4.04/4.05 Applications of Ratios and Percents

Applications of Ratios and Percents

4.04/4.05

Objectives

• After completing this lesson, you will be able to say:

  
  

  •I can use proportional relationships to solve multi-step ratio problems.

  •I can use proportional relationships to solve multi-step percent problems

Poll

Let's start with a simple question!

Did you just wake up?

• Yes

• No

• I haven't slept yet

Poll

Would you rather live without heat and AC or live without social media?

• Live without heat

• Live without social media

Multiple Choice

What is a ratio?

• A comparison of quantities

• A comparison of measurements with a denominator of 1

• A part per hundred

Multiple Choice

What is a unit rate?

• A comparison of quantities

• A comparison of quantities with different units (with a denominator of 1)

• A part per hundred

Let's Start!

Mr. Peabody needs to pour cement for his new time machine, “The Way Back”. Currently, his crew has completed 500 square feet in the last 40 days. The total area will be 800 square feet… at this rate how long will it take to complete the project?

Multiple Choice

How should I set up the ratio? 

"500 square feet in the last 40 days.  The total area will be 800 square feet… "

•  $\frac{500}{40}$  

•  $\frac{500}{80}$  

•  $\frac{40}{800}$  

•  $\frac{800}{40}$  

Now we should find the unit rate...

 $\frac{500\ sq\ feet}{40\ days}=\frac{x}{1\ day}$  

Multiple Choice

What is the unit rate for  $\frac{500}{40}$  ?

•  $\frac{40}{500}$  or 0.08

•  $\frac{500}{1}$  or 500

•  $\frac{40}{1}$  or 40

•  $\frac{500}{40}$  or 12.5

Multiple Choice

How should I use the unit rate  $\frac{12.5}{1}$   to solve for how long it'll take to cover 800 square feet?

• Add 12.5 and 800

• Subtract 12.5 and 800

• Multiply 12.5 and 800

• Divide 12.5 and 800

• Set up a proportion! 

Summary!

 $\frac{500\ ft^{2\ }}{40\ days}=\frac{12.5\ ft^{2\ }}{1\ day}$  

 $\frac{12.5\ ft^{\ 2}}{1\ day}=\frac{800\ ft^{2\ }}{x\ days}$  

Complex Fractions

Mr. Peabody’s Workers have been painting his lab for ¼ of the month… they painted 1/3 of his lab. How many months will it take them to paint the entire lab?

Multiple Choice

How should I set up this complex fraction?

Mr. Peabody’s Workers have been painting his lab for ¼ of the month… they painted 1/3 of his lab.  

•  $\frac{1}{4}\div\frac{1}{3}$  

•  $\frac{1}{3}\div\frac{1}{4}$  

 $\frac{1}{4}\div\frac{1}{3}=\frac{\frac{1}{4}}{\frac{1}{3}}$  

 $\frac{\frac{1}{4}month}{\frac{1}{3}of\ the\ lab}$  is the rate that we are working with. 

We want to find out how much it would take for 1 month so we will divide using KCF (Keep, Change, Flip)

Multiple Choice

What should $\frac{1}{4}\div\frac{1}{3}$  look like after we use KCF (Keep, Change, Flip)

•  $\frac{1}{4}\ \times\ \frac{1}{3}$  

•  $\frac{1}{3}\ \times\ \frac{1}{4}$  

•  $\frac{1}{4}\ \times\ \frac{3}{1}$  

•  $\frac{1}{3}\ \times\frac{4}{1}$  

Multiple Choice

 $\frac{1}{4}\ \times\ \frac{3}{1}$  

Let's multiply

•  $\frac{3}{4}$  

•  $\frac{4}{3}$  

•  $\frac{4}{5}$  

Summary of Complex Fractions

 $\frac{1}{4}\ month\div\ \frac{1}{3}lab\ =\ \frac{1}{4}\ \times\ \frac{3}{1}\ =\ \frac{3}{4}$  month! 

Extra Example

If 3/4 of a ton of sand covers 1/5 of a playground, how many tons of sand are required to cover the entire playground?

Multiple Choice

If 3/4 of a ton of sand covers 1/5 of a playground, how many tons of sand are required to cover the entire playground?

• $\frac{3}{20}$

• $\frac{15}{4}$

• $\frac{4}{9}$

• $\frac{9}{4}$

Multiple Choice

What does the term "Percent" means?

• a part in every hundred

• divide by 100

• every cent

• how many cents

Percent of Increase or Decrease

Daisy calculated that she would spend $250 on school supplies this year. She actually spent $195 on school supplies. What is Daisy’s percent of error?

Multiple Choice

How should I set up this problem based on the formula?

"She spend $250 on school supplies this year. She actually spent $195 on school supplies."

• $\frac{\left|195-250\right|}{195}\times100$

• $\frac{\left|195-250\right|}{250}\times100$

 $\frac{\left|195-250\right|}{195}\times100$  

 $\frac{\left|195-250\right|}{195}\times100$  

 $\frac{\left|-5\right|}{195}\times100$  

 $\frac{5}{195}=0.0256\ \times100=2.56\%$  

Remember that we multiply by 100 to get percents.

How about another one?

Roger spent 20 hours doing yard work last week. This week he spent 26 hours doing yard work. He says that she spent 130% more time doing yard work this week. Is he correct? 

Open Ended

Roger spent 20 hours doing yard work last week. This week he spent 26 hours doing yard work. He says that she spent 130% more time doing yard work this week. Is he correct?

Answer to "Roger spent 20 hours doing yard work last week. This week he spent 26 hours doing yard work. He says that she spent 130% more time doing yard work this week. Is he correct?"

 $\frac{\left|20-26\right|}{20}\times100$  

 $\frac{\left|-6\right|}{20\ }\times100\ =\ \frac{6}{20}\ =\ 0.3\ \times100\ =\ 30\%$ 


No, he is not because he only spent 30% more time.  

Now for something a little different

Jamie read 50 pages of her 400-page book in 6 hours. At this rate, how long will it take her to read the entire book?

Multiple Choice

Marcus painted $\frac{1}{5}$   of his bedroom in  $\frac{1}{3}$  of an hour. At this rate, how long would it take him to finish the room?

•  $\frac{1}{15}$  

•  $\frac{5}{3}$  

•  $\frac{3}{5}$  

•  $\frac{1}{8}$  

Answer!

 $\frac{1}{5}\div\ \frac{1}{3}\ =\ \frac{1}{5}\ \times\ \frac{3}{1}\ =\ \frac{3}{5}hours$