Ratios Review and Basics Ratio A ratio is
Ratios Review and Basics
Ratio • A ratio is a comparison of two numbers. It shows a relationship between two quantities. • What is the ratio of apples to oranges? 6 to 9 6: 9 2 to 3 2: 3
Ratios • What is the ratio of blue squares to red squares? 12 : 8 Simplify? 3: 2 • What is the ratio of red squares to blue squares? 8 : 12 • Rates and Ratios 1 Simplify? 2: 3
Why are people upset with the Venetian and Palazzo casinos in Las Vegas? • The Venetian and Palazzo casinos have changed their Blackjack payout from a 3: 2 payout to a 6: 5 payout. Win Bet
Why are people upset with the Venetian and Palazzo casinos in Las Vegas? • The Venetian and Palazzo casinos have changed their Blackjack payout from a 3: 2 payout to a 6: 5 payout. • What if you bet $10 3: 2 6: 5 6: 4 9: 6 12 : 8 15 : 10 12 : 10
Unit Rates • Per 1 unit • • • Miles per hour Beats per minute (heart rate) Cost per month Points per game Price per pound
Unit Rates • Per 1 unit • Miles per hour A runner runs 16 miles in 2 hours. What is her unit rate (miles per hour)? A walker walks ½ mile in ¼ hour. What is her unit rate (miles per hour)? 16 : 2 ½: ¼ or 16 miles 2 hours 8 miles per hour or ½ miles ¼ hours or 2 miles per hour • Unit Rates Worksheet 1 x 4 miles 2 1 hours
Proportional Relationship • Equal ratio between every x, y pair • y/x • Constant of Proportionality (unit rate) x y y x -1 -2 2 1 2 2 2 4 2 3 6 2 4 8 2
Proportional 1. Linear – the picture is a line 2. Goes through the origin – if x=0, y=0 3. Constant of Proportionality is the same for every data point (y/x) x y 12 y x 10 5 10 8 -1 -2 2 1 2 2 4 2 2 3 6 2 4 8 6 -2 -1 0 -1 -2 -2 -4 3 6 2 4 1 2 0 1 2 3 4 5 6
Proportional or not? • A student is making trail mix. The table below shows the quantities of nuts and fruit used. Is the mix proportional? Serving Size 1 2 3 4 Serving Size 1 2 Cups of Nuts 1 2 3 4 2: 1 Cups of Fruit 2 4 6 8 Ratio of Fruit to Nuts • What is the constant of proportionality? • What is the constant of fruit : nuts? 2 • What is the unit rate? • What is Fruit when Nuts = 1? 2 3 4
Proportional or not? • A student is making trail mix. The graph below shows the quantities of nuts and fruit used per serving. Is the mix proportional? 9 8 Is this a straight line? Yes 4 8 7 Fruit 6 3 6 5 4 2 4 Does it go through (0, 0)? 3 2 1 2 Yes 1 0 0 0, 5 1 1, 5 2 Nuts 2, 5 3 3, 5 4 4, 5
Proportional Handout
Equations from tables • Every proportional relationship can be expressed as an equation. Number of lunches Cost in dollars Dollars Lunches 1 5 5 5 25 5 7 35 5 20 100 5 • Every lunch costs $5 (unit rate). For every lunch I want to buy I have to make sure I have $5 in my account. • Cost (c) = number of lunches (l) x 5. • c = 5 l • If there are 22 days in October that I want to buy lunch, what is my cost? • c = 5(22) = $110
Equations from graphs • Every proportional relationship can be expressed as an equation. The way it should be 12 Detentions 10 8 6 4 2 0 0 1 2 3 Questions missed 4 5 6 • For every question you miss on a test you have to serve 2 detentions (unit rate). • Detentions (d) = questions missed (q) x 2 • d = 2 q • How many detentions do you have to serve if you miss 13 out of 100 questions on your test? • d = 2(13) = 26 detentions
Parts of an equation • d = ci • d = dependent variable • a variable whose value depends on that of another • c = constant • a number that never changes • i = independent variable • a variable whose value does not depend on (is independent of) that of another
Proportional Handout Part 2
Equivalent Ratios Ratio of 3 2 What is n? 4 Ratio of 8 4 x 2 Is equivalent to 6 n x 2 Is equivalent to What is n? 600 What is 8 ? 2 4 1 n 300
Equivalent Ratios Ratio of 4 5 Is equivalent to n 36 What is n? 4 = n 5 36 4 x 36 = 5 x n 144 = 5 n 5 5 28. 8 = n Cross multiply
Equivalent Ratios Ratio of 4 5 Is equivalent to 54 n What is n? 4 = 54 5 n 4 x n = 5 x 54 4 n = 270 4 4 n = 67. 5 Cross multiply
It is 32 miles from my house to school. If I print out this page, it is 6 inches from my house to school. How many miles are represented by one inch? If I divide it up into one inch segments, I have 6 one inch segments. 1 inch = 6 inches ÷ 6 times 1 inch = 6 inches 6 times A map is a proportional relationship between the actual distance and the inches on the map. If 1 inch = 6 inches 6 times then 1 inch = 32 miles 6 times 1 inch = 5. 33 miles
Cross Multiply? 1 = 500 2 1000 2 = 8 1 4 2 x 4=8 x 1 8=8 1 x 1000 = 2 x 500 1000 = 1000 You can do that
Percent
Percent - % Percent = per cent each 100 1% = 1 of each 100 2% = 2 of each 100 30% 63% 100%
Percent - % Ratio Fraction Decimal x : 100 50: 100 x_ 100 50 = 1 100 2. 5
Percent - % Ratio 7: 10 Fraction 7 = 70 x 10 100 = 70% Decimal. 7 x 100 = 70%
Percent - % Each rectangle below equals 100%. What is the total percentage shown? 100% 100 + 100 = 300% 100%
Percent - % Each rectangle below equals 100%. What is the total percentage shaded? 100% 100 + 80 = 180% 4 = 80% 5 1 4 5 2 4 =. 8 x 100 = 80% 5 = ? 80 100 = 80%
Percent - % In the box below, what percent is shaded? 1 2 Fraction of x : 100 6 ? = 30 20 100 = 30% Decimal times 100 6 =. 3 x 100 = 30% 20
Calculate a change in percent Change in % = |final value – initial value| initial value x 100 You started jogging. The first day you jogged you could only go 2 miles. You jogged every day to improve. At the end of a month you could jog 5 miles. What percent improvement did you see? % change = |5 – 2| x 100 2 % change = |3| x 100 2 % change = 3 x 100 2 % change = 1. 5 x 100 = 150% improvement
Calculate a change in percent % change = |final value – initial value| initial value x 100 A pair of shoes was originally $45. They are on sale for $39. 99. What percent off is the store giving you? % change = |39. 99 - 45| x 100 45 % change = |-5. 01| x 100 45 % change = 5. 01 x 100 45 % change =. 11 x 100 = 11%
Apply a change in percent New value = old value + (old value x percent) as a decimal (divide by 100) A pair of shoes was originally $45. They are on sale for 20% off. What is the sale price of the shoes? Sale price = original price – (original price x 20%) Sale price = 45 – (45 x. 20) Sale price = 45 – (9) Sale price = $36
Apply a change in percent New value = old value + (old value x percent) as a decimal (divide by 100) You have a job. You get paid $8. 45 per hour. You get a 3% raise. What is your new hourly rate? New rate = old rate + (old rate x 3%) New rate = 8. 45 + (8. 45 x. 03) New rate = 8. 45 + (. 25) New rate = $8. 70
Working with fractions and percents • Aza and Becky play on the same basketball team. During the last game, Aza scored 3/5 of the team’s points. Becky scored 16% of the team’s points. What percentage of the team’s points were not scored by Aza and Becky?
Percent - % In the box below, what percent is shaded? 1 2 Fraction of x : 100 6 = 30 20 100 = 30% Decimal times 100 6 =. 3 x 100 = 30% 20
Working with fractions and percents • Aza and Becky play on the same basketball team. During the last game, Aza scored 3/5 of the team’s points. Becky scored 16% of the team’s points. What percentage of the team’s points were not scored by Aza and Becky? Aza Becky 3 5 16% . 6 x 100 = 60% Aza + Becky 60% + 16% = 76% Total – (Aza and Becky) 100% - 76% = 24%
Working with fractions and percents • A High School has 756 students. Only 1/10 of the students joined the debate team but 60% play sports. What is the difference in popularity between the debate team and sports? Debate 1 10 . 1 x 100 = 10% Sports 60% Popularity of one – popularity of the other 60% - 10% = 50%
Working with Fractions and Percents 1. Turn the fraction into a decimal 1. Divide the top by the bottom 2. Turn the decimal into a percent 1. Multiply by 100 You now have a %
Apply a change in percent more than once New value = old value + (old value x percent) as a decimal (divide by 100) A pair of shoes was originally $45. They were not selling so the store reduced them by 20%. After that so many people started buying them that the store increased the price by 15%. What is the sale price of the shoes now? First Sale price = original price – (original price x 20%) Current price = First Sale price + (First Sale price x 15%) First Sale price = 45 – (45 x. 20) Current price = 36 + (36 x. 15) First Sale price = 45 – (9) Current price = 36 + (5. 4) First Sale price = $36 Current price = $41. 40
Apply a change in percent more than once New value = old value + (old value x percent) as a decimal (divide by 100) A pair of shoes was originally $45. They were not selling so the store reduced them by 20%. After that so many people started buying them that the store increased the price by 15%. What is the sale price of the shoes now? Same thing as -20% + 15% = -5%? Same thing as original price – (original price x 5%)? 45 – (45 x. 05) = $42. 75 No, Don’t do this First Sale price = original price – (original price x 20%) Current price = First Sale price + (First Sale price x 15%) First Sale price = $36 Current price = $41. 40
Apply a change in percent to more than one thing New value = old value + (old value x percent) There were 24 dogs and 40 cats at the animal shelter last month. During the month the number of dogs increased 25% and the number of cats decreased 40%. What is the percent change in the number of cats and dogs? New dog count = old dog count + (old dog count x 25%) New cat count = old cat count - (old cat count x 40%) New dog count = 24 + (24 x. 25) New cat count = 40 – (40 x. 40) New dog count = 24 + 6 New cat count = 40 - 16 New dog count = 30 New cat count = 24 We’re half way done!!!!
Calculate a change in percent Change in % = |final value – initial value| initial value x 100 There were 24 dogs and 40 cats at the animal shelter last month. During the month the number of dogs increased 25% and the number of cats decreased 40%. What is the percent change in the number of cats and dogs? Initial total = initial dogs + initial cats % change = |54 - 64| x 100 64 % change = |-10| x 100 64 % change = 10 x 100 64 Initial total = 24 + 40 = 64 Final total = final dogs + final cats Final total = 30 + 24 = 54 % change =. 16 x 100 = 16% decrease
Simple Interest
Why put money in the bank (credit union, stock market, etc. )? 1. Bank keeps your money safe 2. Bank loans out your money 3. Bank pays you for letting them do this Interest
Simple Interest • Principal • The amount you started with • Rate • The percent of change • Time • How long this lasts I = Prt
I = Prt Put $1000 in an investment Leave it for 2 years Bank paid 5. 5% a year Principal = Rate = Time = 1000 5. 5% (. 055) 2 I = 1000 x. 055 x 2 Stock market paid 7% a year Principal = Rate = Time = 1000 7% (. 07) 2 I = 1000 x. 07 x 2 I = $110 I = $140 You get = 1000 + 110 = $1110 You get = 1000 + 140 = $1140
I = Prt Put $1000 in an investment Leave it for 2 years Bank paid 5. 5% a year I = 1000 x. 055 x 2 I = $110 You get = 1000 + 110 = $1110 New Value = Old Value+ (Old Value x %) New Value = Old Value + (Old Value x % x time) New Value = Principal + (Principal x rate x time) New Value = Principal + Interest
I = Prt Put $1000 in a savings account Leave it in for 3 years Interest rate is 2% per year I = 1000 x. 02 x 3 I = $60 New Balance = 1000 + 60 = $1060 2 + 3/12 years 2 + 1/4 years 2 +. 25 years 2. 25 years Put $1000 in a savings account Leave it in for 2 years 3 months Interest rate is 2% per year I = 1000 x. 02 x 2. 25 I = $45 New Balance = 1000 + 45 = $1045
I = Prt Borrow $100, 000 for a house Pay it back over 30 years Bank A charges 3. 6% a year I = 100000 x. 036 x 30 I = $108, 000 Bank B charges 4% a year I = 100000 x. 04 x 30 I = $120, 000
Sales Tax
Sales Tax • Just another % added to an amount New shoes = $50 Sales tax rate = 4. 5% New Value = Old Value + (Old Value x %) Total Cost = Price of shoes + (tax) Total Cost = Price of shoes + (Price of shoes x sales tax rate) Total Cost = 50 + (50 x. 045) Total Cost = $52. 25
New shoes = $50 Employee Discount = 10% Sales tax rate = 6. 75% Step 1: What is your price for the shoes? Step 3: What is the total price you must pay? New Value = Old Value+ (Old Value x %) Your Cost = Price of shoes - (Price of shoes x discount rate) Your Cost = 50 - (50 x. 10) Your Cost = $45 New Value = Old Value+ (Old Value x %) Final cost = Your cost + (Your cost x sales tax rate) Final cost = 45 + (45 x. 0675) Final Cost = $48. 04 Final cost = ( Cost of the shoes – your discount) + sales tax
- Slides: 51