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QUESTION EXPLANATIONS

For NEW SAT PRACTICE TEST 3 (Calculator Math Test)

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1. (C) — Dividing 60 minutes by 5 minutes gives us 12 five-minute intervals. During each of these intervals, 8 grams of barite are synthesized, giving us a total of 12 × 8 = 96 grams. 111 grams – 96 grams = 15 grams at the start of the experiment.
 
2. (A) — Graphs (B) and (D) are quadratic functions, which we can determine by their “U” shape and symmetry across the y-axis. Graph (C) cannot be correct, since y should get larger as x gets larger. That leaves graph (A), which is the correct answer.
 
3. (A) — We can set up an equation using ratios of eggs to cupcakes to find the number of eggs, which we’ll call x: e/c = x/40. Solving this equation gives us x = 40 × (e/c).
 
4. (B) — We can find the slope with the expression (y2-y1)/(x2-x1). (-5 – (-3))/(-2 – (-5)) = -2/3.
 
5. (C) — On the first square, there is one (20) grain of sand. On the second, there are two (21), on the third, there are four (22). Continuing this pattern shows us that for the nth square, there will be 2n-1 grains of sand, so on the 64th square, there are 263 grains.
 
6. (A) — The total value of coffee shop gift cards is $31 billion × 13.9% = $4.31 billion. Therefore, the amount that is unused is $4.31 billion × 27% = $1.16 billion.
 
7. (C) — Profit is positively related to the number of loaves produced by a factor of three. This means that every time b increases by 1, p increases by 3. (A) and (D) are incorrect because profit increases as more loaves are produced. (B) is incorrect because fixed costs are independent of the number of loaves produced.
 
8. (D) — Answers (A) – (C) can be found by manipulating the given equation. In (A), each term is multiplied by 12. In (B), each term is multiplied by 2. In (C), each term is multiplied by 4.
 
9. (A) — The graph includes 9 films, so the film with the median ticket sales (x axis) is the 5th from the left or right. Referring to the y axis, we see that this film had a budget of $125 million.
 
10. (C) — Substituting 2/3 for z in the equation we get: f(2/3) = 2/(2/3) + (2/3) × 3 = 2 × (3/2) + 2 = 5.
 
11. (D) — First we convert the snail’s speed to m/min: 13mm/1sec × 1m/1000mm × 60s/1min = 0.78 m/min. The time for the snail to travel 169m is: 169m × 1min/0.78m = 217 min.
 
12. (B) — We can calculate the percent growth as the ratio of the end year to the start year. The percent growth between 2006 and 2008 is 172/86 = 2, which is the highest of the periods given. For 2005-2007, the ratio is 122/65 ≈ 1.88; for 2007-2009, it is 238/122 ≈ 1.95, and for 2008-2010 is it 207/172 ≈ 1.20.
 
13. (C) — When we increase a side by S%, the new length is (1 + S%) times the original length. The area is (1 + L) × (1 + W) times the original area, which expands to 1 + L + W + LW. Since the question asks for the percent increase in the rectangle’s area, we can eliminate the 1 (which represents the original area). The resulting equation can be rewritten as L(W + 1) + W. You can also plug numbers into the equations to check what works, which may be faster for you.
 
14. (B) — We can solve the given equations for b in terms of x. The first equation gives b = 5x + 3 and the second equation gives b = -(1/2)(x + 3). Either of these values of b will satisfy the system of equations, but only the second value is an answer choice, so the correct answer is (B).
 
15. (B) — The slope of the function is rise (difference in y values) divided by run (difference in x values), which in this case is (0 – 9)/(3 – 0) = -3. We can use the slope equation for (a, 3): -3 = (9 - 3)/(0 - a) to find a = 2.
 
16. (C) — The fastest way to solve this problem is to recognize that the x in f(x) corresponds to the x-axis, and that f(x) corresponds to the y-axis. We can take these values as coordinate pairs to determine the slope. Slope is rise (difference in y values) divided by run (difference in x values), which in this case is (-1 – 1)/(-5 + 1) or -2/-4 = 2. Alternatively, by substituting the given values of f(x) and x, we can generate two equations: -5 = -a – c and -1 = a – c. Adding c to both sides in both equations gives us –a = c – 5 and a = c – 1. Multiplying the first equation by -1 gives a = c – 1. We can now add these two equations together to get 2a = 4, so a = 2.
 
17. (D) — Since (C) and (D) cannot both be true, it makes sense to start by checking these answer choices first. Since a is 40% of b, a/b = 40/100 = 2/5. Therefore (D) is not true. Since the question asks which of the answer choices is not true, (D) is the correct answer.
 
18. (D) — We can treat a and b as x and y, respectively, and plot the inequalities on a coordinate plane:

From the plot, we can see that the only given value of a that satisfies both inequalities is -6. Alternatively, we can rewrite the second inequality as a - 8 < b. Since b ≤ 2a - 1, a - 8 < 2a - 1. Subtracting a from both sides gives us -8 < a - 1, and adding 1 to both sides gives us a > -7. Of the given values, only -6 satisfies this inequality.
 
19. (C) — The given expression simplifies to x2abyab. By using exponent rules, we can see that all the answer choices except (C) also simplify to this value. (C) simplifies to xa+3bya. Remember that you can only add exponents when the bases are the same.
 
20. (C) — We can quickly see that since the trend line passes a 10% mark every 4 years, there is a yearly increase of 2.5%. To confirm this, we can find the slope of the trend line, choosing two data points close to the line. If we take, for example, (2004, 68) and (2012, 88), then the slope is (88 – 68)/(2012 – 2004) = 2.5% per year.
 
21. (A) — We can factor the right-hand side of the equation such that y = (3x - 2)(x + 4). For y = 0, the solutions are x = 2/3 and x = -4, which is choice (A).
 
22. (C) — We can rearrange the first equation to find that y = x + 8. Plugging into the second equation gives us x2 – x(x + 8) = -4. Expanding and simplifying gives -8x = -4, so x = ½.
 
23. (C) — The South had 35,700 apartments that were rented for less than $950; the Northeast had only 2,700, the Midwest had 10,200, and the West had 8,800.
 
24. (B) — Since the median rent for the South is below the median rent for the U.S. as a whole, at least 50% of renters in the South paid less than the U.S. median price. (A) is not correct because without more information we cannot say anything about the mean; (C) is not correct because apartments in that category make up 1000/17200 ≈ 5.8% in the Midwest and 3900/42400 ≈ 9.2% in the West, and (D) is not correct because the median price in the South is lower than both the Northeast and the West.
 
25. (D) — The median monthly rent in the Midwest is $857, so John would save $1200 - $857 = $343 per month. Annually, that comes to $343 × 12 = $4,116.
 
26. (B) — We can define the original number of apples as A and the original number of oranges as R. When 6 oranges are removed, A = 3(R – 6). When 11 apples are removed, 4(R – 6) = A – 11. Solving this system of equations, we find that there were originally 12 apples and 10 oranges.
 
27. (A) — To develop trait g, the experiment must develop c, which has a probability of 0.60, and then develop g, which has a probability of 0.20. To find the probability of developing g from a, we use the multiplication rule and multiply the probability of each event: 0.60 × 0.20 = 0.12 or 12%.
 
28. (D) — By multiplying the first equation by x – 2, we get 3y = 2x2 + x – 10. Dividing by 3 gives us y = (2x2 + x – 10)/3. The equations are equivalent, so there are infinitely many solutions.
 
29. (C) — First we draw the right triangles marked with dotted lines (we know they are right triangles since a radius and a line tangent to a circle must form a 90 degree angle if they share a point on the circle). The legs of each triangle is the radius r of the circle, so the hypotenuse is r√(2). Therefore, the distance between the centers of the two circles is the sum of the hypotenuses of the two triangles, which is 1√(2) + 3√(2) = 4√(2). To find the distance between the edges of the circles, we subtract the two radii to get 4√(2) - 4. Factoring out the 4 gives us 4(√(2) - 1).

 
30. (D) — The information in the question tells you that you are 95% sure that between and 65.2% and 74.9% of residents want the sewers. You calculate this interval by adding and subtracting half of 9.7% to the result of 70%. This means that between 25.1% and 34.8% of residents do not want the sewers. The inspector received calls from 1700 people protesting, which is 34% of the residents. The survey predicted that up to 34.8% of the residents might protest, so this outcome was predicted by the survey. The sample size was big enough for a 95% confidence level, so (A) is not correct. Asking fewer people would not have made the results clearer and may in fact have made the results less reliable, so (B) is not correct. A smaller confidence interval is indicative of more accurate results, so 9.7% being “too low” does not make sense, and (C) is incorrect.
 
31. (18) — We can solve the equation by substituting 21 for j: h/21 = 6/7, so h = 18.
 
32. (10) — Defining Nanna, Laurel, and Jennifer’s ages as N, L, and J respectively, we can set up three equations from the given information: J = 2L, L + 4 = N, and N + 2 = J. Solving these equations, we find that N = 10.
 
33. (5) — If the weed initially covers 32 square feet and increases by 50% each week, after one week it will cover an area of 32 × 1.5 = 48 square feet. After two weeks it will cover 48 × 1.5 = 72 square feet. We can model this growth as 32 × 1.5n, where n is the number of weeks; after 5 weeks, the weed will cover 243 square feet.
 
34. (45) — The difference between the silo when it is 40% empty and 40% full is 60% - 40% = 20%. If 20% of the silo’s volume is 9 pounds, the full volume is 9/0.20 = x/1, and x = 45 pounds.
 
35. (233) — The Cessna will travel the 500 km in 500 km/180 km/h = 2.78 hours, or about 167 minutes. The pigeon will take 500 km/75 km/h = 6.67 hours or 400 minutes. 400 – 167 = 233 minutes.
 
36. (10) — Solving the function for x = 1, we get y = 5, so the height of the rectangle is 5 units. The base is 2 units long (the function is symmetrical about the y-axis), so the area is 2 × 5 = 10.
 
37. (30) — The boat travels 1,750 miles, so if 10% of the original population dies every 250 miles, 1,750/250 × 10% = 70% of the population will die, leaving 30 thousand mussels remaining.
 
38. (12) — The total number of mussels in the quarter of the lake is .25 × (69.4 × 109 sq. yd.) × 700,000 mussels per sq. yd. = 1.21 × 1016 mussels. The percent increase in risk of transport is 1.21 × 1016 mussels/10 million mussels × 10-8 percent = 12 percent.


Dividing 60 minutes by 5 minutes gives us 12 five-minute intervals. During each of these intervals, 8 grams of barite are synthesized, giving us a total of 12 × 8 = 96 grams. 111 grams – 96 grams = 15 grams at the start of the experiment.
Graphs (B) and (D) are quadratic functions, which we can determine by their “U” shape and symmetry across the y-axis. Graph (C) cannot be correct, since y should get larger as x gets larger. That leaves graph (A), which is the correct answer.
We can set up an equation using ratios of eggs to cupcakes to find the number of eggs, which we’ll call x: e/c = x/40. Solving this equation gives us x = 40 × (e/c).
We can find the slope with the expression (y2-y1)/(x2-x1). (-5 – (-3))/(-2 – (-5)) = -2/3.
On the first square, there is one (20) grain of sand. On the second, there are two (21), on the third, there are four (22). Continuing this pattern shows us that for the nth square, there will be 2n-1 grains of sand, so on the 64th square, there are 263 grains.
The total value of coffee shop gift cards is $31 billion × 13.9% = $4.31 billion. Therefore, the amount that is unused is $4.31 billion × 27% = $1.16 billion.
Profit is positively related to the number of loaves produced by a factor of three. This means that every time b increases by 1, p increases by 3. (A) and (D) are incorrect because profit increases as more loaves are produced. (B) is incorrect because fixed costs are independent of the number of loaves produced.
Answers (A) – (C) can be found by manipulating the given equation. In (A), each term is multiplied by 12. In (B), each term is multiplied by 2. In (C), each term is multiplied by 4.
The graph includes 9 films, so the film with the median ticket sales (x axis) is the 5th from the left or right. Referring to the y axis, we see that this film had a budget of $125 million.
Substituting 2/3 for z in the equation we get: f(2/3) = 2/(2/3) + (2/3) × 3 = 2 × (3/2) + 2 = 5.
First we convert the snail’s speed to m/min: 13mm/1sec × 1m/1000mm × 60s/1min = 0.78 m/min. The time for the snail to travel 169m is: 169m × 1min/0.78m = 217 min.
We can calculate the percent growth as the ratio of the end year to the start year. The percent growth between 2006 and 2008 is 172/86 = 2, which is the highest of the periods given. For 2005-2007, the ratio is 122/65 ≈ 1.88; for 2007-2009, it is 238/122 ≈ 1.95, and for 2008-2010 is it 207/172 ≈ 1.20.
When we increase a side by S%, the new length is (1 + S%) times the original length. The area is (1 + L) × (1 + W) times the original area, which expands to 1 + L + W + LW. Since the question asks for the percent increase in the rectangle’s area, we can eliminate the 1 (which represents the original area). The resulting equation can be rewritten as L(W + 1) + W. You can also plug numbers into the equations to check what works, which may be faster for you.
We can solve the given equations for b in terms of x. The first equation gives b = 5x + 3 and the second equation gives b = -(1/2)(x + 3). Either of these values of b will satisfy the system of equations, but only the second value is an answer choice, so the correct answer is (B).
The slope of the function is rise (difference in y values) divided by run (difference in x values), which in this case is (0 – 9)/(3 – 0) = -3. We can use the slope equation for (a, 3): -3 = (9 - 3)/(0 - a) to find a = 2.
The fastest way to solve this problem is to recognize that the x in f(x) corresponds to the x-axis, and that f(x) corresponds to the y-axis. We can take these values as coordinate pairs to determine the slope. Slope is rise (difference in y values) divided by run (difference in x values), which in this case is (-1 – 1)/(-5 + 1) or -2/-4 = 2. Alternatively, by substituting the given values of f(x) and x, we can generate two equations: -5 = -a – c and -1 = a – c. Adding c to both sides in both equations gives us –a = c – 5 and a = c – 1. Multiplying the first equation by -1 gives a = c – 1. We can now add these two equations together to get 2a = 4, so a = 2.
Since (C) and (D) cannot both be true, it makes sense to start by checking these answer choices first. Since a is 40% of b, a/b = 40/100 = 2/5. Therefore (D) is not true. Since the question asks which of the answer choices is not true, (D) is the correct answer.
We can treat a and b as x and y, respectively, and plot the inequalities on a coordinate plane:

From the plot, we can see that the only given value of a that satisfies both inequalities is -6. Alternatively, we can rewrite the second inequality as a - 8 < b. Since b ≤ 2a - 1, a - 8 < 2a - 1. Subtracting a from both sides gives us -8 < a - 1, and adding 1 to both sides gives us a > -7. Of the given values, only -6 satisfies this inequality.
The given expression simplifies to x2abyab. By using exponent rules, we can see that all the answer choices except (C) also simplify to this value. (C) simplifies to xa+3bya. Remember that you can only add exponents when the bases are the same.
We can quickly see that since the trend line passes a 10% mark every 4 years, there is a yearly increase of 2.5%. To confirm this, we can find the slope of the trend line, choosing two data points close to the line. If we take, for example, (2004, 68) and (2012, 88), then the slope is (88 – 68)/(2012 – 2004) = 2.5% per year.
We can factor the right-hand side of the equation such that y = (3x - 2)(x + 4). For y = 0, the solutions are x = 2/3 and x = -4, which is choice (A).
We can rearrange the first equation to find that y = x + 8. Plugging into the second equation gives us x2 – x(x + 8) = -4. Expanding and simplifying gives -8x = -4, so x = ½.
The South had 35,700 apartments that were rented for less than $950; the Northeast had only 2,700, the Midwest had 10,200, and the West had 8,800.
Since the median rent for the South is below the median rent for the U.S. as a whole, at least 50% of renters in the South paid less than the U.S. median price. (A) is not correct because without more information we cannot say anything about the mean; (C) is not correct because apartments in that category make up 1000/17200 ≈ 5.8% in the Midwest and 3900/42400 ≈ 9.2% in the West, and (D) is not correct because the median price in the South is lower than both the Northeast and the West.
The median monthly rent in the Midwest is $857, so John would save $1200 - $857 = $343 per month. Annually, that comes to $343 × 12 = $4,116.
We can define the original number of apples as A and the original number of oranges as R. When 6 oranges are removed, A = 3(R – 6). When 11 apples are removed, 4(R – 6) = A – 11. Solving this system of equations, we find that there were originally 12 apples and 10 oranges.
To develop trait g, the experiment must develop c, which has a probability of 0.60, and then develop g, which has a probability of 0.20. To find the probability of developing g from a, we use the multiplication rule and multiply the probability of each event: 0.60 × 0.20 = 0.12 or 12%.
By multiplying the first equation by x – 2, we get 3y = 2x2 + x – 10. Dividing by 3 gives us y = (2x2 + x – 10)/3. The equations are equivalent, so there are infinitely many solutions.
First we draw the right triangles marked with dotted lines (we know they are right triangles since a radius and a line tangent to a circle must form a 90 degree angle if they share a point on the circle). The legs of each triangle is the radius r of the circle, so the hypotenuse is r√(2). Therefore, the distance between the centers of the two circles is the sum of the hypotenuses of the two triangles, which is 1√(2) + 3√(2) = 4√(2). To find the distance between the edges of the circles, we subtract the two radii to get 4√(2) - 4. Factoring out the 4 gives us 4(√(2) - 1).

The information in the question tells you that you are 95% sure that between and 65.2% and 74.9% of residents want the sewers. You calculate this interval by adding and subtracting half of 9.7% to the result of 70%. This means that between 25.1% and 34.8% of residents do not want the sewers. The inspector received calls from 1700 people protesting, which is 34% of the residents. The survey predicted that up to 34.8% of the residents might protest, so this outcome was predicted by the survey. The sample size was big enough for a 95% confidence level, so (A) is not correct. Asking fewer people would not have made the results clearer and may in fact have made the results less reliable, so (B) is not correct. A smaller confidence interval is indicative of more accurate results, so 9.7% being “too low” does not make sense, and (C) is incorrect.
We can solve the equation by substituting 21 for j: h/21 = 6/7, so h = 18.
Defining Nanna, Laurel, and Jennifer’s ages as N, L, and J respectively, we can set up three equations from the given information: J = 2L, L + 4 = N, and N + 2 = J. Solving these equations, we find that N = 10.
If the weed initially covers 32 square feet and increases by 50% each week, after one week it will cover an area of 32 × 1.5 = 48 square feet. After two weeks it will cover 48 × 1.5 = 72 square feet. We can model this growth as 32 × 1.5n, where n is the number of weeks; after 5 weeks, the weed will cover 243 square feet.
The difference between the silo when it is 40% empty and 40% full is 60% - 40% = 20%. If 20% of the silo’s volume is 9 pounds, the full volume is 9/0.20 = x/1, and x = 45 pounds.
The Cessna will travel the 500 km in 500 km/180 km/h = 2.78 hours, or about 167 minutes. The pigeon will take 500 km/75 km/h = 6.67 hours or 400 minutes. 400 – 167 = 233 minutes.
Solving the function for x = 1, we get y = 5, so the height of the rectangle is 5 units. The base is 2 units long (the function is symmetrical about the y-axis), so the area is 2 × 5 = 10.
The boat travels 1,750 miles, so if 10% of the original population dies every 250 miles, 1,750/250 × 10% = 70% of the population will die, leaving 30 thousand mussels remaining.
The total number of mussels in the quarter of the lake is .25 × (69.4 × 109 sq. yd.) × 700,000 mussels per sq. yd. = 1.21 × 1016 mussels. The percent increase in risk of transport is 1.21 × 1016 mussels/10 million mussels × 10-8 percent = 12 percent.