Ratio and Proportion

7-1 RATIONALS - Ratio & Proportion MATH 210 If quantity "a" of item x and "b" of item y are related (so there is some reason to compare the numbers), we say the RATIO of x to y is "a to b", written "a:b" or a/b. Ex 1 If a classroom of 21 student s has 3 computers for student use, w e say the ratio of students to computers is . We also say the ratio of computers to students is 3:21. The ratio, being defined as a fraction, may be reduced. In this case, w e can also say there is a 1:7 ratio of computers to students (or a 7:1 ratio of student s t o comput ers). We might also say " there is one computer per seven student s" . Ex 2 We express gasoline consumption by a car as a ratio: If w e traveled 372 miles since the last fill-up, and needed 12 gallons to fill the tank, w e' d say w e' re getting mpg (miles per gallon—mi/gal). We also express speed as a ratio—distance traveled to time elapsed... Caution: Saying “ the ratio of x to y is a:b” does not mean that x constitutes a/b of the w hole; in fact x constitutes a/(a+ b) of the w hole of x and y together. For instance if the ratio of boys to girls in a certain class is 2:3, boys do not comprise 2/3 of the class, but rather ! A PROPORTION is an equality of two ratios. E.x 3 If our car travels 500 miles on 15 gallons of gas, how many gallons are needed t o travel 1200 miles? We believe the fuel consumption w e have experienced is the most likely predictor of fuel use on the trip. We assume that the mpg ratio of our trip w ill equal mpg of the past. Therefore w e w rite the proportion: 500 miles:15 gallons = 1200 miles:x gallons Or, expressed in the more familiar form: (or 500:15 = 1200:x). 500/15 = 1200/x ...or x/1200 = 15/500 ...). Ex 4 If not ation on the corner of a map indicates a scale of 1/2 inch per 12 miles. A distance of 5 and 1/2 inches on the map corresponds to miles. Ex 5 Tea brand A sells in a 6-ounce container for $3; Brand B sells in a 10-ounce container for $4. Are these equivalent price-per-ounce? Ex 6 There are 105 male babies born for every 100 girl babies. Out of 20,000 babies born in LA last month, approximately how many w ere girls? Ex 7 One-fourth of a company’ s employees w ere laid off. If 552 remained, how many w ere laid off? What w as the company’ s original w ork-force? Ex 8 In 1970, 4 men to every 1 w oman w ere admitted to med-school. Of 850 admissions, how many w ere male? Ex 9 A recipe for champagne punch calls for 5 parts ginger ale, 5 parts champagne and 1 part limeade. How much champagne is needed to make 500 4-ounce servings? Ex 10 The “ standard toll-house” cookie recipe calls for 1 cup chocolate chips & 1 c nuts to 4 c flour. Julia makes hers w ith 2 cups chocolate chips and 1 cup nuts to four cups flour. Jacques composes his of 1 cup chocolate chips and 1 cup nuts to three cups flour. Which recipe is more true to the proportions established in the “ standard” recipe? Whose w ould you rather eat? Ans: E1 21:3 or 7:1 E2 31 E3 x is 36 E4 132 E5 B is cheaper E6 9756 21:20 E7 1656 2208 E8 680 E9 ~909 oz E10 compare the fractions. Jacques. Julia’ s! More ratio problems Math 210 Example 1.6 from the text, p 169: The ratio of boys to girls in a class is 5:3. There are 6 more boys than girls. How many are in the class? Example 1.7: The ratio of Jim’ s money to Peter’ s money w as 4:7 at first. After Jim spent ½ of his money and Peter spent $60, Peter had tw ice as much money as Jim. How much money did Jim have at first? Algebra solution: Let x = Jim’ s original $ Then Peter had 7x/4 After: Jim then had x/2 Peter then had 7x/4 – 60 and that w as 2(w hat Jim had then) 7x )) – 60 4 = 2 x ‰ 2  )) Alternate algebra solution: Let x be a certain tidy sum of money ($). Since the ratio w as 4:7... Jim had 4x and Peter had 7x at first. After Jim spent ½ his money, he then had 2x. After Peter spent $60, he then had 7x – 60. We are told at this point Peter had tw ice as much as Jim, so 7x – 60 = 2 ( 2x ) (Compare that to the equation above. Very similar, but not identical because the x variable above is Jim’ s money. This x is 1/4 of that. ) Example 1.11: Tw o numbers are in the ratio 3:5. After subtracting 11 from each, the new ratio is 2:7. What are the numbers? Example 1.14: A bag contained 6 White & 10 Red marbles. 4 White & 20 Red marbles w ere added to the bag. What is the new ratio of White to Red marbles? D, K & M solved this problem differently: D said K said 3:5 + 1:5 = 4:10 3 + 5 ) 1 5 ) = 4 5 ) M said 6+ 4 : 10+ 20 = 10:30 = 1:3 Ratios do NOT add. We can set ratios = to make a proportion. Ratios DON’T tell us how many total! 7-2 DECIMAL NUMBERS & PERCENTS Math 210 Every rational can be expressed as a ratio (fraction), a decimal or a percent. PERCENT means " per hundred" . Thus 50 percent (50%) means 50 per hundred, or 50/100 or fifty hundredths, or .50. You must be able to express any one of these forms in any of the other. ’Express the follow ing 12% = PERCENTA GES )) = as rational-form and as DECIM A LS : )) = 7 % = ))) = 100% = 1500% = ))) = 7¼% = ))) = ½% = 20% = 250% = ))) = FRA CTIONS ))) = To change "x percent" to decimal form: jSince % means “ per hundred” , X% means X/100.... (w e do w hat “ % ” says... divide by 100). ’Express these .25 = DECIM A LS % .002 = as PERCENTA GES : .04 = % 1.5 = % % .0425 = % .4 = % 320 = % .0003 = % To change a decimal to percent form: j % means “ per hundred” or “ hundredths” so if you can w rite as 100ths, conversion is intuitive. j Use “ patterning” : if .07 = 7% , then .7 must be 70% , and .007 must be .7% . j Since dividing by 100 converts % ÷ decimal, so multiplying by 100 should convert decimal ÷ % . ’Express these 1/2 = % FRA CTIONS as 3/5 = PERCENTA GES : % 1/3 = % 1/8 = % 4/3 = To change a fraction to "percent": j % means hundredt hs, so w rite as hundredt hs: 3/8 = .375 = 37.5/100 = 37.5% j Divide to convert the fraction to decimal form, then convert to % as above. E1 % word problems: Susan bought a house in 1994 for 140,000. The value of the house has increased 250% . How much has the value increased? What is the value of the house now ? E2 John w as making 25,000 per year, until he got a raise to 26,000 per year. By w hat percent w as his salary raised? E3 Jackie bought a dress on sale at 30% off for $150. What w as the original price? Thirty percent of 500 is . One hundred tw enty is w hat percent of eighty? Thirty is tw enty percent of . Equation: Equation: Equation: % 7-2A PERCENTS IN USE Math 210 0. Express 10 !5 as a percent. 1. Generic problems: a. Thirty percent of 500 is . Equation: b. One hundred tw enty is w hat percent of eighty? c. Thirty is tw enty percent of . Equation: 2. Quick! 9.25% is: a. .925 b. .0925 Equation: c. 9.25 What is greater: .10, or 3% of 2.98? 3. If the sales tax rate is 8.25% , and the sales t ax on my new car is $559.35, w hat is the selling price of the car? 4. In another state, the sales tax on a $350 it em is $14.00. What is that state' s sales tax rate? 5. If the sales tax rate is 7% , how much money is needed to purchase an item w hose price is $3.71? 6. Thirteen of thirty-five students in a calculus class earned A' s. What percent of the class earned A' s? 7. If the property tax rate is 1.2% , & the tax assessment on a home is $2,398, w hat is its assessed value? 8. Thirty percent of immigrants to LA become entrepreneurs; of those, 45% are successful. What percent age of immigrant s to LA become successful ent repreneurs? 9. Zeta is currently making $34,500 per year; she needs to earn at least $40,000 per year t o buy a tow nhome. What percentage raise does she need? 10. A dress originally priced at $550.00 is already marked dow n 30% , and the red-tag sale sign says " Additional 50% off marked price" . What is the total percent discount? 11. A school lunch program has been selling food w ithout sales tax. The St Board of Equalization demands payment of tax. The school collected $8,432,200 in all for hot lunches; the tax rate is 7% . How much must the school fork over? 12. Marco received a ten percent raise in January, but now that the company is “ dow nsizing” , all remaining employees are receiving a ten percent pay cut. What is the effect on Marco’ s salary, relative to last year’ s? (Overall, is Marco now making more than, less than, or the same as, last year’ s pay?) Explain. Answ ers: 0. .001% 1. 150;150;150 2. .0925 ; .10 > .03x3 > ... 3. $6780 4. 4% 5. $3.97 6. .37% 7. $199833 8. 13.5% 9. 15.942% 10. 65% 11. $551639.25 (It’ s not 7% of 8432200.) (Here’ s an example that’ s closer to Home! Suppose you are at a Dodger game, and you buy a hot dog from the vendor for $3. Was that transaction subject to sales tax? How does that w ork; do you think the vendor is going to send 7% of $3 to the state?) 12. Marco w ill end up w ith a 1% decrease in pay compared to last year. (That’ s because he received a ten perent pay raise, making his new salary 110% of last year’ s... then lost 10% of 110% = 11% of last year’s pay. Marco w ould be better off if he had said forget about giving me a raise!)