Chapter 7 – Applying Fractions

Learning Emphasis:Chapter 7 has a lot of math that students will already know how to do.

Ratios and proportions Solving fractional equations Mixture and work problems

May want to consider teaching dimensional analysis during this chapter if time.

Where there is a D: in the notes it denotes a discussion item.

Book Deviations: 7-9 (negative exponents) is taught after chapter 4. 7-5 and 7-6 are combined into one lesson. 7-10, scientific notation can be skipped for 7th grade but 8th graders should have it at some point during the year. It does not appear on the algebra final but is part of 8th grade standards.

Assignments – I assign a lot of odd numbered problems because I want students to check their answers to see if they are doing the problems correctly.

Plan to work some extra mixture and work problems after the two sections. Test/Quiz –two quizzes (one after 7-6 and one for work and mixture problems) and one chapter test

7-1 Ratios

Ratios are division expressions and are written:

Simplifying a ratio is the same as simplifying a fraction.

To compare ratios, both items must be in the same units

Example:

Write the ratio 3 hours: 30 minutes in simplest form.

Hint: Change to the smallest unit for easier division

3 hours = 180 minutes.

Note: also

You try: What is the ratio of a 500 g baseball to a 30kg bowling ball?

The order comes from the problem statement.

We can use ratios to solve problems:If the perimeter of a rectangle is 68 feet and the ratio of length to width is 9:8, what are the dimensions?

Method 1 – solve using substitutionLet L= length and w=width What are the equations?

2L + 2w = 68 or L =

2( ) + 2w = 68

ft and L=18ft

Method 2 – solve using ratio factor

Let 9x = length and 8x = width

but this is the factor to multiply by

L = 9(2)= 18 ft. w = 8(2) = 16 ft

If , what is the ratio of x to y ?

Do the math, do not guess. Get on one side by itself

Divide by y: Divide by 7:

Assignment: Page 290 Multiples of 3, Page 291 #8-12

7-2 Proportions

Proportion: An equation that states two ratios are equal.

D: What else is true about a, b, c and d ?

The cross products are equal:

a and d are the “extremes” ; b and c are the “means”

We use this information to solve problems:

Example: Solve for x if

= 15

We can also solve word problems:

Example: An investment paid $102 interest on a $1200 investment. How much would it have paid on a $1600 investment?

x =

You Try: Solve for w if:

so the answer is ?

“all numbers” which we call “Identity”

Try another:

so

add 20 and 7x to each side: -28x = 24

Assignment: Page 295 #12-36 Multiples of 3, Page

296/297 # 6-11

7-3 Equations with Fractional Coefficients

; ; ; are terms with fract. coeff.

D: How would you solve:

Method 1 : Multiply every term by the LCD

The LCD is 12 so

Method 2: Simplify each side and cross multiply

Example:

n=5

You try:

x=4

Word Problem Example: Three numbers are in the ratio

3:5:6. One fourth of their sum is equal to more than the

smallest. What are the numbers?

Write equations based on the problem statement.Let 3x=first; 5x=second; 6x=third

What is x?

The numbers are 3x, 5x, 6x = 9, 15, 18

Assignment: Page 299/300 #11-27 odd;

Page 300/301 #8-12

7-4 Fractional Equations ( equations with a variable in the denominator)

It is important to look for restrictions as we do these

We are going to use the same method as equations with fractional coefficient. Either simplify each side and cross multiply or multiply by the LCD.

Example: Solve What is the LCD?

Example: Solve

Cross Multiply: 0 = 7 ????

Since the result is not true, there is “no solution”.

You Try:

Cross Multiply: 0 = 0 ? Identity?

“All numbers except 2”. 2 makes the denominator = 0

Try a word problem: The sum of the reciprocals of 2 consecutive even integers

is . What are the integers?

Let n=smaller and n+2 = larger

how do you want to solve?

Factor:

so the solutions are: 10, -

Choose 10 since the problem asked for integers.

The numbers are 10 and 12 .

See Number 10 on page 307

Assignment: Page 306 #13-35; Page 307 #5-11 odd

7-5 / 7-6 Percents and Percent Problems

D: How do you change from fractions to decimals to percents and back?

Fraction to decimal is a division problem

Decimal to percent is multiplication by 100

Decimal to fraction: Rewrite the number as a fraction

and simplify: Example: =

D: What is 3% of 25? What % of 40 is 7? 30 is 6% of what? How do you solve these types of problems?

Method 1: Translate into an equation and solve

Replace “is” with =, “of” with multiply, “what” with a variable.

Method 2: use the equation and replace

with numbers from the problem statement.

D: How do you find percent change?

Make sure you read problems carefully and find what

is asked.

Example: Walnut Springs had a 10% increase in enrollment this year. If there are 902 students this year, how many were there last year?

Let n = number of students last year.

Cross Multiply:

n = 820 students

We also can do interest problems using percents:

Example: Josh earned $36 interest on a bank account that paid 3% interest. How much did he have in the account?

Let a=amount invested

he had $1200 invested

You try: Bob invested $4000, part in stocks and part in bonds. His stocks earned 6% and his bonds earned 4%. How much was invested in each if he earned $220?

Let t = stocks and n = bonds

Substitute and solve: t = $3000 n = $1000Assignment: Page 312/313 # 13-53 prime, 319#8-13

7-7 Mixture Problems

Mixture problems are often solved using charts and drawing pictures. Focus on defining variables and relationships when setting up these type problems.

D: How much water would it take to dilute one liter of a 5% acid solution to make a 2% acid solution.

How much acid is in the solution?

5% of one liter is 0.05 liters

0.05 liters is 2% of what?

liters However, you already have

1 liter, so you need to add 1.5 liters of liquid.

We will look at a problem using a chart:

A grocer mixes 5 lb. of noodles that cost $0.80 per pound with 2 lbs. of chicken that cost $1.85 per pound. What should the cost per pound be for the mixture?

Total Amount (lbs)

Cost($) = per pound

total cost($)

Noodles 5 0.80 5(0.80)=4.00

Chicken 2 1.85 2(1.85)=3.70Mixture ? ? ?

How are we going to fill in the question marks?The total amount is 7 lbs. Do we know the cost per pound? What else do we know?

Total Amount (lbs)

Cost($) = per pound

total cost($)

Noodles 5 0.80 5(0.80)=$4

Chicken 2 1.50 2(1.85)=$3.70

Mixture 7 x 7x

What is the relationship that helps us solve this?The cost of the mixture = the cost of the other two items

so x = 1.10 or the cost is $1.10 per pound

Additional examples page 323

Assignment: Page 324 / 325#9-17 odd, 20

7-8 Work Problems

Work problems are done using the formula:

The biggest obstacles to solving these problems is understanding work rate. If Joe can do a job in 6 hours, what is his work rate?

Think of work rate as how much of the job you can do in one unit of time. Since Joe takes 6 hours to do a job, his

work rate is = of a job per hour.

If Katy can do 10 problems in 40 minutes, what is her work rate? Hint: The rate is how much she can do in one minute.

problem per minute

Work problems are often solved using charts like :work rate time = work done

Thing 1

Thing 2

Thing 3 sometimesJust like other chart problems, you need to look for relationships in the chart that help you solve the problem.

Example: an installer can carpet a room in 3 hours.

His assistant takes hours. How long will it take

them working together?D: What do you know and what will be variable?

The time is the variable, we know the work rates

and the amount of work being done (1 job).

work rate time = work doneInstaller

Assistant =

Together + t 1

What is the relationship that helps us solve this?

t=?

Additional examples on board – page 328

Assignment: Page 329 #9-14

7-9 Negative Exponents

Consider the following patterns:

D: What happens when you decrease by one exponent from the same base?

Decreasing by one exponent from the same base is the same as dividing by the base.D: What do you think

Since

Then

=

The definition of where n is a positive integer is:

Our rules for positive exponents are true for negative exponents:o When multiplying exponents with the same base,

you add the exponents.o When dividing exponents with the same base, you

subtract the exponentso When raising an exponent to a power, you

multiply the exponents.

Examples:

Always simplify to positive exponents.

What would ?

Simplify

Assignment: Page 333/334 Primes

7-10 Scientific Notation

See Scientific Notation activity