Understanding Ratios and Proportional Reasoning

Unit 3 - Ratios & Proportional Relationships Review

An equation stating that two ratios are EQUAL.

Is a fraction that compares two different things. The top and the bottom numbers represent different things.

Ratios & Proportions

Ratio​

​Proportion

How do you determine if two ratios are proportional?

Cross-multiply, if the cross-products are the same, the ratios are proportional. If the cross-products are different, the ratios are not proportional

Multiple Choice

Which two ratios form a proportion?

• $$\frac{8}{12}\ and\ \frac{4}{8}$$  

• $$\frac{8}{12}\ and\ \frac{1}{4}$$  

• $$\frac{8}{12}\ and\ \frac{4}{6}$$  

• $$\frac{8}{12}and\ \frac{4}{1}$$  

Multiple Choice

Suzanne can work 10 math problems in 15 minutes. How many can she work in an hour and a half?

• 45

• 60

• 70

• 75

How to identify proportional relationships in EQUATIONS

​Must be in the form y = kx

- NO adding or subtracting

​- if it is division, variable must be on the top, NOT THE BOTTOM

Multiple Choice

Which equation represents a proportional relationship?

• $$y=\frac{1}{2}x$$  

• $$y=\frac{1}{2}x+1$$  

• $$y=\frac{1}{2}x+2$$  

• $$y=\frac{1}{2}x+\frac{1}{2}$$  

How to identify proportional relationships in TABLES

​Cross-products or quotients must be the same!!

Look at the examples:

Multiple Choice

Is this table proportional or non-proportional?

• Proportional
• Non-Proportional
• both
• neither

How to identify proportional relationships in GRAPHS

Graphs must do BOTH of these two things:

1. Be a straight line

2. Go through the origin (0,0)​

Multiple Choice

Which graph is proportional?

• A
• B
• C and D
• All of the graphs are proportional

Multiple Choice

Is the graph proportional or non proportional?

• proportional
• non proportional

Constant of Proportionality

is the value that is consistently multiplied by the independent variable (x) of a situation

Can also be understood as the unit rate of a situation (the amount per 1 unit)

​Constant of proportionality = k

Divide any non-zero y-value by its corresponding x-value

Tables

k. The number that gets multiplied by x.

y =kx

Example: y = 5x

k = 5 ​

​

Equations

How to find (k) constant of proportionality

Graphs

Find a coordinate pair (x , y) on the line and divide the y by the x​

​Remember that k = y/x

Multiple Choice

In the equation y = 14x, what is the constant of proportionality?

• y
• x
• 14

Examples:

​Remember that k = y/x

​1. What is the constant of proportionality in this table?

2. What is the constant of proportionality in this graph?

​- choose any y-value and its corresponding x-value y=80 x=5

​

- divide y/x 80/5 k=16​

​

​- choose any coordinate pair (point on the graph (x,y) (2,20)

- divide y/x 20/2 k=10​

Multiple Choice

What is the constant of proportionality for this table?

• 1/9
• 9
• 8

Multiple Choice

What is the constant of proportionality for the line on the graph below?

• 2

• 1

• -1

• $$\frac{1}{2}$$

Multiple Select

Check all the words that mean the same thing...

• constant of proportionality

• k

• unit rate

• direct variation

Writing Equations for Proportional Relationships

1.Find k. Remember:

​

Plug in k into the equation y = kx ​

Some text here about the topic of discussion

Multiple Choice

Write an equation for this relationship.

• y=1/5x
• y=1/2x
• y=2x
• y=5x

Multiple Choice

Which equation matches the graph?

• y = 8x
• y = 1x
• y = 3x
• y = 16x

Unit Rate

Is just another word for constant of proportionality

We use rates to COMPARE ratios by dividing by the unit we want to know one of

- Unit rate tells us the rate PER ONE UNIT​

- In unit rates, the denominator is always 1!!!​

Some text here about the topic of discussion

Multiple Choice

Which of these describes the unit rate?

• a unit of measure
• a fraction with a numerator of one
• a fraction with a denominator of one
• a proportion

Multiple Choice

Albert wants to buy some pens. He compares 4 different packs of pens. Which two packs of pens have the same cost per pen?

• Packs 1 and 2

• Packs 1 and 3

• Packs 2 and 4

• Packs 1 and 4

Multiple Choice

Elijah earn $200 for 8 hours of work. What is the unit rate (in dollars per hour)?

• $50 per hour
• $12 per hour
• $192 per hour
• $25 per hour

Multiple Choice

Walmart sells 24 bottles of water for $3 and 36 bottles of water for $4. Which is the better buy and by how much?

• 24 bottles of water for $3 by 1.5 cents per bottle

• 24 bottles for $3 by 0.70 cents per bottle

• 24 bottles for $3 by 0.75 cents per bottle

• 36 bottles for $4 by 1.4 cents per bottle

Multiple Select

A store had 2 blue chairs for $22.78 or 6 red chairs for $67.80. Which color chair has the lower unit price?

• 2 blue chairs for $22.78

• 6 red chairs for $67.80

Multiple Choice

A candy store can make one piece of candy in 15 seconds. What is the rate per minute?

• 2 candies per minute

• 3 candies per minute

• 4 candies per minute

• 5 candies per minute

Unit Rates with Complex Fractions

When you divide a fraction by a fraction that is called a complex fractions

Some text here about the topic of discussion

Multiple Choice

Simplify the complex fraction shown.

• 9/40
• 40/9
• 10/36
• 5/18

Multiple Choice

It took Julius 1/3 of an hour to ride his bike 2 & 3/4 of a mile. How many miles per hour can he ride his bike?

• 3 & 1/2
• 6 & 3/4
• 8 & 1/4
• 16 & 1/2

Multiple Choice

Mr. Roman drove from his hometown to another city. He used 12 & 1/4 gallons of gas for the 318 & 1/2 mile trip. On average, how many miles did Mr. Roman travel per gallon of gas?

• 26
• 26 & 1/2
• 30 & 5/8
• 38 & 4/25

Interpreting Graphs

What does the point (1 , 4) represent?

​Remember:

- a point is written as (x, y)

​- the x-values are on the horizontal axis

​

​

​Answer: In 1 hour, they travel 4 miles.

Multiple Choice

Choose the best explaination

• Taylor can write 6 words per minute
• Taylor can write 25 words per minute
• Taylor can write 50 words per minute
• Taylor can write 150 words per minute

Multiple Choice

What is the unit rate for the rain fall?

• 2 cm per hour
• 1 cm per hour
• ¹/₂ cm per hour
• 4 cm per hour

Finding Percent

Multiple Choice

60% of the animals at a pet shop are dogs. If there are 42 dogs at the shop, then how many animals are there at the shop?

• 70 animals

• 18 animals

• 60 animals

• 42 animals

Multiple Choice

In a neighborhood, there are 9 families that own a poodle. If these owners make up 50% of all dog owners in the neighborhood, how many dog owners are there?

What part of the percent proportion are you missing?

• Part

• Whole

• Percent

​

​

- What did they pay after discount is applied?

- What is the sale price of the item?​

​

Some will ask you to find the final price after the discount.

Some problems will ask you to find discount amount only.

Discounts & Markdowns

​- How much of a discount did they receive?

- How much did they save?​

​You have to pay attention to what the question is asking!!!

Multiple Choice

A bottle of soda is on sale for 15% off. The soda is $3.50. How much is the bottle of soda after the discount?

• $4.03

• $2.97

• $0.53

• $0.75

Multiple Choice

A clothing store is having a sale for 35% off all clothes. A shirt original price is $69.99. What is the sale price of the shirt?

• $45.49

• $24.50

• $94.49

• $32.59

​

​

Some you have to find the total cost (add the % amount to the original price):

Some you have to find % only:

Tax, Tip, Mark-up, Commission

​Again, you have to pay attention to what the question is asking!!!

​- How much did they pay in sales tax or tip?

- What is the markup amount?

- How much did they pay in commission?​

​- What was the total cost?

- How much did they have to pay back total with the commission?

- What is the profit?​

Multiple Choice

Martin went shopping at a bookstore. The price of the book he selected was $25.99 and the sales tax rate at the bookstore was 4%. What was the sale price of the book?

• $10.40

• $1.04

• $27.03

• $29.99

Multiple Choice

The original price of a collectible train set is $345.75. The sales tax for this train set is 5.5%. What is the sale price of the train set?

• $19.02

• $326.73

• $364.77

• $22.93

Simple Interest

​Use the formula:

Multiple Choice

Maria borrowed $3,000 at a simple interest rate of 4% per year.  How much did she have to repay after 4 years?

• $480
• $3,480
• $4,800
• $7,800