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Question Explanations For

Practice Test 4 (Math Calculator)

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1.

The Correct Answer is (C) — If every 30 minutes the mass of the substance increases by 48 grams, then after 1 hour (or 2 periods of 30 minutes) it must increase by (2)(48) = 96 grams. If the final weight is 111 grams, then the original weight was 111 – 96 = 15 grams.

2.

The Correct Answer is (B) — The total number of manuscripts Bartleby edits can be found by adding up the number of manuscripts he edits each day: x + 2.5x + 3.5x + 4x = 11x. Since he edits 33 manuscripts in total, 11x = 33, so x = 3.

3.

The Correct Answer is (C) — If Skye spends $1.75 per day, then her total spending over a period of time is $1.75 times the total number of days. Since there are 31 + 31 + 30 = 92 days in July, August, and September combined, Skye spends ($1.75)(92) = $161.00.

4.

The Correct Answer is (C) — Because of the associative property, $\frac{6x}{y}=6\frac{x}{y}$ . If $\frac{x}{y}=4$ , then $6(\frac{x}{y})=6(4)=24$ .

5.

The Correct Answer is (A) — To check which options are equivalent to 5x² + 13x – 6, you can use FOIL to expand each factored expression. Option I gives 5x² + 15x – 2x – 6 = 5x² + 13x – 6. Option II gives 5x² + 3x – 10x – 6 ≠ 5x² – 7x – 6. Option III gives 5(x² + 3x - 2x – 6) ≠ 5x² + 5x – 30. So, the only option which is equivalent to 5x² + 13x – 6 is option I.

6.

The Correct Answer is (C) — Option (A) is incorrect because 32 out of 94 people aged 35–54 oppose the regulation, which is a greater proportion (and therefore a higher likelihood to oppose) than the 32 out of 103 people aged 55–75 who oppose the regulation. Option (B) is incorrect for the same reason: people aged 35–54 are more likely to oppose the regulation than people aged 55–74. However, 53 out of 77 people aged 18–34 oppose the regulation, which is a higher proportion than the 32 out of 103 people aged 55–74 who oppose the regulation, so (C) is correct.

7.

The Correct Answer is (D) — Since the left sides of these equations are both equal to y, you can set the right sides equal to each other:

$\begin{align*} 6x+2 &= x^2+5x-4 \\ x^2-x-6 &= 0 \\ (x-3)(x+2) &= 0 \\ \end{align*}$

This means that x = 3 or x = -2. Plugging these values of x into either equation shows you that the two solutions are (3, 20) and (–2, –10). This gives two possible values of xy: 60 and 20. Since only 60 is an answer choice, (D) is correct.

8.

The Correct Answer is (C) — If a line passes through (x,y), then its equation is true for those values of x and y. Plugging in 3 for x and 6 for y gives k(3) – 6(6) = 24. Solving for k gives k = 20.

9.

The Correct Answer is (A) — You can solve for x as follows:

$\begin{align*} 3(x+5)+7 &=22 \\ 3x+15+7 &=22 \\ 3x+22 &= 22 \\ 3x &= 0 \\ x &= 0 \end{align*}$

10.

The Correct Answer is (A) — You can separate the absolute value inequality into two different inequalities: x – 4 ≤ 6 and –(x – 4) ≤ 6. Solving for x gives x ≤ 10 and –2 ≤ x, which can be rewritten as –2 ≤ x ≤ 10.

11.

The Correct Answer is (A) — First, you should notice that all answer choices are in the form y = mx+b. You can see in the table that when x = 0, f(x) = 2. Therefore, the y-intercept (the constant b) is 2. You can also calculate the slope m using the slope formula: $m=\frac{y_1 - y_2}{x_1 - x_2}$ . Plugging in two points, (x₁, y₁) and (x₂, y₂), from the frequency table will give you the slope; for example, $\frac{6-2}{3-0}=\frac{4}{3}$ . Since $m=\frac{4}{3}$ and b = 2, $f(x)=\frac{4}{3}x + 2$ .

12.

The Correct Answer is (C) — This problem can be solved by process of elimination. Answer choices (A) and (D) are incorrect because they both represent exponential growth instead of quadratic growth; the ladybug population grows by a factor of 2, and the investment increases by a factor of 0.05 (or 5%). Answer choice (B) is incorrect because the height of a falling rock would decrease, and so the coefficient in front of the x² term would be negative, not positive. However, a hot air balloon would increase in height, so (C) is correct.

13.

The Correct Answer is (B) — The number of employed workers over 25 who have not attended college is equal to the sum of the number of employed workers who have “Less than a High School Diploma” and the number of employed workers who are “High School Graduates”: 198 + 510 = 708. The total number of employed workers over 25 is 1,782. Therefore, the likelihood that an employed worker over 25 has not attended college is 708/1,782 = ~39.7%.

14.

The Correct Answer is (A) — You can examine the line of best fit to find that when T = 200, S is about 40, and when T = 400, S is about 80. Since $\frac{80-40}{400-200}=0.2$ , this yields a slope of 0.2. If the line of best fit were extended, it would intersect the y-axis at about 0, meaning the equation of the line is S = 0.2T + 0, or S = 0.2T. You could have also found the y-intercept by setting up the equation y = 0.2x + b, substituting a point on the line for (x, y), and solving for b.

15.

The Correct Answer is (A) — You can split the first expression into terms, and multiply the second expression by each term individually, and finally combine like terms: (x² + 2x - 2)(3x² - x - 1) = (x²)(3x² - x - 1) + 2x(3x² - x - 1) - 2(3x² - x - 1) = 3x⁴ - x³ - x² + 6x³ - 2x² - 2x - (6x² - 2x - 2) = 3x⁴ + 5x³ - 9x² + 2.

16.

The Correct Answer is (B) — In one week, an armadillo will eat (7)(3) = 21 lbs of food. Three armadillos will eat (21)(3) = 63 lbs. Since only 10% of that consists of plant matter, three armadillos will eat (0.10)(63) = 6.3 lbs of plant matter in a week.

17.

The Correct Answer is (B) — If a total of 17 females had litters of 4, there were (17)(4) = 68 armadillos born in 2012 and 2013. Including the 20 armadillos counted in 2012, there would be 68 + 20 = 88 armadillos in 2014 if they all survived. Since there are only 54 armadillos in 2014, you can conclude that 88 – 54 = 34 armadillos did not survive.

18.

The Correct Answer is (A) — Using the slope formula and plugging in two points, for example (6, 4) and (2, 2), you can see that the slope of the line is $\frac{4-2}{6-2}=\frac{1}{2}$ . Only answer (A) has a slope of ½. To double check, you can see that the line intersects the y-axis at (0, 1), so b = 1.

19.

The Correct Answer is (C) — The mean of Data Set A is $\frac{1+1+2+4+4+6}{6}=3$ , and the mean of Data Set B is $\frac{2+2+3+3+5}{5}=3$ . Their means are the same, but Data Set A has a much larger range with more values that are distant from the mean, so you can conclude that Data Set A has a larger standard deviation than Data Set (B). So, the answer is (C).

(A) and (B) are incorrect because the means are the same. (C) is incorrect because Data Set A has a larger standard deviation that Data Set (B), not smaller.

20.

The Correct Answer is (A) — If 8% of the population is 56 students, then $\frac{8}{100}=\frac{56}{x}$ where x is the total number of students. Solving for x shows that there are 700 students in total. If 7% of the students are affected indirectly, as shown in the diagram, then (0.07)(700) = 49 students are affected indirectly.

21.

The Correct Answer is (C) — Car 1 has a pre-sale price of $4,500. 5% off brings the price to ($4,500)(0.95) = $4,275, and 8% of the pre-sale price (for 80,000 miles) is $360, so the final price of Car 1 is $4,275 – $360 = $3,915. Car 2 has a pre-sale price of $5,200; 5% off gives $4,940, and 2% off the pre-sale price (for 20,000 miles) is $104, so the sale price of Car 2 is $4,940 – $104 = $4,836. The difference between these prices is $4,836 – $3,915 = $921.

22.

The Correct Answer is (B) — You can simplify the first equation by dividing both sides by 2 to get $y=\frac{x}{10}-\frac{1}{2}$ . You can substitute the expression for y in the first equation into the second equation to get $\frac{2x+8}{3}=\frac{x}{10}-\frac{1}{2}$ . Multiplying both sides by 30 to cancel the denominators gives 20x + 80 = 3x – 15, which we can simplify to 17x = –95, so x = –95/17. Plugging this into either equation gives y = –18/17.

23.

The Correct Answer is (C) — The volume of Package 1 is (10 in)(6 in)(2 in) = 120 in³; the volume of package 2 is (6 in)(6 in)(4 in) = 144 in³. The efficiency of package 1 is 120/184 = 0.65; the efficiency of package 2 is 144/168 = 0.86. So, Package 2 is more efficient than Package 1 by approximately 0.20 cubic inches per square inch.

24.

The Correct Answer is (B) — You can write out every possible result of the coin toss using “H” and “T” for heads and tails: HH, HT, TH, TT. Since two out of the four options have exactly one tails, the probability of any toss having exactly one tails is 1/2.

25.

The Correct Answer is (B) — Since pre-experiment levels were the same for all groups, you can see that the control group has the lowest post-experiment activity level, and therefore comparatively the other methods showed more improvement.

(A) is incorrect because sharing goals showed a greater increase in physical activity than participation on a sports team. (C) is incorrect because the control group did not decrease in physical activity over time. (D) is incorrect because although it may be true, this conclusion is not related to the data presented in the graph.

26.

The Correct Answer is (B) — You can use your calculator to see that $\frac{413545}{301231207}$ is more than double $\frac{182911}{287625193}$ .

Using process of elimination, you can see that (A) is incorrect because although there were fewer establishments in 2007 than in 2002, the value of shipments increased. (C) is incorrect because although there were fewer employees in 2007 than in 2002, the average pay increased. (D) is incorrect because although there were fewer establishments in 2007 than in 2002, there were more people employed.

27.

The Correct Answer is (A) — The ratio of value of annual shipments to annual payroll in 1997 was $\frac{173985}{20798}=8.365$ . The ratio in 2007 was $\frac{413525}{40687}=10.164$ . Percentage growth from 1997 to 2007 is the value in 2007 divided by the value in 1997, minus 100% (since growth is the value above 100%): $\frac{10.164}{8.365}$ =121.5%, so the percentage growth was 21.5%.

28.

The Correct Answer is (B) — First, you should recognize that 480 minutes is equal to 8 hours, or five 1.6-hour periods. This means that the salbutamol loses half of its therapeutic activity five times. After one 1.6-hour period, its activity is 240; after two periods, its activity is 120, and so on. After five 1.6-hour periods, its activity is 15.

29.

The Correct Answer is (D) — In 2000, approximately 3% of people walked and approximately 0.5% biked, and 3% is six times 0.5%.

Using process of elimination, (A) is incorrect because the chart only presents some of the data collected, and does not incorporate other modes of transportation (subways, for example). You can confirm this by showing that no more than 7% of people combined walked or biked to work, so the other 93% of people must use other transportation.
(B) is incorrect because although the percentage of commuters who biked to work stayed about the same, the data does not show the number of commuters.
(C) is incorrect because by looking at the slope of each line segment, you can see that the percentage of commuters who walked to work actually decreased more slowly between 1990 and 2000 than between 1980 and 1990.

30.

The Correct Answer is (C) — If you draw a radius from the center of the circle to point A, then the triangle OAB (where O is the center of the circle) is isosceles, since two of its sides are radii of the circle (and so have length 10). This means that the angle BAO is 50°, and so angle AOB is 180° – 50° – 50° = 80°. Therefore, the angle AOC is 180° – 80° = 100°. The circumference of the circle is 2πr = 20π, so you know that $\frac{AC}{20\pi}=\frac{100\degree}{360\degree}$ . You can solve this to find that AC = $\frac{50\pi}{9}$ .

31.

The Correct Answer is (4) — When y = 6, you can write 6 = 0.25x + 5, or 1 = 0.25x. Multiplying both sides by 4 gives x = 4.

32.

The Correct Answer is (70) — The missing angle on the left-most triangle is 180° – 90° – 15° = 75°. Since the bases of the triangles lie on a line (180°), you can find the bottom-left angle of the rightmost triangle by finding 180° – 75° – 50° = 55°. Since this is an isosceles triangle (as indicated by the marks on the sides), the bottom-right angle is also 55°, so 180° – 55° – 55° = 70°, so x = 70.

33.

The Correct Answer is (1.2) — You can see that 16 crates of 5 pounds each means there are (16)(5) = 80 pounds of oranges on the truck. If one truckload is 80 pounds for $96, then $96/80 pounds = $1.20 per one pound of oranges.

34.

The Correct Answer is (19) — First, f(3) = 6(3) + 1 = 19, so the numerator of $\frac{f(3)}{g(f(0))}$ is 19. Then, f(0) = 6(0) + 1 = 1. Plugging that in to g(x), you get g(1) = 2(1) – 1 = 1, so the denominator is 1. Finally, 19/1 = 19.

35.

The Correct Answer is (487) — The time it takes to orbit the sun today is (24 hours/day)(365 days) = 8,760 hours. If a day were only 18 hours long, there would be 8,760/18 = 486.7 or about 487 days in a year.

36.

The Correct Answer is (18) — You can rewrite this equation as $3\sqrt{m} = \sqrt{162}$ . Squaring both sides gives 9m = 162, and dividing by 9 gives m = 18.

37.

The Correct Answer is (12) — First, calculate the pace Susan ran in Week 3 and in Week 4. To calculate the difference in seconds per mile, you must convert the minutes of Week 3 and Week 4 into seconds: in Week 3, Susan took 4,080 seconds to run 8 miles, and in Week 4, she took 5,220 seconds to run 10 miles. Her pace in Week 3 was 4080/8 = 510 seconds per mile, and her pace in Week 4 was 522 seconds per mile. Therefore, she ran each mile 12 seconds faster in Week 3 than in Week 4.

38.

The Correct Answer is (8) — First, convert the given time and Susan’s Week 2 time to seconds: 4 hours and 46 minutes = 17,160 seconds, and 108 minutes = 6,480 seconds. Her pace in Week 6 is 17,160/26 = 660 seconds per mile, and her pace in Week 2 is 6,480/12 = 540 seconds per mile. Therefore, she must improve her pace from 660 to 540 seconds per mile, a difference of 120 seconds per mile. If she can improve her pace at a rate of 15 seconds per mile per week, then it will take her 120/15 = 8 weeks to improve her pace.