Solving and Applying Proportions Name Core pg. 1
L. 4.1 Ratio and Proportion Notes Ratio- a comparison of 2 numbers by -written. a:b, a to b, or a/b. For example if there are twice as many girls in this class, we say the ratio of girls to boys is 2 to 1. We could also write or. Rate-If two numbers represent in different units, then the ratio of the two numbers is a rate. Unit rate- Is a rate with a of 1. To find a unit rate: divide both the numerator and the denominator by the value of the denominator Proportion- an that states that are equal. a c read as a is to b as c is to d b d for b 0 and d 0 of the proportion- a and d of the proportion- b and c Cross Products of a Proportion- if a c then ad = bc b d -You can use cross products to solve for an unknown variable in a proportion -To solve a multi-step Proportion: use cross products, distribute, and solve for the unknown variable. Examples Ex. 1 Find the Unit Rate Ex. 2 Find x 40miles x 25 5hours 4 12 Ex. 3 x+2 x Ex. 4 You are riding your bicycle. It takes you 12 min. to 14 10 go 2.5 miles. If you continue traveling at the same Rate, how long will it take you to go 7 miles? pg. 2
Finding and Comparing Unit Rate Example 1) You are shopping for PS3 Games and you want to know which store offers the best deal. Store A Store B Store C $25 for 2 games $45 for 4 games $30 for 3 games Notice: We must divide the ratios to figure out the cost for 1 game (THIS IS CALLED THE UNIT RATE) Example 2) You are buying doughnuts for your family for breakfast. Which package is the best deal? Package A Package B Package C $3.50 for 6 doughnuts $5.00 for 8 doughnuts $6.00 for 12 doughnuts Converting Rates Example 1) A cheetah ran 300 feet in 2.92 seconds. What was the cheetah s speed in miles per hour? We need to convert feet to miles and seconds to hours. Example 2) A sloth travels at.15 miles per hour. Convert this speed to feet per minute. Practice 1. Convert 140 hours to days 2. Convert 21 yards to feet pg. 3
4-2 Proportions and Similar Figures NOTES Similar Figures: have the same but not necessarily are the same. In similar triangles: E ABC ~ DEF B 12 15 8 10 C D 18 F Ex: #1: Find the Length of a Side: E ABC DFE C 21cm 18cm A x 10cm B D 15cm F QRS WXY Q 32cm 24cm W R S X m Y 20cm pg. 4
Solving Similar Figures pg. 5
Finding Distances on Maps 4.2 Proportions and Similar Figures (cont d) Example 1: How many miles is it from Tampa to Jacksonville? The scale of the map is 1 cm : 100 miles If you measure from Tampa to Jacksonville on the map you get Map distance = 1.72 cm Write a proportion to figure out the actual distance. Example 2: The scale of a map is 1 inch : 12.5 miles. Find the actual distance for each given map distance. a. 7 inches b. 3.25 inches c. 15.6 inches pg. 6
L. 4.3 Proportions and Percent Equation Notes REMEMBER: a percent is a that compares a number to. Using Proportions to solve percent problems: plug in for what you have and solve for the unknown a % is % b 100 of 100 Examples 1. What is 85% of 320? 2. 393 is 60% of what number? 3. Find 75% of 320. 4. 58% of 60 Using the Percent equation to solve: plug in for what you have and solve for the unknown a = p% * b when using the percent equation you must o change the percent to a decimal if it is part of the known or o change the decimal to a percent if it is the unknown Examples a = p% * b 1. What % of 170 is 68? 2. What is 85% of 320? (change 85% to.85 for equation) 3. 49% of 280 is what number? 4. 45 is what percent of 60? First change 49%.49 pg. 7
4.3 Application: Finance The formula for is I = p r t Simple interest principal amount interest rate (per year) time in decimal form (in years) What does this mean???? Interest you earn = amount of money interest rate your number of years on your money you deposit in bank money will earn you leave your $ in the bank OR SIMPLE INTEREST YOU PAY ON A LOAN..LIKE TO BUY A CAR Interest you pay = amount borrowed interest rate length of the loan to the bank from the bank (in dec. form) (in years) Examples: 1. You invest $2000 in a bank account. Find the amount of simple interest you earn in two years for an annual interest rate of 5.5%. 2. You require a loan to buy your first car. The current interest rate for used cars is 5.61% if you finance your loan for 60 months. The car you want costs $12,000 and you have saved $2000 to pay upfront. a. What is the principal amount of the loan you need? b. What is the length of the loan (in years)? c. How much interest will you pay in total on your loan? d. What is the total amount you pay for the car after the loan is paid off? 3. Find the missing value: I = $1540, p = $2200, r = 3.5%, t =? pg. 8
4.5 Applying Ratios to Probability probability (of an event): P(event) outcome: event: sample space: **Apply the vocabulary to finding the probability of rolling an even number on a number cube event sample space favorable outcome theoretical probability: P(event) = number of favorable outcomes number of possible outcomes P(rolling an even number) = less likely more likely 0 0.5 1 (impossible event) (equally likely/unlikely) (certain event) pg. 9
EX1: Finding Theoretical Probability Suppose you write the names of the days of the week on identical pieces of paper. Find the theoretical probability of picking a piece of paper at random that has the name of a day that starts with the letter T. complement of an event: possible outcomes for outcomes for rolling complement of rolling rolling a number cube an even number an even number ( unfavorable ) The sum of the probabilities of an event and its complement is 1. P(event) + P(not event) = 1 experimental probability: P(event) = number of times the event occurs number of times the experiment is done Ex3: Finding Experimental Probability: The manufacturer decided to inspect 2500 skateboards. There are 2450 skateboards that have no defects. Find the probability that a skateboard selected at random has no defects. EX4: Using Experimental Probability: A manufacturer inspects 700 light bulbs. She finds that the probability that a light bulb works is 99.6%. There are 35,400 light bulbs in the warehouse. Predict how many light bulbs are likely to work. (HINT: turn % into a decimal total number) Guided Practice: You have a bag containing 3 red, 4 blue, 5 white, and 2 black marbles. Find the theoretical probability. 1.) P(not white) 2.) (red or blue) 3.) P(orange) 4.) P(blue or white) pg. 10
4.6 Investigation: Compound Events Suppose you draw cards at random from the following collection: R R A N N N D O M M 1.) You draw an R card and replace it. What is the probability that the next card you draw will be an R card? 2.) You draw an R card and do not replace it. What is the probability that the next card you draw will be an R card? 3.) Copy and complete each table: Probability With Replacement Probability Without Replacement First Card Second Card P(R) = P(R) = P(A) = P(A) = P(N) = P(N) = P(D) = P(D) = P(O) = P(O) = P(M) = P(M) = First Card Second Card P(R) = P(R) = P(A) = P(A) = P(N) = P(N) = P(D) = P(D) = P(O) = P(O) = P(M) = P(M) = 4.) For each letter, the probability of drawing the first cards is the same with replacement and without replacement. Explain why the probability of drawing the second card is not the same. pg. 11
4.6 Probabilitgy of Compound Events Independent event: events that influence each other If A and B are independent events, P(A and B) = Ex1: Independent Events 1.) You have two number cubes, one red and one blue. What is the probability that you will roll a 5 on the red cube and a 1 or 2 on the blue cube? 2.)P(odd and even) 3.) P(1 or 2 and less than 5) Ex2: Selecting With Replacement: The number of items that you are selecting from will change because each time the object is replaced. Therefore your denominator will remain the same. Ex2a: I U I A O O E A O U E A O A E 1.) P(U then I) with replacement 2.) P (I then O) with replacement Dependent event: events that influence each other If A and B are independent events, P(A then B) = Ex3: Selecting Without Replacement The numbers of items that you are selecting from will because each time the event takes place; the object is not replaced or put back. Therefore your denominator as the event is carried out from time to time. Ex3a: I U I A O O E A O U E A O A E 1.) P(U then O) without replacement 2.) P (I then O) without replacement pg. 12
Ex4a: Suppose a teacher must select 2 high school students to represent their school at a conference. The teacher randomly picks names from a hat that contains the names of 3 freshman, 2 sophomores, 4 juniors, and 4 seniors. 1.) P(sophomore then junior) 2.) P(junior then sophomore) Guided Practice 1. Suppose you roll two number cubes. P(odd and multiple of 3) 2. Suppose you have 3 quarters and 5 dimes in your pocket. You take out one coin, and then put it back. Then you take out another coin. P(dime then quarter) 3. A teacher must select students for a conference. The teacher randomly picks names from among 3 freshman, 2 sophomores, 4 juniors and 4 seniors. P(Junior then senior) 4. Suppose you have 3 quarters and 5 dimes in your pocket. You take out one coin from your pocket. Without replacing the coin, you select a second coin. P(dime then quarter) A A B B B C D D E F G G G 5. You select a letter at random and do not replace them. 1.) P(A and then B) 2.) P(vowel then G) 3.) P(A then A) pg. 13