it have on Saturday night?

12. Changing Ratio with Similar Totals
Use equivalent ratios and given totals to solve problems. The school band has middle school students and high school students, but it always has the same maximum number of members. Last year the ratio of the number of middle school to high school students was 1:8. This year the ratio of the number of middle school students to high school students changed to 2:7. If there are 18 middle school students in the band this year, how many fewer high school students are in the band this year compared to last year? Explain.

13. At the beginning of 6th grade, the ratio of the number of advanced math students to the number of regular math students was 3:8. However, after taking placement tests, students were moved around changing the ratio of the number of advanced math students to the number of regular math students to 4:7. How many students started in regular math and advanced math if there were 92 students in advanced math after the placement tests?

14. A sporting goods store ordered new bikes and scooters
A sporting goods store ordered new bikes and scooters. For every 3 bikes ordered, 4 scooters were ordered. However, bikes were way more popular than scooters, so the store changed its next order. The new ratio of the number of bikes ordered to the number of scooters ordered was 5:2. If the same amount of sporting equipment was ordered in both orders and 64 scooters were ordered originally, how many bikes were ordered as part of the new order?

15. Comparing Ratios
Using tables
Comparing the ratio directly
Unit rates

16. Which ratio produces a darker grey? Explain your evidence.
Let’s compare two methods of mixing grey paint. One method has a 2/5 ration of black to white paint. The second method has a 3/7 ratio of black to white paint. Complete the tables.
black
white 2 5 35
black
white 3 7 35
Which ratio produces a darker grey? Explain your evidence.

17. Let’s compare two methods of mixing grey paint
Let’s compare two methods of mixing grey paint. One method has a 2/5 ration of black to white paint. The second method has a 3/7 ratio of black to white paint.
black
white 2 5 14 35 6 15
black
white 3 7 15 35 6 14
There are several ways to explain. When the white is equal at 35 the first mixture has 14 parts black which is less than 15parts in the second mixture, so the second mixture will be darker. Or if the black is emade equal at 6 parts you should see that the first mixture has more white making it lighter.

18. Value of the ratio
Comparing ratios. Which ratio is larger? One way to compare ratios is to use tables as we just did Another way is to compare ratios like fractions. Which is larger 2/5 or 3/7? Use common denominators 14/35 or 15/35?

19. Unit rates A third way to compare ratios is to use unit rates
A ratio of two quantities, with different units, such as 5 miles per 2 hours, a rate.
A unit rate is simply a rate with 1 as the denominator, (per 1 unit)
UNITS and ORDER matter!
“per” indicates the divisor. Miles per gallon means Miles ÷ gallon; km per hour means km ÷ hour
Unit price is a unit rate $ per unit. Dollars ÷ units

20. Teagan went to Gamer Realm to buy new video games
Teagan went to Gamer Realm to buy new video games. Gamer Realm was having a sale: $65 for 4 video games. He bought 3 games for himself and one game for his friend, Diego, but Teagan does not know how much Diego owes him for the one game. What is the unit price of the video games? What is the divisor? What is the dividend? What are the units?
$65 ÷ 4 = $16.25 dollars per game

21. You may use a calculator but be sure to estimate your answer first
A publishing company is looking for new employees to type novels to be published. The publishing company wants to find someone who can type at least 45 words per minute. Dominique discovered she can type at a constant rate of 704 words in 16 minutes. Does Dominique type at a fast enough rate to qualify for the job? Explain why or why not.

22. Dave can clean pools at a constant rate of 3/5 pools per hour.
Write the ratio in a different format.
How many pools can Dave clean in 10 hours?
How long does it take Dave to clean 15 pools?
How many pools does Dave clean in 1 hour?
3 to 5 , 3:5
6 pools in 10 hours
25 hours to clean 15 pools
3/5 or 0.6 pools in 1 hour

23. Graphing ratios
A ratio of two quantities, such as in 2 hours to travel 5 km hours, can be written as another quantity called a rate. Make a table of equivalent ratios These become a table of ordered pairs
hours
distance 2 5 x y 2 5

24. Graphing the ratio Choosing intervals for the graph
Since the x values are increasing by 2, we could choose 2 for the interval of the x axis
Since the y values are increasing by 3, we can use 3 for the y interval

25. Label the axis, Label the intervals, and plot the points from the table
Distance in Km
Time in Minutes

26. Watch out for:
Title
Axes
Interval
Labels
Scale T A I L S

27. Graphing Example
A TV station broadcast a specific website so viewers could find information on adopting pets. The website, two hours after the broadcast, had 24 views. An additional hour later, the website had 36 views. Assuming a constant rate of hits:
Write the ratio of hours to views ________
Make a table of equivalent ratios
From the table, create ordered pairs
Make a graph of the relationship of views to time. Remember TAILS.
Hours
Views

28. Review Be able to write a ratio in all formats and use ratio language
Be able to find equivalent ratios
Be able to use tables and diagrams to support your reasoning
Be able to extract all the information in a ratio including using the total and difference
Be able to compare ratios using tables, values of ratios and unit rates
Be able to graph ratios