Sat - MC Grawhill part 30 pps

280 MCGRAW-HILL’S SAT 1. If 4 quarts of apple juice are mixed with 5 quarts of cranberry juice and 3 quarts of grape juice, what part of the total mixture is apple juice? (A) (B) (C) (D) (E) 2. For what value of y does ? (A) −3 (B) (C) (D) 3 (E) 6 3. Which of the following is greatest? (A) (B) (C) (D) (E) 4. If , then x = (A) (B) (C) (D) (E) 10 5. If , then m = (A) (B) (C) (D) 3 (E) 5 6. If x ≠ y and x + y = 0, then (A) −1 (B) 0 (C) 1 ⁄2 (D) 1 (E) 2 x y = 2 3 2 5 1 3 1 2 15 6 += m 5 2 2 5 1 5 1 10 1 2 5 x = 2 3 1÷ 2 3 2 3 ÷ 1 2 3 ÷1 2 3 − 2 3 1× 1 3 − 1 3 1 3 y =− 2 3 4 9 1 3 1 4 1 6 7. If , then m + 2n = (A) (B) (C) 5 (D) 7 (E) 14 8. If z ≠ 0, which of the following is equivalent to ? (A) 1 (B) (C) (D) (E) 9. Five-eighths of Ms. Talbott’s students are boys, and two-thirds of the girls do not have dark hair. What fraction of Ms. Talbott’s students are girls with dark hair? (A) (B) (C) (D) (E) 10. If , then x = 2 3 6 7 1 6 2 3 1 6 1 +−=−+ x 1 3 1 4 1 8 1 10 1 24 z z 2 1+ z z +1 2 z 1 2z 1 1 z z + 7 2 7 4 mn 42 7 8 += SAT Practice 3: Fractions 1 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 11. If n > 1, and , then x = (A) (B) (C) (D) (E) m n + + 1 1 m n + − 1 1 m n +1m n +1 m n −1 nx mx+ = 1 CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 281 Concept Review 3 1. 19/35 (Use zip-zap-zup: it’s better than using your calculator!) 2. 20/3 (Change to 5/2 × 8/3.) 3. 8/3 (Divide numerator and denominator by their common factor: 7.) 4. 8/45 (Change to 2/9 × 4/5.) 5. (12x + 14)/21x (Use zip-zap-zup.) 6. 4x/3z (Change to 3x/2 × 8/9z and simplify.) 7. (3m + 1)/(2m + 1) (Factor and cancel 4 from the numerator and denominator.) 8. −1/(6x) (Change to −2/(9x 2 ) × 3x/4 and simplify.) 9. 13/12 (Change to 6/12 + 4/12 + 3/12.) 10. (3 − 2x)/4 (Use zip-zap-zup and simplify.) 11. x + 5 (As long as x ≠ 5.) (Factor as and cancel the common factor. For factoring re- view, see Chapter 8, Lesson 5.) xx x − () + () − () 55 5 12. (2n + 3)/4n (Divide numerator and denominator by the common factor: 3. Don’t forget to “distrib- ute” the division in the numerator!) 13. 1 ⁄ 4 14. 1 ⁄ 5 15. 1 ⁄ 10 16. 1 ⁄ 3 (Knowing how to “convert” numbers back and forth from percents to decimals to fractions can be very helpful in simplifying problems!) 17. Multiply by the reciprocal of the fraction. 18. “Cross-multiply” to get the new numerators, and multiply the denominators to get the new denominator, then just add (or subtract) the numerators. 19. Only common factors. (Factors = terms in products.) 20. Just divide the numbers by hand or on a calculator. 21. 4/9 (Not 4/5! Remember the fraction is a part of the whole, which is 27 students, 12/27 = 4/9.) 22. 27 (If 2/3 are girls, 1/3 are boys: t/3 = 9, so t = 27.) 23. It must have a value between 0 and 1 (“bottom- heavy”). Answer Key 3: Fractions SAT Practice 3 1. C 4/(4+5+3) = 4/12 = 1/3 2. B Multiply by y: 1 =−3y Divide by −3: 3. C 4. A Multiply by x: Multiply by : 1 10 = x 1 5 1 2 5= x 1 2 5 x = 1 2 3 3 3 2 3 1 3 0 333 1 2 3 1 3 2 15 2 3 2 3 2 3 −=−= = ÷=× = ÷= . ××= = ÷= × = 3 2 6 6 1 2 3 1 2 3 1 0 666. 2 3 1 0 666×=. −= 1 3 y 1 3 y =− 5. D Multiply by 6m: 3m + 6 = 5m Subtract 3m: 6 = 2m Divide by 2: 3 = m 6. A The quotient of opposites is always −1. (Try x = 2 and y =−2 or any other solution.) 7. B To turn into m + 2n, we only need to multiply by 4! 8. E You can solve this by “plugging in” a number for z or by simplifying algebraically. To plug in, pick z = 2 and notice that the expression equals 2 ⁄5 or 4. Substituting z = 2 into the choices shows that only (E) is .4. Alternatively, you can simplify by just multiply- ing numerator and denominator by z: 9. C 5 ⁄8 are boys, so 3 ⁄8 must be girls. Of the girls, 2 ⁄3 do not have dark hair, so 1 ⁄3 do. Therefore, 1 ⁄3 of 3 ⁄8 of the class are girls with dark hair. 1 ⁄3 × 3 ⁄8 = 1 ⁄8. 1 1 1 1 1 2 z z z zz z z z + = × + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = × mn mn += + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ==24 42 4 7 8 28 8 7 2 mn 42 + 1 2 15 6 += m 11. A Multiply by m + x: nx = m + x Subtract x: nx − x = m Factor: x(n − 1) = m Divide by (n − 1): x m n = −1 nx mx+ = 1 10. 7/6 or 1.16 or 1.17 Begin by subtracting 2/3 from both sides and adding −1/6 to both sides, to simplify. This gives . Just “reciprocate” both sides or cross-multiply. 6 7 1 = x 282 MCGRAW-HILL’S SAT CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 283 Working with Ratios When you see a ratio—such as 5:6—don’t let it confuse you. If it is not a part-to-part ratio, then just think of it as a fraction. For instance, 5:6 = 5/6. If it is a part-to-part ratio, just divide each number by the sum to find the fraction of each part to the whole. For instance, if the ratio of boys to girls in a class is 5:6, then the sum is 5 + 6 = 11, so the boys make up 5/11 of the whole class, and the girls make up 6/11 of the whole class. (Notice that these fractions must add up to 1!) Example: If a $200 prize is divided up among three people in a 1:4:5 ratio, then how much does each person receive? The total of the parts is 1 + 4 + 5 = 10. Therefore, the three people receive 1/10, 4/10, and 5/10 of the prize, respectively. So one person gets (1/10) ϫ $200 = $20, another gets (4/10) ϫ $200 = $80, and the other gets (5/10) ϫ $200 = $100. Working with Proportions A proportion is just an equation that says that two fractions are equal, as in 3/5 = 9/15. Two ways to simplify proportions are with the law of cross-multiplication and with the law of cross- swapping. The law of cross-multiplication says that if two fractions are equal, then their “cross-products” also must be equal. The law of cross-swapping says that if two fractions are equal, then “cross-swapping” terms will create another true proportion. Example: If we know that then by the law of cross-multiplication, we know that 7x = 12, and by the law of cross-swapping, that . In a word problem, the phrase “at this rate” means that you can set up a proportion to solve the problem. A rate is just a ratio of some quantity to time. For instance, your reading rate is in words per minute; that is, it is the ratio x 3 4 7 = x 4 3 7 = , of the number of words you read divided by the number of minutes it takes you to read them. (The word per acts like the : in the ratio.) IMPORTANT: When setting up the propor- tion, check that the units “match up”—that the numerators share the same unit and the de- nominators share the same unit. Example: A bird can fly 420 miles in one day if it flies con- tinuously. At this rate, how many miles can the bird fly in 14 hours? To solve this, we can set up a proportion that says that the two rates are the same. Notice that the units “match up”—miles in the nu- merator and hours in the denominator. Now we can cross-multiply to get 420 ϫ 14 = 24x and divide by 24 to get x = 245 miles. Similarity Two triangles are similar (have the same shape) if their corresponding angles all have the same measure. If two triangles are similar, then their corresponding sides are proportional. Example: In the figure above, When setting up proportions of sides in simi- lar figures, double-check that the correspond- ing sides “match up” in the proportion. For instance, notice how the terms “match up” in the proportions above. m k n l r m n k l ==o m n kl 420 miles 24 hours miles hours = x 14 Lesson 4: Ratios and Proportions 1. A speed is a ratio of __________ to __________. 2. An average is a ratio of __________ to __________. 3. Define a proportion: __________________________________________________________________________________________________ __________________________________________________________________________________________________ 4. Write the law of cross-multiplication as an “If . . . then . . .” statement: If _________________________________________________________________________________________________ then ______________________________________________________________________________________________ 5. Write three equations that are equivalent to . 5. a)__________ b)__________ c)__________ 6. Three people split a $24,000 prize in a ratio of 2:3:5. What is the value of each portion? 6. a)__________ b)__________ c)__________ 7. A machine, working at a constant rate, manufactures 25 bottles every 6 minutes. 7. ____________ At this rate, how many hours will it take to produce 1,000 bottles? 8. If m meteorites enter the Earth’s atmosphere every x days (m > 0), then, at this rate, 8. ____________ how many meteors will enter the Earth’s atmosphere in mx days? 9. In the diagram above, ᐉ 1 ⏐⏐ ᐉ 2 , AC = 4, BC = 5, and CE = 6. What is DE? 9. ____________ 10. There are 12 boys and g girls in Class A, and there are 27 girls and b boys in Class B. 10. ____________ In each class, the ratio of boys to girls is the same. If b = g, then how many students are in Class A? A B C D E ᐉ 1 ᐉ 2 2 3x y = Concept Review 4: Ratios and Proportions 284 MCGRAW-HILL’S SAT CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 285 1. If x is the product of .03 and .2, then x is equiv- alent to the ratio of 6 to what number? 6. If 3,600 baseball caps are distributed to 4 stores in the ratio of 1:2:3:4, what is the maximum num- ber of caps that any one store receives? (A) 360 (B) 720 (C) 1,080 (D) 1,440 (E) 14,400 7. David’s motorcycle uses of a gallon of gaso- line to travel 8 miles. At this rate, how many miles will it travel on 5 gallons of gasoline? 2 5 SAT Practice 4: Ratios and Proportions 1 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 2. Jar A contains six red marbles and no green marbles. Jar B contains two red marbles and four green marbles. How many green marbles must be moved from Jar B to Jar A so that the ratio of green marbles to red marbles is the same for both jars? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 3. 90 students are at a meeting. The ratio of girls to boys at the meeting is 2 to 3. How many girls are at the meeting? (A) 30 (B) 36 (C) 40 (D) 54 (E) 60 4. If , then 5x + 1 = (A) 3y + 1 (B) 3y + 2 (C) 3y + 3 (D) 3y + 4 (E) 3y + 5 5. On a map that is drawn to scale, two towns that are x miles apart are represented as being 4 inches apart. If two other towns are x + 2 miles apart, how many inches apart would they be on the same map? (A) 4(x + 2) (B) 6 (C) (D) (E) 6 x 42x x + () 4 2 x x + x y + = 1 3 5 1 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 8. On a blueprint that is drawn to scale, the draw- ing of a rectangular patio has dimensions 5 cm by 7.5 cm. If the longer side of the actual patio measures 21 feet, what is the area, in square feet, of the actual patio? (A) 157.5 (B) 294.0 (C) 356.5 (D) 441.0 (E) 640.5 9. To make a certain purple dye, red dye and blue dye are mixed in a ratio of 3:4. To make a cer- tain orange dye, red dye and yellow dye are mixed in a ratio of 3:2. If equal amounts of the purple and orange dye are mixed, what frac- tion of the new mixture is red dye? (A) (B) (C) (D) (E) 1 1 27 40 18 35 1 2 9 20 286 MCGRAW-HILL’S SAT Concept Review 4 1. distance to time 2. a sum to the number of terms in the sum 3. A proportion is a statement that two fractions or ratios are equal to each other. 4. If two fractions are equal, then the two “cross- products” must also be equal. 5. a) 6 = xy b) c) 6. $4,800, $7,200, and $12,000. The sum of the parts is 2 + 3 + 5 = 10, so the parts are 2/10, 3/10, and 5/10 of the whole. 7. 4. Cross-multiply: 25x = 6,000 Divide by 25: x = 240 minutes Convert to hours: 240 1 4mins hour 60 mins hrs× ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 25 bottles 6 minutes = 1,000 bottles minutesx 2 3y x = 3 2x y = 8. m 2 . “At this rate . . .” implies a proportion: Cross-multiply: m 2 x = ?x Divide by x: m 2 = ? 9. 12.5. Because ᐉ 1 ⏐⏐ ᐉ 2 ; ΔABC is similar to ΔADE. Thus, . Substituting x for DE gives . Cross-multiply: 4x = 50 Divide by 4: x = 12.5 10. 30. Since the ratios are the same, . Cross-multiply: bg = 324 Substitute b for g: b 2 = 324 Take the square root: b = 18 So the number of students in Class A = 12 + 18 = 30. 12 27g b = 4 46 5 + = x AC AE BC DE = m xmx meteorites days ? meteorites days = Answer Key 4: Ratios and Proportions SAT Practice 4 1. 1,000. .03 × .2 = 6/x Simplify: .006 = 6/x Multiply by x: .006x = 6 Divide by .006: x = 1,000 2. D Moving three green marbles from Jar B to Jar A leaves three green marbles and six red marbles in Jar A and one green marble and two red marbles in Jar B. 3:6 = 1:2. 3. B 2:3 is a “part to part” ratio, with a sum of 5. Therefore 2/5 of the students are girls and 3/5 are boys. 2/5 of 90 = 36. 4. D Cross-multiply: 5(x) = 3(y + 1) Simplify: 5x = 3y + 3 Add 1: 5x + 1 = 3y + 4 5. D Since the map is “to scale,” the correspond- ing measures are proportional: Cross-multiply: ?x = 4(x + 2) Divide by x: 6. D The ratio is a “ratio of parts” with a sum of 1 + 2 + 3 + 4 = 10. The largest part, then, is 4/10 of the whole. 4/10 of 3,600 = .4 × 3,600 = 1,440. ? = + () 42x x xx 4 2 = + ? 7. 100. “At this rate” implies a proportion: Cross-multiply: Multiply by : 8. B Set up the proportion: Cross-multiply: 7.5x = 105 Divide by 7.5: x = 14 feet Find the area: Area = 21 × 14 = 294 ft 2 9. C Each ratio is a “ratio of parts.” In the purple dye, the red dye is 3/(3 + 4), or 3/7, of the total, and in the orange dye, the red dye is 3/(3 + 2), or 3/5, of the total. If the mixture is half purple and half orange, the fraction of red is 1 2 3 7 1 2 3 5 3 14 3 10 72 140 18 35 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =+= = 75 5 21. = x 7.5 cm 5 cm 21 feet x feet x =× = = 5 2 40 200 2 100 miles 5 2 2 5 40x = 2 5 5 gallon 8miles gallons miles = x CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 287 Word Problems with Percents The word percent simply means divided by 100. Word problems are easy to solve once you know how to translate sentences into equa- tions. Use this key: Example: What number is 5 percent of 36? Use the translation key to translate the question into Then simplify to get x = 1.8. Example: 28 is what percent of 70? Use the translation key to translate the question into Then simplify to get 28 = .7x and divide by .7 to get x = 40. To convert a percent into a decimal, just re- member that percent means divided by 100 and that dividing by 100 just means moving the decimal two places to the left. Example: 35.7% = 35.7÷100 = .357 .04% = .04÷100 = .0004 Finding “Percent Change” Some word problems ask you to find the “per- cent change” in a quantity, that is, by what per- cent the quantity increased or decreased. A percent change is always the percent that the change is of the original amount. To solve these, use the formula 28 100 70=× x x =× 5 100 36 Percent change = final amount – starting amount × 100% starting amount Example: If the population of Bradford increased from 30,000 to 40,000, what was the percent increase? According to the formula, the percent change is Increasing or Decreasing by Percents When most people want to leave a 20% tip at a restau- rant, they do two calculations: First, they calculate 20% of the bill, and then they add the result to the orig- inal bill. But there’s a simpler, one-step method: Just multiply the bill by 1.20! This idea can be enormously helpful on tough percent problems. Here’s the idea: When increasing or decreasing a quantity by a given percent, use the one-step shortcut: Just multiply the quantity by the final percentage. For instance, if you decrease a quantity by 10%, your final percentage is 100% – 10% = 90%, so just multiply by 0.9. If you increase a quantity by 10%, your final percentage is 100% + 10% = 110%, so just multiply by 1.1. Example: If the price of a shirt is $60 but there is a 20% off sale and a 6% tax, what is the final price? Just multiply $60 by .80 and by 1.06: $60 ϫ .80 ϫ 1.06 = $50.88 Here’s a cool fact that simplifies some percent problems: a% of b is always equal to b% of a. So, for instance, if you can’t find 36% of 25 in your head, just remember that it’s equal to 25% of 36! That means 1/4 of 36, which is 9. 40 000 30 000 30 000 100 33 1 3 ,, , %% − ×= Lesson 5: Percents percent means ÷100 is means = of means × what means x, y, n, etc. 288 MCGRAW-HILL’S SAT Concept Review 5: Percents 1. Complete the translation key: 2. Write the formula for “percent change”: 3. To increase a quantity by 30%, multiply it by _____ 4. To decrease a quantity by 19%, multiply it by _____ 5. To increase a quantity by 120%, multiply it by _____ 6. To decrease a quantity by 120%, multiply it by _____ Translate the following word problems and solve them. 7. 5 is what percent of 26? Translation: ____________________ Solution: __________________ 8. 35% of what number is 28? Translation: ____________________ Solution: __________________ 9. 60 is 15% of what number? Translation: ____________________ Solution: __________________ 10. What percent is 35 of 20? Translation: ____________________ Solution: __________________ 11. What percent greater than 1,200 is 1,500? 11. ___________ 12. If the price of a sweater is marked down from $80 to $68, what is the percent markdown? 12. ___________ 13. The population of a city increases from 32,000 to 44,800. What is the percent increase? 13. ___________ 14. What number is 30% greater than 20? 14. ___________ 15. Increasing a number by 20%, then decreasing the new number by 20%, is the same as multiplying the orig- inal by _____. 16. Why don’t the changes in problem 15 “cancel out”? __________________________________________________________________________________________________ 17. If the sides of a square are decreased by 5%, by what percent is the area of the 17. ___________ square decreased? 18. 28% of 50 is the same as _____percent of _____, which equals _____. 19. 48% of 25 is the same as _____percent of _____, which equals _____. Word(s) in problem Symbol in equation what, what number, how much of = percent SAT Practice 5: Percents CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 289 1. David has a total of $3,500 in monthly expenses. He spends $2,200 per month on rent and utilities, $600 per month on clothing and food, and the rest on miscellaneous expenses. On a pie graph of his monthly expenses, what would be the degree measure of the central angle of the sector representing miscellaneous expenses? (A) 45° (B) 50° (C) 70° (D) 72° (E) 75° 2. In one year, the price of one share of ABC stock increased by 20% in Quarter I, increased by 25% in Quarter II, decreased by 20% in Quarter III, and increased by 10% in Quarter IV. By what percent did the price of the stock increase for the whole year? (Ignore the % symbol when gridding.) 5. The cost of a pack of batteries, after a 5% tax, is $8.40. What was the price before tax? (A) $5.60 (B) $7.98 (C) $8.00 (D) $8.35 (E) $8.82 6. If the population of Town B is 50% greater than the population of Town A, and the popu- lation of Town C is 20% greater than the pop- ulation of Town A, then what percent greater is the population of Town B than the popula- tion of Town C? (A) 20% (B) 25% (C) 30% (D) 35% (E) 40% 7. If the length of a rectangle is increased by 20% and the width is increased by 30%, then by what percent is the area of the rectangle increased? (A) 10% (B) 50% (C) 56% (D) 65% (E) It cannot be determined from the given information. 8. If 12 ounces of a 30% salt solution are mixed with 24 ounces of a 60% salt solution, what is the percent concentration of salt in the mixture? (A) 45% (B) 48% (C) 50% (D) 52% (E) 56% 9. The freshman class at Hillside High School has 45 more girls than boys. If the class has n boys, then what percent of the freshman class are girls? (A) (B) (C) (D) (E) 100 45 45 n n + () + % 100 45 245 n n + () + % 100 245 n n + % n n + + 45 245 % n n + 45 % 1 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 3. On a two-part test, Barbara answered 60% of the questions correctly on Part I and 90% cor- rectly on Part II. If there were 40 questions on Part I and 80 questions on Part II, and if each question on both parts was worth 1 point, what was her score, as a percent of the total? (A) 48% (B) 75% (C) 80% (D) 82% (E) 96% 4. If x is of 90, then 1 − x = (A) −59 (B) −5 (C) 0 (D) 0.4 (E) 0.94 2 3 % . cross-multiply. 6 7 1 = x 282 MCGRAW-HILL’S SAT CHAPTER 7 / ESSENTIAL PRE-ALGEBRA SKILLS 283 Working with Ratios When you see a ratio—such as 5:6—don’t let it confuse you. If it is not a part- to -part. 1 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 1 0 2 3 4 5 7 8 9 6 3. On a two -part test, Barbara answered 60% of the questions correctly on Part I and 90% cor- rectly on Part II. If there were 40 questions on Part I and 80 questions on Part II, and if. cross-multiplication and with the law of cross- swapping. The law of cross-multiplication says that if two fractions are equal, then their “cross-products” also must be equal. The law of cross-swapping