Section 4-1 Ratio and Proportion• A ratio is a comparison of two numbers by

division• The two numbers must have the same units in

a ratio• A rate is a comparison of two numbers by

division where the units are different• For example: 55 miles per hour is a rate

because it compares the number of miles to the number of hours

• Rates with a denominator of 1 are called unit rates (so the 55 mph is a unit rate)

• Ex1. Find the unit price of grapes if you can buy 32 ounces of grapes for $3.96 (hint: you are looking for price per ounce)

• When converting from one unit to another, you must figure out which conversion factor must be used to produce the desired outcome. This process is called unit analysis or dimensional analysis.

• Ex2. Bob drives 292.5 miles in 6.5 hours. How many miles per second would this be?

• A proportion is an equation that states that two ratios are equal

• for b ≠ 0 and d ≠ 0

Proportions are read “a is to b as c is to d”

a and d are called the extremes

b and c are called the means

To solve a proportion you cross-multiply (using the Means-Extremes Property) ad = bc

It can also be written a:b = c:d

• Word problems can be given in this format and are often used on the SAT

a c

b d

• Solve the following proportions• Ex3.

• Ex4.

• Ex5. Complete the following statement: $56/day = _____/week

• See page 748 for conversions you may not know (but you will need to before the quiz/test)

8

9 12

x

3 2 5

9 4

m m

Sect 4-2 Proportions & Similar Figures• Similar figures have the same shape, but not

necessarily the same size (one is usually an enlarged or shrunken version of the other)

• With similar figures all corresponding angles are congruent

• With similar figures all corresponding sides are proportional

• The symbol ~ means “is similar to”

• The symbol means “is congruent to”

• When you are writing similar figures, you must put corresponding angles in corresponding order

• ∆ABC ~ ∆DEF would mean that <A corresponds to <D, <B and <E are corresponding, and <C and <F are corresponding

• Turn your book to pg 190, we are going to examine the info. at the top of the page

• You’ll no doubt figure out that the ratios and proportions can be written in numerous ways, and many would be correct

• A scale drawing is an enlarged or reduced drawing that is similar to the original object

• The ratio of a distance in the drawing to the corresponding actual distance is the scale of the drawing

• Ex1. Find x

68 in

75 in

112 in

x

• Ex2. The scale of a map is 1 in : 20 miles. If two cities are 3.5 inches apart on the map, how far apart are they in real life?

• Ex3. The dimensions on a blueprint are 1 in : 3 feet. If the real dimensions of the room are 12 feet by 15 feet, what will the dimensions be on the blueprint?

Section 4-3 Proportions and Percent Equations

• Ratios can be used to solve percent questions using the formula is = %

of 100

• Another way to think of this same formula ispart = %

whole 100

• Ex1. Find 40% of 200.

• Ex2. 39 is what percent of 158?

• Ex3. 85% of a school’s population has been surveyed. If there were 324 students surveyed, how many total students are there at that school?

• You can also use an equation to solve percent problems using the equation

% · of = is or % · whole = part as long as the percent has been changed to a decimal first

• Ex4. Solve using an equation. 56 is 95% of what number?

• Percents greater than 100% will be greater than 1 when they are changed to a decimal

• Ex5. What is 325% as a decimal?• Percents less than 1% will have more than

one zero following the decimal point when they have been changed to a decimal

• Ex6. What is .35% as a decimal?• Study and know the chart on page 199• Checking the reasonableness of your

answer is always important (see example 6 on page 199)

Section 4-4 Percent of Change• Percent of change = amount of change

original amount• When a value increases, like when you factor

in tax or a markup, the percent is called the percent of increase

• When a value decreases, like when it goes on sale or is marked down, the percent is called the percent of decrease

• Ex1. A TV was originally $350, but it went on sale for $295. What was the percent of decrease (nearest tenth)?

• The greatest possible error in a measurement is ½ of that measuring unit

• Ex2. The length of a room is 14.5 feet. What is the greatest possible error?

• The room was measured to the nearest .5 feet, so the greatest possible error is half of that. ½ of .5 feet is .25 feet, which is the greatest possible error (a.k.a. 3 inches)

• Open your book to page 205, let’s look at example 4

• Percent error = greatest possible error measurement

• Ex3. You measure a friend’s height to be 72.5 inches. What is the percent of error?

• Read example 6 on page 206

Section 4-5 Applying Ratios to Probability

• Probability of an event tells how likely it is to occur, it is written P(event)

• An outcome is what happened in one trial (i.e. if you are rolling a die, it is what you rolled on one single roll)

• An event is any one outcome or a group of outcomes

• Sample space is all possible outcomes

• Theoretical probabilityP(event) = number of favorable outcomes

number of possible outcomes

• Ex1. In a standard 52-card deck, find P(king) =

• Probabilities can be written as simplified fractions, decimals, or percents

• Probability (as a fraction or decimal) can be any number from 0 to 1

• A probability of 0 means it can’t happen• A probability of 1 means it definitely will

happen

• The complement of an event consists of all the outcomes not in the event (the event + the complement make up all possible outcomes)

• P(event) + P(complement) = 1• Experimental probability is based on an actual

experiment and it deals with what happened, not what might happenP(event) = number of times an event occurs

number of times the experiment is done

• Ex2. A quality control inspector looked at 250 bolts. Of those, 14 were unacceptable. What is the probability of choosing a bad bolt if you were to chose one at random from this group?

Section 4-6 Probability of Compound Events

• Independent events are events that do not influence one another

• Probability of Two Independent Events: If A and B are independent events, P(A and B) = P(A) · P(B)

• Ex1. Suppose you roll a 6-sided die twice. The first time you are hoping to roll a 5, and the second time you want to roll a number less than 3. What is the probability?

• Dependent events are events that DO influence each other (the occurrence of one affects the probability of the second)

• (optional to write) For instance, if you were to choose a colored marble from a bag of marbles, and then not place it back in the bag before you drew the next one, those are dependent events

• Probability of Two Dependent Events: If A and B are dependent events, P(A then B) = P(A) · P(B after A)

• Ex2. Tiles containing the letters: A L G E B R A I C are placed into

a bag, find the probability of randomly choosing an A followed by a R, without replacement.