Week 2: Ratio and Proportional Relationships

Ratio: a comparison of two numbers. A ratio can compare a part to a part, a part to a whole, or a whole to a part. Ratios should always be written in simplest terms and labeled.

Write the ratio in the same order it is asked for in the question If no order is given, write the ratio as it appears in the problem. Remember, you can always write the ratio as a fraction to do your calculations.

Part to PartThere are 21 girls and 14 boys in the class. Write the ratio of boys to girls:

be careful: the problem stated boys first:

14boys21girls or 14:21 or 14 to 21.

This can be reduced to:23 or 2:3 or 2 to 3

Part to WholeThere are 21 girls and 14 boys in the class. Write the ratio of boys to the whole class.

add up the two parts to get the whole

14boys35 total or 14:35 or 14 to 35

This can be reduced to:25 or 2:5 or 2 to 5

Rates: a comparison of two numbers with different units.

For example, if we want a rate for 5 cars for 20 people, we write: 5cars

20 people

Unit Rate – a rate in which the second term is 1! (how much for 1? How much per?) Divide Always label Round all answers to the nearest cent (\ Find “better buy” by looking at LOWEST unit

rate

You Try: 1) Twelve notebooks cost $16.68. Find the unit rate.

2) A 28oz can of tomatoes cost $1.89. A 16 oz can of tomatoes cost $.99. Find the better buy.

Ratios Involving Complex Fractions

Jana is training for a triathlon that includes a 112 bike ride. Today, she rode her bike 12 miles in 45 minutes. What is Jana’s rate in miles per hour.

The units are miles and hours. 45 minutes is the same thing as ¾ hour.

number of milesnumber ofhours =

1234

This fraction is a complex fraction.

You can always simplifying by dividing. Top ÷ Bottom

1234

= 121÷ 3

4=12

1× 4

3=48

3=16miles per hour

You Try:1. Jose’s mother is trying to decide whether or not she should buy a 12 ounce package of coffee on

sale for $7.50. She knows that she can buy the same coffee for $9.00 per pound. Which is the better buy?

2. Oliver is training for a marathon. In practice, he runs 15 kilometers in 72 minutes. What is his speed in kilometers per hour?

Complex Fraction: A fraction where either the numerator is a fraction, the denominator is a fraction, or both the numerator and denominator are fractions.

3. Alexis washes 1012 windows in

34 hour. At this rate, how many windows can she wash in one hour?

4. A restaurant uses 814 pounds of carrots to make 6 carrot cakes. Frank wants to use the same

recipe. How many pounds of carrots does Frank need to make one carrot cake?

5. Two friends worked out on treadmills at the gym.

Crystal walked 2 miles in 34 hour.

Shannon walked 134miles in 30 minutes.

Who walked at a faster rate? Explain your reasoning.

ProportionsProportions: shows that two ratios are equal.

Cross Product: the product of the denominator of one fraction and the numerator of the other. If the cross products of two ratios are equal, the form a proportion

You Try: Determine if the ratios form a proportion

414 =

1035

4824 =

242

Solve for a missing number in a proportion –

m120 =

730

620 =

27y

To solve proportion word problems – Set up 2 ratios equal to each other Make sure the labels match (top label matches top label) - always set up a proportion key for

yourself. If it is a TOTAL problem, ADD the ratio to find the total, then solve

Example:

The ratio of boys to girls in your new class is 5 : 2. The sum of the kids in the class is 28. How many boys are in the class?

You Try: 1) Joe uses 6 pencils every 5 days. How many pencils will Joe use in 25 days?

2) The ratio of horses to chickens on a farm is 9 : 4. If there are a total of 169 animals, how many chickens are on the farm?

3) It takes 618 brick to build 100 square feet of wall. A crew of masons on the project laid 2,163 bricks on Monday. How many square feet of wall did they build?

4) The ratio of cats to dogs at the animal shelter is 7 : 5. There are a total of 108 cats and dogs in the animal shelter. Determine how many cats are in the animal shelter.

Proportional Relationships Groups of ratios that are equivalent When a group of ratios are all equivalent – they will all have the same unit rate. In a proportional relationship, the unit rate is called the constant of proportionality.

- Equation will always be in the form y=kx where k is the constant of proportionality- Graph will always be a straight line that passes through the origin (0,0)

Proportional Relationships can be represented in a table, an equation, and a graph.

Example: Apples cost $2.00 per pound. The following example shows a table showing the cost for each number of apples purchased. Note If you buy 0 apples, the cost is 0. For all proportional relationships, if the x value is 0, the y value will also be 0. All proportional relationships will have the point (0,0) on its graph, so the graph will always start at the origin.

Equations in the form y=kx represent proportional relationship. The variables x and y represent the values of each ratio in the

proportion. The variable k represents the constant of proportionality.

(will be the same no matter what the values of x and y are)

The equation y = kx can also be re-written as yx=k

The equation y = kx can be read “y is proportional to x for some constant k”

Proportional Relationships Non-Proportional Relationships- The graph can be represented by a straight

line- The line will go through the origin (0,0)

- The graph may or may not be a straight line- If it is a straight line, it does not go through

the origin.

The equation for this apple situation can be thought of as:

Cost = 2 x number of apples

C = 2a or ca=2

Where 2 is the constant of proportionality.

You Try:Jesse is making punch. For every 3 cups of juice, he needs 6 cups of seltzer. Represent this proportional relationship using a table, a graph, and an equation and identify the constant of proportionality. What does the constant of proportionality represent in this situation?

Table: Juice 1 2 3 4 5 6 7Seltzer

Graph : Constant of Proportionality: _______________

Equation: _____________________________

What does the point (1,2) represent on this graph?

What does the point (2,4) represent on this graph?

For a different recipe, the equation s=4c represents the number of cups of seltzer, z that you need for c cups of cranberry juice. What is the constant of proportionality? Explain.

Multiple Choice Practice: 1. A ferry can hold 200 cars. On average, each car weighs 2,300 pounds. What is the average number of

tons the ferry can hold? (1 ton = 2000 lbs.) (MCC.7.RP.2)

A. 230 tons B. 520 tons C. 600 tons D. 1,400 tons

2. Ron makes and sells greeting cards. He charges $5.00 for 2 cards, $7.50 for 3 cards, $12.50 for 5 cards, and $20.00 for 8 cards. Which equation shows the relationship between the number of cards, n and the total cost, c?

A. C = 25n B .C = 0.4 C. C = 2.5n D. C = 2.5n + 5

3. . Wally and his parents are going on a vacation. The table shows the number of hours they drive and the miles they travel. Use one of the data points to find k (the constant of proportionality). (MCC.7.RP.2b)

Hours (x)

2 2.5 3 4

Miles(y)

100 125 150 200

A. 50 mi. B. 100 mi. C. 125 mi. D. 200 mi.

4. $7.47 for 9 cans. What is the unit rate? (MCC.7.RP.1)

A. $.57 per can B. $.63 per can C. $.74 per can D. $.83 per can

5. Which represents a proportional relationship? (MCC.7.RP.2a)

6. Which represents a proportional relationship? (MCC.7.RP.2)

A. n=6/ p B. n / p=9 C. np=3 D. n=7

7. Which equation shows the proportional relationship in the table below? Use m for miles and h for hours. (MCC.7.RP.2b)

Hours 12 13 14Miles 48 52 56

A. mh=48 B. 12h=48m C. m=4h D. h=4m

0

5

10

15

20

25

0 1 2 3 4

Dollars

Hours

Use the following information and graph to answer questions 8 and 9

The graph shows the amount of money Benjamin earned per hour for a summer job.

8. What is the unit rate? (MCC.7.RP.2d)

A. $6.00 B. $7.00 C. $7.25 D. $7.75

9. What is the meaning of point A on the graph? (MCC.7.RP.2d)

A. Benjamin worked 7 hours and earned $7 per hour.B. Benjamin worked 1 hour and earned $3 per hour.C. Benjamin worked 7 hours and earned $3 per hour.D. Benjamin worked 1 hour and earned $7 per hour.

Use the graph to answer questions 10 and 11

10. What is Debra’s constant of proportionality? (MCC.7.RP.2d)

A. 1.5 B. 2 C. 2.5 D. 3

11. What is Dan’s constant of proportionality? (MCC.7.RP.2d)

A. 1.5 B. 2 C. 2.5 D. 3

A

12) The equation r=34b represents the number of cups of raisins, r, that you need to make b batches

of trail mix. Which point is on the graph that represents this proportional relationship?

a. ( 34 , 1)

b. (4,3)c. (3,4)

d. (0, 34

¿

13)

14)

Extended Response from Test 2012 Solve by finding a unit rate: The item measures 7.RP.1 because it involves computing unit rates associated with ratios of fractions.

First find out how much the tree had grown all together. Then find out how many feet per year it had grown.