Practice Questions Answers and Explanations 1. Choice a is correct. The word and indicates a decimal point. Therefore, the decimal point should go after five hundred twelve and before sixteen thousandths. The number 16 must end in the thousandths place, which is three digits to the right of the decimal. The correct answer is 512.016. Choice b is “five hundred twelve and sixteen hundredths.” Choice c is “five hundred twelve thousand, one hundred sixty.” Choice d is “fifty one and two hundred sixteen thousandths.” Choice e is “five hundred twelve and sixteen ten thousandths.” 2. Choice f is correct. First, change the fractional parts of the problem to have the common denominator of 12. 4 ᎏ 1 4 2 ᎏ − 1 ᎏ 1 9 2 ᎏ Subtract the numerators. Since 4 is less than 9, you must borrow one whole from the whole number 4. This means that you are adding ᎏ 1 1 2 2 ᎏ to the first fraction. 3 ᎏ 1 1 6 2 ᎏ − 1 ᎏ 1 9 2 ᎏ . Subtract the fractional parts then the whole numbers. The final answer is 2 ᎏ 1 7 2 ᎏ . 3. Choice b is correct. The correct order of operations must be used to simplify the expression. You may remember this as PEMDAS or “Please Excuse My Dear Aunt Sally.” The P stands for parentheses or any grouping symbol. Absolute value is a grouping symbol, so it will be done first. |−8| + 4 × 2 3 = 8 + 4 × 2 3 Next, perform the exponent part. 8 + 4 × 8 Then, the multiplication. 8 + 32 Last, the addition. The final answer is 40. 4. Choice g is correct. This problem can be approached a couple of different ways. The simplest way might be to look at multiples of 4 and 5 until the multiples add to 18. If both 4 and 5 are multiplied by 2, they become 8 and 10. 8 plus 10 is 18. Therefore, there are 8 boys and 10 girls in the class. The problem can also be done with an equation. 4x + 5x = 18 When solved, x = 2. Multiply 4 by 2 to find that there are 8 boys. – ACT MATH TEST PRACTICE– 184 5. Choice c is correct. To find the median, place the numbers in order from least to greatest and find the middle number. In order, the numbers are: 0.008, 0.024, 0.024, 0.095, 0.1, 0.3 Since there are an even number of numbers, there are two middle numbers (0.024 and 0.095). Take the average of these two middle numbers by adding them and dividing the sum by two. The answer is 0.0595. 6. Choice h is correct. Use the vertical line test to see if each graph is a function. A graph is NOT a func- tion if vertical lines drawn through the graph hit the graph more than once. 7. Choice d is correct. When multiplying by 10 5 you move the decimal point 5 places to the right. The answer is 460,000. Another way to look at the problem is to recognize that 10 5 = 100,000 and multiply 4.6 by 100,000. The answer is 460,000. 8. Choice g is correct. 3 5 = 3 × 3 × 3 × 3 × 3 = 243. The answer is 243. 9. Choice a is correct. Consecutive odd integers starting with 3 are being added to find the next number. Therefore, 11 must be added to 24 to find the next number. The answer is 35. You may also notice that the numbers can be found using the expression n 2 − 1 where n is the place in the pattern. We are looking for the sixth number, so n = 6. 6 2 − 1 = 35. The answer is 35. 10. Choice h is correct. First, you can eliminate choices f and i because they contain numbers that are not prime. Next, use a factor tree to determine the prime factorization. The prime factorization of 84 is 2 × 2 × 3 × 7, which can be written in exponential notation as 2 2 × 3 × 7. The answer is 2 2 × 3 × 7. 84 2 42 76 3 2 0, 3, 8, 15, 24, . . . +3 +5 +7 +9 +11 4.60000. = 460,000 – ACT MATH TEST PRACTICE– 185 11. Choice c is correct. To easily see the slope, change the equation into the form y = mx + b. The equation is then y = ᎏ 7 3 ᎏ x + 3. The coefficient of x is the slope. ᎏ 7 3 ᎏ is the answer. 12. Choice f is correct. The perimeter is twice the width plus twice the length: P = 2w + 2l. Insert 20 for P and 4 for w, then solve for l. 20 = 2(4) + 2l 20 = 8 + 2l 20 − 8 = 2l 12 = 2l 6 = l 6 is the length. 13. Choice e is correct. Find the lengths of the unlabeled sides by comparing them to the given sides. Divide the shape into two rectangles as shown below. Find the area of each of the regions and add together to find the total area. 30 + 28 = 58 sq in. 14. Choice g is correct. Find the cost for one can (unit rate) by dividing the cost of five cans by 5. $6.50 ÷ 5 = $1.30 per can. Multiply the cost per can by 9 cans. $1.30 × 9 = $11.70. Nine cans cost $11.70. A proportion can also be used: ᎏ $6 5 .50 ᎏ = ᎏ 9 x ᎏ . To solve the proportion, cross-multiply and divide. 5x = $58.50 ᎏ 5 5 x ᎏ = ᎏ $58 9 .50 ᎏ x = $11.70 15. Choice c is correct. Find a common denominator (15x). Multiply the first fraction by ᎏ 5 5 ᎏ and the second fraction by ᎏ 3 3 x x ᎏ . The result is ᎏ 1 1 5 0 x ᎏ + ᎏ 1 3 5 x x ᎏ = ᎏ 10 1 + 5x 3x ᎏ . The answer is ᎏ 10 1 + 5x 3x ᎏ . 16. Choice i is correct. The endpoints are on −1.5 and 0, so the possible choices are i and h. The endpoints are open dots (not solid) and, therefore, only < or > signs can be used (not ≤ or ≥). This information narrows the answer choices down to only i. 17. Choice c is correct. First, raise everything in the parentheses to the second power: −(6 2 (x 4 ) 2 y 3 ) 2 ). When you have a power to a power you multiply the exponents. Thus, −(36x 8 y 6 ). Apply the negative for the final answer of − 36x 8 y 6 . 10 in 7 in 4 in 3 in 30 28 – ACT MATH TEST PRACTICE– 186 18. Choice h is correct. Use substitution to solve for x and y. First, solve the second equation for y. y = 4x − 47 Next, substitute the above value for y into the first equation and solve for x. 2x + 3(4x − 47) = 55 2x + 12x − 141 = 55 14x − 141 = 55 14x = 196 x = 14 Now, substitute 14 for x in the second equation and solve for y. 4(14) = y + 47 56 = y + 47 y = 9 The solution to the system of equations is (14, 9). The problem asks you to find x − y. 14 − 9 = 5. The answer is 5. 19. Choice e is correct. First, eliminate choice d because the dot on the graph is open and therefore the inequality sign must be either < or > (not ≤ or ≥). Next, solve each inequality for x remembering that the inequality symbol must be flipped when multiplying or dividing by a negative. The answer choices become: a. x > 0 b. x < − ᎏ 1 4 ᎏ c. x < −4 e. x > −4 The endpoint is on −4, so the only possibilities are choices c and e. The arrow points to numbers greater than −4. The answer is choice e. 20. Choice f is correct. Notice that you are taking the cube root, not the square root. Break up the expres- sion under the radical into perfect cubes. ͙ 3 (8)(2)x ෆ 3 x 2 y 3 y ෆ Any exponent divisible by 3 is a cube root. Take out the perfect cubes and leave everything else under the radical. 2xy͙ 3 2x 2 y ෆ is the answer. 21. Choice c is correct. Substitute the value 63° for C. F = ᎏ 5 9 ᎏ (63) + 32 F = 35 + 32 F = 67 The answer is 67°F. – ACT MATH TEST PRACTICE– 187 22. Choice h is correct. The equation is quadratic, so there are two ways to solve it. First, try to factor the left-hand side of the equation. Since it is factorable, solve the equation using factoring. x 2 + 8x + 15 = 0 (x + 5)(x + 3) = 0 Set each of the factors equal to zero and solve for x. x + 5 = 0 x + 3 = 0 x = − 5 x = −3 The solution set is {−5, −3}. The quadratic equation can also be used to solve the equation. x = x = x = x = ᎏ −8 2 ±2 ᎏ x = ᎏ −8 2 +2 ᎏ x = ᎏ −8 2 − 2 ᎏ x = ᎏ − 2 6 ᎏ = −3 x = ᎏ − 2 10 ᎏ = −5 The solution set is {−5, −3}. 23. Choice b is correct. Solve the equation for m using inverse operations. 5k = 9m − 18 5k + 18 = 9m ᎏ 5k + 9 18 ᎏ = m Since this answer does not appear as one of the choices, you must determine if any of the choices are equivalent to it. If you divide each of the numerator terms by 9 you get ᎏ 5 9 ᎏ k + 2 = m, which is choice b. 24. Choice h is correct. Solve the equation by moving all x terms to one side. 5x − 5x − 7 = 5x − 5x + 10 − 7 = 10 − 7 ≠ 10 Ø The x’s cancel, leaving −7 = 10, which is not true. Since −7 never equals 10, there is no solution. 25. Choice e is correct. Factor the numerator. ᎏ (4x − x 1 + )( 3 x+3) ᎏ Use the denominator as a clue when factoring the numerator. Most likely, the denominator will be one of the factors in the numerator. Cancel the x + 3 in the numerator with the x + 3 in the denominator. This leaves 4x − 1. −8 ± ͙64 − 60 ෆ ᎏᎏ 2 −8 ± ͙(8) 2 − ෆ (4)(1)( ෆ 15) ෆ ᎏᎏᎏ 2 −b ± ͙b 2 − 4a ෆ c ෆ ᎏᎏ 2a – ACT MATH TEST PRACTICE– 188 26. Choice f is correct. Subtract the numbers in y from the corresponding numbers in x. [] = [] 27. Choice b is correct. log 3 x = 2 is equivalent to 3 2 = x. Therefore, x = 9. 28. Choice h is correct. Factor the numerator. Use the denominator as a clue. Most likely, one of the fac- tors in the numerator will be the same as the denominator. Also, notice that the numerator is the dif- ference of two squares. ᎏ (x − x 3) − (x 3 +3) ᎏ The x − 3 in the numerator cancels with the x − 3 in the denominator leaving an answer of x + 3. 29. Choice b is correct. Use the distance formula or the Pythagorean theorem to find the distance. The dis- tance formula is d = ͙(x 2 − x ෆ 1 ) 2 + (y ෆ 2 − y 1 ) 2 ෆ . Substitute the x and y values for points A and C and solve. d = ͙(−2 −− ෆ 1) 2 + ( ෆ −1 − 3 ෆ ) 2 ෆ d = ͙(−1) 2 + ෆ (−4) 2 ෆ d = ͙1 + 16 ෆ d = ͙17 ෆ To use the Pythagorean theorem (which is what the distance formula is derived from), draw the seg- ment on a coordinate plane and create a right triangle where A ෆ C ෆ is the hypotenuse. The legs of the right triangle are 1 and 4. Use the Pythagorean theorem to find the length of the hypotenuse. a 2 + b 2 = c 2 1 2 + 4 2 = c 2 1 + 16 = c 2 17 = c 2 ͙17 ෆ = c (−1,3) (−2,−1) 1 4 50 66 3 − ( − 2) 4 − 4 5 − ( − 1) 6 − 0 – ACT MATH TEST PRACTICE– 189 . by 2 to find that there are 8 boys. – ACT MATH TEST PRACTICE– 184 5. Choice c is correct. To find the median, place the numbers in order from least to greatest and find the middle number. In order,. × 7. The answer is 2 2 × 3 × 7. 84 2 42 76 3 2 0, 3, 8, 15, 24, . . . +3 +5 +7 +9 +11 4.60000. = 460,000 – ACT MATH TEST PRACTICE– 185 11. Choice c is correct. To easily see the slope, change the. 5 = $1.30 per can. Multiply the cost per can by 9 cans. $1.30 × 9 = $11.70. Nine cans cost $11.70. A proportion can also be used: ᎏ $6 5 .50 ᎏ = ᎏ 9 x ᎏ . To solve the proportion, cross-multiply