Chapter 2.6. Percent and Mixture Problem Solving

Transcription

1 Chapter 2.6 Percent and Mixture Problem Solving

2 Learning objectives This section is devoted to solving problems in percentages and other ratios Solve percent equations Solve discount and mark-up problems Solve percent increase and percent decrease problems Solve mixture problems Vocabulary: discount, mark-up

3 Solving percentage equations Recall the General Strategy for (word) Problem Solving: UNDERSTAND the problem, propose a solution TRANSLATE your proposal into an equation SOLVE the equation INTERPRET your results into the original problem statement This checklist appears (again) on the top of page 142 Percentage problems are important to understand, because companies and the media tend to report statistics in terms of percentages: stock market trends, sports results, interest rates, nutritional packaging, voting results

4 Example 1, pg. 142 The number 63 is what percent of 72? UNDERSTAND: take a reasonable guess, maybe 80% To check, multiply 72 by 80% (0.80) = 57.6 Close, but no cigar. We can guess that 63 is greater than 80% of 72. Let x be the unknown percent. State the problem: 63 is x percent of 72 TRANSLATE: 63 = x * 72 SOLVE: divide both sides by 72: Solve the fraction: x = or 87.5% INTERPRET: 63 is 87.5% of 72 63/72 = x

5 Example 2: The number 120 is 15% of what number? UNDERSTAND: let x be the unknown number, which is going to be much larger than is only 15% (0.15) of the larger number TRANSLATE: 120 = 0.15 * x SOLVE: divide both sides by /0.15 = x 800 = x INTERPRET: 120 is 15% of 800

6 Example 3: trips made by American travelers Pie chart on the bottom of page % of trips are business only, 4% are business-pleasure, and 13% are personal-other. The rest are pleasure. On a flight with 253 Americans onboard, how many can we expect to be traveling solely on business? UNDERSTAND: 17% are business only. Let x be the number of passengers we expect to be solely on business TRANSLATE: x = 0.17 * 253 SOLVE: x = INTERPRET: this is not meant to be an exact figure, only an approximation based on known averages. We expect there to be around 43 business-only passengers

7 Find each number described: 5% of 300 is what number? 207 is 90% of what number? 15 is 1% of what number? What percent of 350 is 420?

8 Solving discount and mark-up problems So, who doesn t love a discount? Discounts are reductions in an original price On the other hand, mark-ups are increases in an original price You calculate mark-ups when you leave a tip in a restaurant or bar The discount is not the new price, it is the difference between the old (higher) price and the new (lower) price A 20% discount means the new price is = 0.80 = 80% of the old price

9 Example 4: discounting a cell phone A $140 cell phone was discounted 20%. Determine the discount and the new price. UNDERSTAND: the discount is the old price ($140) times the discount percentage (20%). The new price is the old price minus the discount. TRANSLATE: discount = $140(0.20) SOLVE: discount = $140(0.20) = $28 New price = $140 $28 = $112 INTERPRET: the discount on the phone is $28, and the new price is $112

10 Solve the following problems With a 75% discount, how much would an item originally priced $130 now be? A dinner bill came to $145.23, and you tip 15%. How much will your total cost be?

11 Solving percent increase and decrease problems Most things in life increase in price, and decrease in value, over time Most times, you receive raises, or an increase in salary Lately, though, companies have been decreasing salaries by a certain percentage Tuition costs and fees increase by a percentage each year Other percent increases or decreases you may hear about: crime rates, unemployment rates

12 Example 5: % increase in college cost The cost of public college rose from $1454 in 1990 to $2928 in Calculate the percent increase UNDERSTAND: we can see that the new cost is more than double the old cost. Therefore, the increase is more than 100% of the old cost TRANSLATE: calculate the increase, then divide it into the old cost to find the percent increase Increase = = 1474; x = increase/old cost SOLVE: x = 1474/1454 = = 101.4% INTERPRET: the new cost is 101.4% more than the old cost

13 Example 6: movie screens This year, the number of digital screens in the US is 124, which is a 175% increase over last year. How many screens had digital last year? UNDERSTAND: last year s screens is x. This year, 124 equals last year s screens plus the 175% increase INTERPRET: x x = 124 SOLVE: 2.75x = 124; x = 124/275 = 45 INTERPRET: There were 45 digital screens last year.

14 Solve the following increase/decrease problems The available minutes on a cellphone plan increased from 1200 to Find the percent increase. College enrollment decreased from 8900 to Find the percent decrease. What was the original price on a pair of boots if the sale price is $120, after a 20% discount?

15 Solving mixture problems Budding nurses and car mechanics take note Mixture problems involve two or more different quantities being combined to form a new mixture Usually you are given either a pure substance (which is regarded as 100%) or lesse percentage substance to mix Alcoholic beverages are listed in terms of percentages; some beers list percentage, bottle liquors list proof, which is twice the percentage of alcohol content 80 proof rum is 40% alcohol; 80 proof divided by 2 = 40% Another type of percentage: BAC (blood-alcohol content) Other applications: Mixing oil to gas for small motors; mixing concentrated pesticides to water

16 Example 7 lab experiment A chemist needs to make 12 liters of a 50% acid solution. She only has 40% and 70% solutions. How much of each should she use? UNDERSTAND: we have to understand how much acid is required. 12 liters of a 50% solution means 12 total liters of solution, 6 of which must be acid. We must be aware of not only the amount of solution, but the amount of acid. In this problem, we will write the 12 liters of 50% solution as 12(0.50), which actually represents the amount of acid We will flip a coin and let x be the number of liters of 40% solution, which we will write as x(0.40) The number of liters of 70% will then be 12 x, which will be written as (12 x)(0.70)

17 Example 7 lab experiment (cont) TRANSLATE: 12(0.50) = x(0.40) + (12-x)(0.70) SOLVE: distribute: 6 = 0.4x x combine like terms: 6 = -0.3x Isolate the variable: = -0.3x Simplify the variable: -2.4 = -0.3x; x = 8 12 x = 12 8 = 4 INTERPRET: she will use 8 liters of 40% solution and 4 liters of 70% solution.

18 Solve the mixture problem How much pure alcohol should be mixed with 4 gallons of 30% alcohol solution to produce an 80% solution?