= 20. = 0.2. In other words, there s a 0.2, or 1/5 probability you hit any given pitch during batting practice. = hits

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1 1 Math 1 - Proportions & Probability Part 1 IDENTIFYING THE PARTS AND THE WHOLE The key to solving most fraction and percent word problems is to identify the part and the whole. Usually, you'll find the part associated with the verb is/are and the whole associated with the word of. In the sentence, "Half of the boys are blonds," the whole is the boys ("of the boys"), and the part is the blonds ("are blonds"). Sometimes, the problem will give you just the parts: "The ratio of boys to girls is 2 to 3." In this case, we have a part:part relationship. To find the ratio of boys to the total, you would add the parts: = 5, so the ratio of boys to total (part:whole) is 2 to 5. Percentages, probabilities, and ratios all come from the same basic concept: parts to wholes. Let's say you're taking batting practice. You are thrown 100 pitches and you hit 20 of them. Ignoring the face that you're probably nor ready for the MLB, let's put this into some math Language. Once we get the basics down, we'll try some more advanced problems. First, what percent of the pitches did you hit? Well, that's easy on this one because we're dealing with 100. But you can always find percentages with this simple part-to-whole formula: part whole x 100% Once we put the numbers in, we'll see this: 20 you hit 20% of them. 100 x 100% = 0.2 x 100% = 20%. If you hit 20 of the 100 pitches, Next, what is the probability that you were to hit any given pitch? Remember, this is just a matter of parts to wholes, so we can find this probability as follows: part = hits = 20 = 0.2. In other words, there s a 0.2, or whole pitches 100 1/5 probability you hit any given pitch during batting practice. Ratios are a little different. Usually these will ask for the relationship of some part to some other part. If we're still using our batting practice statistics, we might want to know something like, What's the ratio of the pitches you hit to the pitches you missed? Even though we're not dealing with the whole, we'll find the ratio the same way, but instead of part, we ll use part = hits = 20 = 1. The ratio of hits to misses is ¼, or 1 to 4, or 1:4. If it feels like we just did the same thing part misses 80 4 three times, it was supposed to. As we've seen a few times already, just because things have different names doesn't mean that they are unrelated. 1. Abagail has a jar of jelly beans. It has 5 red jelly beans, 6 yellow jelly beans and 8 blue jelly beans. What is the ratio of red jelly beans to the total? whole 2. Two-fifths of the student body wore khaki pants to school today. What is the ratio of students who did not wear khaki pants to school today?

2 PERCENT FORMULA In percent questions, whether you need to find the part, the whole, or the percent, use the same formula: Part = Percent x Whole When you use the formula, be sure to convert the percent into decimal form: Example: What is 12% of 25? Solution: Part = 0.12 x 25 Example: 15 is 3% of what number? Solution: 15 =.03 x Whole Example: 45 is what percent of 9? Solution: 45 = Percent x 9 Example: At a restaurant, diners enjoy an "early bird" discount of 10% off their bills. If a diner orders a meal regularly priced at $18 and leaves a tip of 15% of the discounted meal (no tax), how much does he pay in total? a. $13.50 b. $16.20 c. $18.63 d. $18.90 e. $20.70 Solution: Step 1: Know the question. We want the price of the discounted meal plus tip. Step 2: Let the answers help. We are reducing a number by 10%, then increasing it by 15%, so it's not likely that the final number will be much less or much greater than $18. Let's eliminate choices (A) and (E). Step 3: Break the problem into bite-sized pieces. First, we'll need to figure out what the discounted price of the meal is. There are a number of ways to do this, but if you find this part method useful, you could find the discount this way: 10% = discount 10 = discount 100 $18 Solving, the discount = $1.80 whole The price of the discounted meal, then, is $18 - $1.80 = $ Let's find the tip the same way. 15% =tip 15 = tip 100 $16.20 Tip = $2.43 The price of the discounted meal plus tip, therefore, is $ $2.43 = $ The correct answer is choice (C). ANOTHER WAY TO DEAL WITH PERCENTS A percentage is a fraction in which the denominator equals 100. In literal terms, the word percent means "divided by 100," so any time you see a percentage in an ACT question, you can punch it 4into your calculator quite easily. If a question asks for 40 percent of something, for instance, you can express the percentage as a fraction: 40/100. Any time you are looking for a percent, you can use your calculator to find the decimal equivalent and multiply the result by 100. If four out of five dentists recommend a particular brand of toothpaste, you can quickly determine the percent of doctors who recommend it by typing 4/5 x 100 and hitting the ENTER key. The resulting "80" just needs a percent sign tacked onto it. To properly translate all percent questions, it is helpful to have a decoding table for the various terms you'll come across. 2

3 3 Math Equivalent Percent /100 Of Multiplication (x) What Variable (y, z) Is, are, were = What percent y/100 PERCENTAGE SHORTCUTS In the last problem, we could have saved a little time if we had realized that 1/5 = 20 percent. Therefore, 4/5 would be 4 x 20 percent or 80 percent. Below are some fractions and decimals whose percent equivalents you should know. 1/5 = 0.2 = 20% ¼ = 0.25 = 25% 1/3 = 0.33 = 33 1/3% ½ = 0.5 = 50% Another fast way to do percents is to move the decimal place. To find 10 percent of any number, move the decimal point of that number over one place to the left. 10% of 500 = 50 10% of 50 = 5 10% of 5 =.5 To find 1 percent of a number, move the decimal point of that number over two places to the left 1% of 500 = 5 1% of 50 =.5 1% of 5 =.05 You can use a combination of these last two techniques to do even very complicated percentages by breaking them own into easy-to-find chunks. 20% of 500: 10% of 500 = 50, so 20% is twice 50, or % of 70: 10% of 70 = 7, so 30% is three times 7, or 21. Example: 32% of 400: Solution: 10% of 400 = 40, so 30% is three times 40, or 120. And 1% of 400 = 4, so 2% is two times 4, or 8. Therefore, 32 percent of 400 = = 128. Example: When 15% of 40 is added to 5% of 260, the resulting number is: a. 19 b. 40 c. 95 d. 180 e. 260 Solution: Let's try this one using the decoding table. Although this isn't really a word problem, that doesn't mean we shouldn't be careful and break it into bite-sized pieces.

4 4 First, 15% of 40. Remember, % translates 100 and "of" translates to multiplication. Therefore, we can rewrite 15% of 40 as x 40. Put this expression in your calculator to find that x 40 = Now, 5% of 260. Use the same translations to find x 260 = 13. We ve done the tough part, so let's substitute what we've found back into the problem: When 6 is added to 13, the resulting number is: = 19, choice (A). 3. If 20% of a number is 125, what is 28% of the number? 4. What percent of 24 is 96? 5. If 30% of a given number is 12, then what is 75% of the given number? 6. What is 1/4 of 20% of $50,000? 7. In the process of milling grain, 3% of the original is lost because of spillage, and another 5% of the original is lost because of mildew. If the mill starts out with 490 tons of grain, how much (in tons) remains to be sold after milling? SETTING UP A RATIO To find a ratio, put the number associated with the word of on top (as the numerator) and the quantity associated with the word to on the bottom (as the denominator), and reduce. The ratio of 20 oranges to 12 apples is 20, which reduces to 5. Be sure to keep the parts in the same order-so if a second ratio of oranges 12 3 to apples was given, the oranges should be on top and the apples on the bottom. Example: If the ratio of 2x to 5y is 1, what is the ratio of x to y? 20 a. 1/40 b. 1/20 c. 1/10 d. 1/8 e. 1/4 Solution: The difficulty of this problem is all in the setup. Just remember that you're comparing parts to wholes. 2x 5y = 1 20.

5 5 To isolate on the left side of chis equation, let's multiply both sides by 5/ x 2x 5y = 1 20 x 5 2 x y = /40 reduces to 1/8. The answer is choice (D). Example: If 3.7 inches of rain fall on Vancouver during the first 4 days of November, and the rain continues to fall at this pace for the rest of the month, approximately how many feet of rain will fall during December? (Note: The month of December has 31 days.) a. 2.4 b. 3.7 c. 9.5 d e Solution: Step 1: Know the question. It's in the last line: how many feet of rain will fall in December? But be careful: The value given in the problem is in inches. We're going have to do some converting! Step 2: Let the answers help. Since we know the problem is asking how many feet of rain will fall, and the problem tells us that 3.7 inches of rain will fall in 4 days, we can do some Ballparking. The number of inches will increase, but it will be reduced again when we convert it into feet. Choice (E) is definitely too big, and choice (D) probably is, too. Step 3: Break the problem into bite-sized pieces. There are going to be two main parts to this problem: First, we want to figure out how much rain will fall in days, then we want to convert it to feet. We can do this by using ratios. The ratio given in the problem is 3.7 inches every four days, or in math terms, 3.7 in 4 days The ratio will remain the same no matter how many days there are, so let's use this ratio to find how many inches of rain will fall in December. 3.7 in =? in (3.7 in)(31days). We can rearrange the terms to find:? in = = in. But we can't stop 4 days 31 days 4 days here! Remember, the problem is asking for a value in feet. We'll use the same process, and this time we already know the ratio of inches to 12 inches to feet: 12 in 1 ft 12 in 1 ft Let's use this ratio to find our answer: in in (1 ft) =. Rearrange the equation to find:? in = = 2.4 ft, choice (A)?ft 1 ft 8. On the line segment below, the ratio of lengths AB to BC is 1:5. What is the ratio of AC to AB? a. 1:3 b. 1:7 c. 1:6 d. 6:1 e. Cannot be determined from the given information

6 6 9. In a shipment of cell phones, 1/25 of the cell phones are defective. What is the ratio of non-defective to total cell phones? 10. The oxygen saturation of a lake is found by dividing the amount of dissolved oxygen the lake water currently has per liter by the dissolved oxygen capacity per liter of the water, and then converting that number into a percent. If the oxygen saturation level of the lake, to the nearest percent, is 45%, and the and the dissolved oxygen capacity is 60 milligrams per liter, what is the lake s current milligrams of dissolved oxygen per liter of water? 11. The ratio of boys to girls at the Milwood School is 4 to 5. If there are a total of 27 children at the school, how many boys attend the Milwood School? RATE A rate is any "something per something"-days per week, miles per hour, dollars per gallon, etc. Pay close attention to the units of measurement, since often the rate is given in one measurement in the question and a different measurement in the answer choices. This means you need to convert the rate to the other measurement before you solve the question. Example: If snow is falling at the rate of one foot every four hours, how many inches of snow will fall in seven hours? Solution: 1 foot 4 hours = x inches 7 hours 4x = 12 x 7 x = inches 4 hours = x inches 7 hours 12. Robert rides his scooter an average of 5 miles per hour, and Cindy rides her roller blades an average of 6 miles per hour. At these rates, how much longer does it take Runner A than Runner B to travel 3 miles? 13. A train traveling south travels 50 miles per hour for 3 hours; A second train traveling south travels 105 miles per hour for 2 hours. What is the difference, in miles between the number of miles traveled by the second bus and the number of miles traveled by the first bus?

7 SIMPLE PROPORTION Solving proportions is simply a matter of stating the ratios as fractions, setting the two fractions equal to each other, cross-multiplying, and solving the resulting equation. Example: Find the unknown value in the proportion: 2 : x = 3 : 9. Solution: First, convert the colon-based odds-notation ratios to fractional form: 2 x = Then solve the proportion: 2 x = 3 9 9(2) = x(3) 18 = 3x 6 = x Note: Always remember to set your ratios up the same way every time. Example: Bountiful Baskets includes 3 oranges and 4 apples in every basket. How many oranges would there be in 4 baskets? Solution: Set up ratios with the number of oranges in the numerator and the number of baskets in the denominator. For example, keep the original (known) numbers together in the same ratio and the new (unknown) numbers together in a second ratio and keep oranges on top and baskets on bottom. 3 oranges 1 basket = x oranges 4 baskets x = (3 * 4)/1 = 12 A second way to set up the proportion would be: x oranges = 4 baskets 3 oranges 1 basket x = (4*3)/1 = 12 Notice that the numerators contains the numbers from the new (unknown) situation and the denominators contain numbers from the original (known) situation AND the oranges are in the first ratio and the baskets are in the second ratio. There are actually 4 ways to correctly set up the proportions. The other two ways are: 3 oranges = 1 basket and 1 basket = 4 baskets x oranges 4 baskets 3 oranges x oranges Proportions are usually set with x in the numerator since they are usually easier to solve, however since all proportions are correct, the answer will be the same. 14. An oil refinery produces gasoline from crude oil. For every 10,000 barrels of crude oil supplied, the refinery can produce 5,500 barrels of gasoline. How many barrels of gasoline can be produced from 3,500 barrels of crude oil? 15. The price of a cantaloupe is directly proportional to its weight. If a cantaloupe that weighs 1.5 pounds costs $2.55, approximately how much will a 4.17-pound cantaloupe cost? 16. As a salesperson, your commission is directly proportional to the dollar amount of sales you make. If your sales are $800, your commission is $112. How much commission would you earn if you had $1,400 in sales?

8 8 PROPORTION WITH UNIT CONVERSIONS Proportions can also be used to convert units. Also, may problems will have unit conversions as part of the problem. How many feet in 7 yards? Remember that there are 3 feet per 1 yard. Set up the proportion as x feet = 3 feet 7 yards 1 yard or 7 yards 1 yards = x feet 3 feet or x = (3*7)/1 = 21 feet 1 yards = 3 feet 7 yards x feet or 7 yards x feet = 1 yard 3 feet Since all proportions are set up correctly, the answer will be 21 feet every time. 17. There are 16 ounces in one pound. If 4.1 pounds of fish cost $9.95, what is the cost per ounce, to the nearest cent? 18. A proofreader can read 75 pages in one hour. How many pages can this proofreader read in 100 minutes? 19. Jordan went for a 5-mile jog on Wednesday that took him 75 minutes. If on Tuesday Jordan jogs at the same rate of speed, how far will he jog in 45 minutes? 20. There are 32 ounces in a quart. If 3 quarts of milk costs $5.25, what is the cost of milk per ounce to the nearest cent? COUNTING THE POSSIBILITIES The fundamental counting principle: If there are m ways one event can happen and n ways a second event can happen, then there are m x n ways for the two events to happen. For example, with 7 shirts and 5 pairs of pants to choose from, you can put together 7 x 5 = 35 different outfits. For more items, multiply all the ways together. For example, with 7 shirts, 5 pairs of pants, 8 ties and 2 pairs of shoes to choose from, you can put together 7 x 5 x 8 x 2 = 560 different outfits. COMBINATIONS: THE SLOT METHOD Combination problems ask you how many different ways a number of things could be chosen or combined. The rules for combination problems on the ACT are straightforward.

9 9 Figure out the number of slots you need to fill. Fill in those slots. Find the product. Example: At the school cafeteria, students can choose from 3 different salads, 5 different main dishes, and 2 different desserts. If Isabel chooses one salad. one main dish, and one dessert for lunch, how many different lunches could she choose? a. 10 b. 15 c. 25 d. 30 e. 50 Solution: There are three slots to fill, one for each item: salad, main dish, dessert. And the number of possibilities for each is pretty clear. Set up the slots and take the produce as your answer. 3 x 5 x 2. = 30 The correct answer here is choice (D). Salad Main Dessert 21. In the school cafeteria, students choose their lunch from 4 sandwiches, 2 soups, 3 salads, and 6 drinks. How many different lunches are possible for a student who chooses exactly 1 sandwich, 1 soup, 1 salad, and 1 drink? 22. Reggie knows how to make 5 different entrees, 4 different side dishes, and 6 different desserts. How many distinct complete meals, each consisting of an entrée, a side dish, and a dessert, can Reggie make? 23. Chris owns 10 different dress shirts, 4 different pairs of pants, and 7 different ties. How many distinct outfits, each consisting of a shirt, a pair of pants, and a tie, can Justin make? 24. At summer camp, one boy and one girl will be selected to lead the weekly activities. If there are 130 boys and 145 girls at the camp, how many different 2-person combinations of 1 boy and 1 girl are possible? 25. Rana and Tom own a pizza shop, which offers 3 kinds of cheese, 4 kinds of meat toppings, and 5 kinds of vegetable toppings. Each type pizza on the menu has a combination of exactly 3 ingredients: 1 cheese, 1 meat, and 1 vegetable. How many types of pizzas are possible?

10 26. A hiking group will go from a certain town to a certain village by van on 1 of 3 roads, from the village to a waterfall by riding bicycles on 1 of 8 bicycle paths, and then from the waterfall to their campsite by hiking on 1 of 5 trails. How many routes are possible for the hiking group to go from the town to the village to the waterfall to their campsite? The bowling league selects its 4 officers by first selecting the president, then the vice president, then the secretary, then the treasurer. If there are 40 bowlers who are eligible to hold office and no member can hold more than one office, which of the following gives the number of different results of the election? PROBABILITY Probability is always the number of desired outcomes divided by the number of possible outcomes. It can be expressed as a fraction, or sometimes, as a decimal. Probability = Favorable outcomes Total possible outcomes If you have 12 shirts in a drawer and 9 of them are white, the probability of picking a white shirt at random is 9.This probability can also be expressed as 0.75 or 75%. 12 = 3 4 Example: Herbie's practice bag contains 4 blue racquetballs. one red racquetball, and 6 green racquetballs. If he chooses a ball at random, which of the following is closest to the probability that the ball will NOT be green? a b c d e Solution: Step 1: Know the question. Make sure you read carefully! We want the probability that the chosen ball will NOT be green. Step 2: Let the answers help. Green balls account for slightly more than half the number of balls in the bag, so the likelihood that the ball will NOT be green could be slightly less than half. That eliminates choices (A), (B), and (E). Step 3: Break the problem into bite-sized pieces. This is a pretty straightforward part NOT green all = problem. part = whole whole The only slight difficulty is that the answers are not listed as fractions, but it's nothing a calculator can't help. Find 5 11 = 0.45, choice (D).

11 28. If a gumball is randomly chosen from a bag that contains exactly 7 yellow gumballs, 3 green gumballs, and 2 red gumballs, what is the probability that the gumball chosen is green? There are 15 balls in a box: 7 balls are green, 3 are blue and 5 are white. Then 1 green and 1 blue balls are taken from the box and put away. What is the probability that a blue ball will NOT be selected at random from the box? 30. Seth has 4 plaid shirts and 5 solid-colored shirts hanging together in a closet. In his haste to get ready for work, he randomly grabs 1 of these 9 shirts. What is the probability that the shirt Seth grabs is plaid? 31. A partial deck of cards was found sitting out on a table. If the partial deck consists of 6 spades, 3 hearts, and 7 diamonds, what is the probability of randomly selecting a red card from this partial deck? (Note: diamonds and hearts are considered red while spades and clubs are considered black )? 32. A bag contains 6 red marbles, 5 yellow marbles, and 7 green marbles. How many additional red marbles must be added to the 18 marbles already in the bag so that the probability of randomly drawing a red marble is 3/5? 33. A bag contains 12 red marbles, 5 yellow marbles, and 15 green marbles. How many additional red marbles must be added to the 32 marbles already in the bag so that the probability of randomly drawing a red marble is 3/5? 34. To make a 750-piece jigsaw puzzle more challenging, a puzzle company includes 5 extra pieces in the box along with the 750 pieces, and those 5 extra pieces do not fit anywhere in the puzzle. If you buy such a puzzle box, break the seal on the box, and immediately select 1 piece at random, what is the probability that it will be 1 of the extra pieces? 35. An integer from 10 through 99, inclusive, is to be chosen at random. What is the probability that the number chosen will have 9 as at least 1 digit?

12 12 Answer Key 1. 5/ / % : / mg per liter hour , $ $ $ miles 20. $ , x 39 x 38 x ¼ / / / / /90 = 1/5