Unit 4 Notes

Unit 4 – Ratio, Proportion, and Percentage Proportions: A proportion is an equation that states that two __________ are equal. Ex1.

4 1 = 20 5 (4)(5)=(20)(1) 20 = 20

4:20 = 1:5 Can be written as.

We can ________ _________ to prove that the two fractions are _______.

We can also ___________ _____________ to solve a proportions.

Ex2. 4 is to 7 as x is to 56. Solve for x.

Ex3.

4 x = 7 56

12 32 = ( x − 2) ( x + 8)

Ex4. The denominator of a fraction exceeds the numerator by 7. If 3 is subtracted from the numerator of the fraction and the denominator is unchanged, the value of the resulting 1 fraction becomes . Find the original fraction. 3

Ratios:

A ratio is a way of comparing 2 numbers by dividing them. We can express a ratio in a few different ways. The ratio 2 to 3 can be expressed as: 2 2 ÷3 or or 2:3 3 •

The order of the numbers matters. The number before the colon in always the numerator and the number after is the denominator.

EX 1: What is the ratio of boys to girls in this class?

State your answer in simplest form:

EX 2: What is the ratio of girls to students in this class?

State your answer in simplest form:

EX 3: The length of a rectangle is represented by 6x and its width by 3x. Find the ratio of the width of the rectangle to its perimeter.

•

In order to compare to quantities, they must be in the same units.

Ex4: Compute the ratio of 6.4 ounces to 1 pound:

Ex5:

Express the ratio 1

3 1 to 1 in simplest terms: 4 2

Ex6:

A student did six of ten problems correctly. a. What is the ratio of the number right to the number wrong?

b. For every two answers that were wrong, how many answers were right?

Continued Ratio: Continued ratio is the ________________ of __________ or more quantities in a definite order. In the accompanying diagram, the length of the rectangle is is 20 cm, the width is 15 cm and the height is 10 cm. What is the ratio of length to width? What is the ratio of width to height? What is the ratio of length to width to height?

Ex7:

A woodworker is fashioning a base for a trophy. He starts with a block of wood whose length is twice its width and whose height is one-half its width. Write, in simplest from, the continued ratio of length to width to height.

Ex8:

Three angles of a triangle are in a 2:3:5 ratio. Find the measure of each angle.

Unit Rate To find unit rate you can _____________ or set up a ____________________. Ex1. If a plane flies 1,920 miles in 3 hours, how many miles will it fly in one hour?

Ex2.

A vacationer traveled 230 miles in 4 hours. What was his average rate of speed expressed in mph.

Ex3.

Tom drove 290 miles from his college to home and used 23.2 gallons of gasoline. His sister, Ann, drove 225 miles from her college to home and used 15 gallons of gasoline. Whose vehicle had better gas mileage? Justify your answer.

Ex4:

A writer was paid $35,000 for a 5,000-word article. Find the rate per word. [A] $70.00 per word [C] $7.00 per word

[B] $0.14 per word [D] $1.43 per word

Verbal Problems Involving Ratios Ex1. The profits in a business are to be shared by the three partners in the ratio of 3 to 2 to

5. The profit for the year was $176,500. Determine the number of dollars each partner is to receive.

Ex2. There are 357 seniors in Harris High School. The ratio of boys to girls is 7:10. How many boys are in the senior class? (1) 210 (2) 147

Ex3.

(3) 117 (4) 107

A hockey team played n games, losing four of them and winning the rest. The ratio of games won to games lost is

n−4 4 4 (2) n−4 (1)

4 n n (4) 4 (3)

Ex4.

A total of $450 is divided into equal shares. If Kate receives four shares, Kevin receives three shares, and Anna receives the remaining two shares, how much money did Kevin receive? (1) $100 (3) $200 2) $150 (4) $250

Ex5.

During a recent winter, the ratio of deer to foxes was 7 to 3 in one county of New York State. If there were 210 foxes in the county, what was the number of deer in the county? (1) 90 (3) 280 (2) 147 (4) 490

Ex6.

Sterling silver is made of sterling silver ingot is 600 (1) 48.65 g (2) 200 g

Ex7.

At the Phoenix Surfboard Company, $306,000 in profits was made last year. This profit was shared by the four partners in the ratio 3:3:5:7. How much more money did the partner with the largest share make than one of the partners with the smallest share?

an alloy of silver and copper in the ratio of 37:3. grams, how much silver does it contain? (3) 450 g (4) 555 g

If the mass of a

Direct Variation:

When building a roof, carpenters place posts every 2 feet along the horizontal support beam starting at the eave. Complete the table post

Horizontal distance from post to eave (d).

Height of post (h)

Ratio h/d

1

2

1.5

0.75

2

4

3

...

3

...

4.5

...

4

...

6

...

When two variable quantities have a constant (unchanged) ratio, their relationship is called a __________ ____________. It is said that one variable "__________ ____________" as the other. The constant ratio is called the constant of variation. The formula for direct variation is y = kx, where k is the ___________ ___________. "y varies directly as x" Solving for k: (y = numerator; x = denominator) In a direct variation, the two variables change at the same rate. If one increases, so does the other.

Ex1: In the following chart, does one variable vary directly with the other?

M

3

4

5

6

7

N

6

8

10

12

14

Ex2: One variable (A) varies directly as the other (C). Find the missing numbers x and y. Write the formula which relates the variables.

A

1

2

x

C

3

y

15

Ex3: In the following table, one variable varies directly as the other. Find the missing numbers and write the formula that relates the variables. X Y

1 5

2 ?

? 25

EX 4: If a is directly proportional to b and a = 2 when b = 8, find a when b = 12.

EX 5: If x is directly proportional to y and x = 1.2 and y = 7.2, find y when x = 4.

Direct Variation Word Problems You can replace the words “varies directly” with the words ___ ____________ ____.

EX 1: There are about 90 calories in 20 grams of a cheese. Reggie ate 70 grams of this cheese. About how many calories were there in the cheese she ate if the number of calories varies directly as the weight of the cheese?

EX 2: A house that is assessed for $12,000 pays $960 in realty taxes. At the same rate, what should be the realty tax on a house assessed for $16,500?

3 Cases of Percent There are 3 types of problems dealing with percent. You can use the following proportion to solve any of the three cases:

part percent is percent = or = whole 100 of 100 1.

Finding the part of the whole:

Find 7% of 250:

2.

Finding the whole. 30 is 15% of what number:

3.

Finding the percent. 3 is what percent of 12?

Try these: 1. 20 is 10% of what number?

4. What is 2% of 36

2. What is 6% of 150

3. 3%of what number is 1.86?

5. 18 is what percent of 12?

6. What is 150% of 18

Ex1.

Find the amount of tax on a $60 radio when the tax rate is 8%. What is the final cost of the radio?

Ex2.

There are 60 pages is a book. 20 of the pages have pictures on them. a. What percent of the book has pictures?

b. What percent of the book has no pictures?

Ex3.

Juan missed 6 out of 92 questions on a test. To the nearest percent, what percent of the questions did he solve correctly?

Ex4.

A real estate company pays commissions to their sales people for selling property. A well known company paid 6% commissions last year to their sales staff, totaling $480,000. What was the dollar value of the real estate sold by the company that year?

Ex5.

Walter is a waiter at the Towne Diner. He earns a daily wage of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?

Ex6.

The owner of a music store received a shipment of stereos at a cost of $160.00 each. What will the selling price be if he applies a 45% markup?

Ex.7. A bookstore offers a 15% discount to its employees. Suppose an employee has a coupon worth $5 off any item and wants to buy a book that costs $45. What is the final cost of the book if the employee discount is taken first and then the coupon is subtracted?

Percent of Increase or Decrease A percent of increase is the ratio or the amount of ______________ or decrease to the ___________ amount.

Percent of increase or decrease =

the difference the original and new amount x100 the original amount

Ex1.

A store reduced the price of a television from $840 to $504. What was the percent of decrease in the price of the television?

Ex2.

Joshua's normal body temperature is 98.5ºF. Due to a cold, his temperature went up 3ºF. To the nearest percent, what was the percent of increase in his body temperature?

Percent of Error = Ex1.

the difference of the measured and true value x100 true value

The length and width of a retangle are 13 inches and 84 inches, the true length of the diagonal, found by using the Pythagorean Theorem, is 85 inches. A student drew this retangle and, using a ruler, found the measure of the diagonal to be 84

7 inches. 8

What would the error of measurement be?

What would the percent of error be? Ex.2

You estimated your monthly car payment to be $315. The actual car payment turned out to be $300. What would the percent of error be?