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## Question Explanations For

##
*Practice Test 1 (Calculator)*

## 1.

The Correct Answer is (D) —
Notice that the equation 8

*c*+ 4 = 44 is simply 2*c*+ 1 =*n*multiplied by a factor of 4. Therefore, you can write that 4*n*= 44, or*n*= 11.## 2.

The Correct Answer is (B) —
You can rewrite this problem in fractions to make it easier to visualize: . In words, this means that if a farmer buys 8 pigs per 1.5 acres, she buys

*p*pigs for 6 acres. Simply multiply both sides by 6 to find that*p*= 32.## 3.

The Correct Answer is (B) —
To find the value of
Therefore,

*x*, you must eliminate*y*from the equations. You can do this by substituting the first equation into the second:*x*= 0.## 4.

The Correct Answer is (B) —
First, notice that all the answer choices are linear functions. Therefore, you are looking for a function in the form of

*y*=*mx*+*p*, where*m*is the slope and*p*is the*y*-intercept. To find the*y*-intercept of*S*(*b*), simply find the value of*S*(*b*) when*b*= 0 in the table, which is 3, so you can eliminate (C) and (D). Next, you need to find the slope, which is the change in shelves divided by the change in boxes, or “rise over run.” Every time*b*goes up by 2,*S*(*b*) goes up by 1; therefore, the slope is^{1}⁄_{2}, and you can combine this with the*y*-intercept to find that answer is*S*(*b*) =^{1}⁄_{2}*b*+ 3.## 5.

The Correct Answer is (B) —
Set the left side of both equations equal to each other, and solve:

## 6.

The Correct Answer is (B) —
To find points to calculate the slope, first notice the scale of the graph: the

*y*-axis increases by units of 3 and the*x*-axis increases by units of 2. The*y*-intercept of this graph is therefore the point (0, –3) and the*x*-intercept is the point (2, 0). Find the difference between these points to find the slope: .## 7.

The Correct Answer is (D) —
You can find the population of moths by constructing an exponential equation and then plugging in the given values, but it is faster to use a simpler method. If the moth population doubles every 5 years, then you can find how many doubling periods (5-year periods) there are between 2005 and 2020; 2020 – 2005 = 15, which is 3 doubling periods. You can then multiply 300 by 2 three times to find that the population is 300 × 2 × 2 × 2 = 2400.

## 8.

The Correct Answer is (B) —
Convert the information in this question into algebraic form to get $0.05

*v*+ $0.10*t*= $3.10 and*v*+*t*= 54, which means that I and II are true. You can see that III has replaced*t*with 54 –*v*, but has misplaced the coefficients ($0.05*t*+ $0.10*v*does not equal $3.10). Therefore, III is false, and the correct answer is (B).## 9.

The Correct Answer is (C) —
Because lines

*BC*and*DE*are parallel, line*AE*is a transversal, which means that angle*BCE*is equal to angle*x*. Angle*BCE*is also supplementary to angle*ACB*. There are 180 degrees in a triangle, so angle*ACB*= 180 – 30 – 40 = 110°. Supplementary angles add to 180°, so the measure of angle*BCE*is 70°, and*x*= 70.## 10.

The Correct Answer is (B) —
To find a possible length of an eel in inches, you need to convert the range of possible lengths from centimeters to inches. The minimum possible length is 50 cm × (

^{1 in}⁄_{2.54 cm}) = 19.7 in, and the maximum possible length is 90 cm × (^{1 in}⁄_{2.54 cm}) = 35.4 in. Therefore, (A) is too small and (C) and (D) are too large, leaving you with the correct answer choice, (B).## 11.

The Correct Answer is (D) —
Essentially, this question asks you to choose the largest number of random samples, which is 1,000. There is no calculation required here, only the knowledge that the larger the sample size, the more likely it is to accurately reflect the population, which in this case means the larger the number of random samples, the more accurate the result derived from them will be.

## 12.

The Correct Answer is (A) —
“Least likely to drink any cups of coffee” means the same thing as “most likely to drink no cups of coffee.” The group of students who is most likely to drink no cups of coffee is the Freshman year, because 50% of these students drink 0 cups of coffee per day.

## 13.

The Correct Answer is (B) —
To answer this question, you must evaluate each of the statements on its own and determine if it is supported by the table, and then choose the statement that is not supported by the table. When you have to examine every answer option, it can be faster to quickly examine the shorter answer options first. (C) is true, since

^{40}⁄_{200}= 20%. (D) is also true, since^{25}⁄_{50}= 50%. To determine if (A) is true, you must find the percentage of juniors who drink 2 or more cups of coffee per day (^{50}⁄_{66}= 76%) and the percentage of sophomores who drink 2 or more cups of coffee per day (^{26}⁄_{50}= 52%). Therefore, (A) is true, which leaves (B) as the answer. You could double-check that it’s false, and indeed, 76% is not greater than 94%.## 14.

The Correct Answer is (D) —
If height,

*h*, must be at least 3.5 feet, then*h*is greater than or equal to 3.5, or*h*≥ 3.5. This fact eliminates (A), (B), and (C), which leaves you with (D). To verify (D), you could check that to be at least 6.5 feet tall means that*h*is less than or equal to 6.5, so*h*≤ 6.5, which fits with (D).## 15.

The Correct Answer is (C) —
To find the average number of books Anthony reads per day in terms of

*x*, add up all the numbers of books and divide by the number of days: . Remember that “*x*” without a coefficient simply means “1*x*.”## 16.

The Correct Answer is (D) —
Look at what the answer options are telling you to avoid wasting time by rearranging and factoring. To determine what values make the inequality true, remember that the cube of a negative number remains negative. Because of this, the square of a negative number, which is positive, is greater than the cube of a negative number, and so (A), (B), and (C) are incorrect. Plug 0 into the inequality to verify that (D) is true.

## 17.

The Correct Answer is (B) —
The solid line represents the population of Bacteria

*A*, which is growing exponentially, since its growth rate becomes even more rapid over time. The dotted line represents the population of Bacteria*B*, which is growing linearly, since its growth rate is always at the same constant rate. Therefore, (B) is true.## 18.

The Correct Answer is (C) —
To find the largest value of

*y*most efficiently, think about the characteristics of this parabola before plugging in the answer options. The parabola is negative, so it opens down. Its vertex, which is also its maximum, occurs at (2, 4). Therefore, there is no*y*-value greater than 4 on this parabola, and*y*= 4 results from*x*= 2.## 19.

The Correct Answer is (C) —
Before solving for the value of

*x*^{2}*y*, you must determine the value of*y*. Substitute*x*= 12 into the second equation to get 36 = 4*y*^{2}, then divide both sides by 4 to get 9 =*y*^{2}. If*y*> 0, then the only possible solution is*y*= 3. Since*x*= 12 and*y*= 3,*x*^{2}*y*= (12^{2})(3) = (144)(3) = 432.## 20.

The Correct Answer is (D) —
Since the two numbers are consecutive and even, they can be represented by

*x*and*x*+ 2. They cannot be*x*and*x*+ 1, because then one would be even and one would be odd. If the product of these two numbers is 168, you can create an algebraic expression that looks like this:*x*(*x*+ 2) = 168. Distributing the*x*gives you*x*^{2}+ 2*x*= 168, and rearranging the equation gives you*x*^{2}+ 2*x*– 168 = 0. You can factor this equation to get (*x*+ 14)(*x*– 12) = 0, which means that*x*= 12 or*x*= –14. Both of these numbers are even, but only 12 is positive. You can check that 12 is correct by calculating 12(12 + 2) = (12)(14) = 168.If you chose (C), you may have calculated the number of fish per pond, the larger of the two numbers.

## 21.

The Correct Answer is (C) —
First, notice that all the answer choices are linear functions. Therefore, you are looking for a function in the form

*y*=*mx*+*b*, where*m*is the slope and*b*is the*y*-intercept. Each answer choice is written in reverse order as*y*=*b*+*mx*. The*y*-intercept on the graph is 1500, which eliminates (A) because it has a*y*-intercept of 200 and (B) because it has a*y*-intercept of 150. Next, you must find the slope of the line of best fit, which is the change in the number of passengers divided by the change in months. Choose two clear and unambiguous points on the line to find its slope: for example, (1, 1600) and (0, 1500). Without doing any calculation, you can see that for every month, which is*m*in the function, the number of passengers increases by 100. Therefore, the slope of the equation is 100, so (C) is correct.## 22.

The Correct Answer is (D) —
There are two main ways to determine how many passengers will be using the commuter line after one year, which is 12 months. The first method is to trust your equation from #21 and simply plug 12 into it:

*f*(12) = 1,500 + 100(12) = 1,500 + 1,200 = 2,700. The second method is to notice how the number of passengers changes per month (find the slope of the line of best fit) and add eight monthly changes to the four-month number of passengers: 1,900 + 8(100) = 2,700. Be sure to use the line of best fit to make your estimate and not the individual points on the scatterplot.## 23.

The Correct Answer is (C) —
If

*g*(*x*) =*f*(*x*) – 1, then*g*(*x*) is the same function as*f*(*x*) translated one unit down. It has the same shape as*f*(*x*), but its minimum is now at (0, –2). Therefore,*g*(*x*) is always greater than or equal to –2. (A) is false, because*g*(*x*) will extend above the*x*-axis. (B) is false because*g*(*x*) takes on some values between –2 and 0 (for example,*g*(1) = –1). (D) is false because*g*(*x*) is not always greater than –1 (for example,*g*(0) = –2).## 24.

The Correct Answer is (B) —
The total number of Ms. Feldman’s students is 26 + 15 = 41. The total number of students who receive either an A or a B is equal to the sum of the students who received an A (3% of the total) and the students who received a B (9% of the total): 41(0.03) + 41(0.09) = 1.23 + 3.69 = 4.92, which is closest to 5. Be sure to use the right number of decimal places when converting the percentages to decimals.

## 25.

The Correct Answer is (A) —
For the average of five positive numbers (their ages) to be 85,

*/*^{n}_{5}= 85, where*n*is the sum of these 5 ages. Solving for*n*in this equation gives you*n*= 425. The question tells you that the oldest person is 100. (A) is false because if one person were 20, the remaining four people's ages would need to add up to 405, but since none can be over 100, this is impossible. (B) is possible since their ages could be 100, 100, 100, 100, 25, in which case the range would be 100 – 25 = 75 years. (C) is possible using the same example as above, in which the median is 100 years. (D) is possible because their ages could be 100, 85, 85, 85, 70, meaning that 85 years would be the mode because it appears the most often.## 26.

The Correct Answer is (C) —
You can find the ratio of

*n*to*j*+*n*expressed as a percentage of*j*+*n*by setting up a system of equations with the information given in the question. Use the ratio of (*j*+*n*):*k*= 1:3 to solve for*n*: Next, you can write the ratio of*n*to*j*+*n*as a fraction, setting it equal to*x*out of 100, where*x*is the percentage*n*is of*j*+*n*:## 27.

The Correct Answer is (C) —
When Amelia hits the water, her height will be

*f*(*t*) = 0. You can set 0 = –2*t*^{2}+ 4*t*+ 30. Dividing by –2 gives you 0 =*t*^{2}– 2*t*– 15. Factoring the rest of the equation gives you 0 = (*t*+ 3)(*t*– 5), which means*t*= –3 or 5. Of these two answers, 5 seconds makes sense because time cannot be a negative number.## 28.

The Correct Answer is (B) —
Rewrite the ratio of

*s*:*b*= 3:1 as the equation , and rewrite the second sentence as*s*+*b*=*w*. To find the value of*s*in terms of*w*, you need to eliminate*b*from the equations. Multiply both sides of the first equation by*b*to get*s*= 3*b*, and then isolate*b*by dividing both sides by 3:*s*/3 =*b*. Now you can substitute this into the second equation, and isolate*s*:## 29.

The Correct Answer is (A) —
First, find the mean number of languages offered across the 20 schools, which is

^{((1 × 0) + (3 × 1) + (5 × 2) + (8 × 3) + (2 × 4) + (1 × 5))}⁄_{20}= 2.5. Therefore, any school that offers 2 languages or fewer offers fewer than the mean. You can see from the graph that 1 school offers 0 languages, 3 offer 1, and 5 offer 2. This makes a total of 1 + 3 + 5 = 9 schools that offer fewer languages than the mean.## 30.

The Correct Answer is (C) —
The number of schools that offer at least 3 foreign language courses is the sum of the number of schools that offer 3, 4, and 5 language courses: 8 + 2 + 1 = 11 schools. Since there are 20 schools in total, the probability that the superintendent randomly visits a school that offers 3 or more foreign language courses is

^{11}⁄_{20}= 55%.## 31.

The Correct Answer is (18) —
Rewrite the information given in the question algebraically: 2

*x*= 11 + 12 + 13 = 36, where*x*is the number of liters of water in one of the pools. Divide both sides by 2 to find that*x*= 18 liters.## 32.

The Correct Answer is (142) —
Distribute the coefficients on each side of the equation to get –30 – 15

*n*= –16*n*+ 112, then collect like terms: –15*n*+ 16*n*= 112 + 30. Simplify to find that*n*= 142.## 33.

The Correct Answer is (18) —
Rodrigo eats 60%, or 0.6 times, the amount of the pizza the students take. If the students take 30% of the pizza, then Rodrigo eats 60% of 30%, which is equal to 0.6 × 0.3 = 0.18 = 18 %.

## 35.

The Correct Answer is (16) —
You know that the angles at the corners of a rectangle are all right angles, so the larger triangles (with a vertical leg of 4 and a horizontal leg of 3) created by the diagonal lines are right triangles. Because of this, the right triangles are Pythagorean triples with side lengths in a 3:4:5 ratio, which means the diagonal lines both have a length of 5. Since there are two solid lines with a length of 5 and two with a length of 3, the total length of the solid lines is 2(5) + 2(3) = 16.

## 36.

The Correct Answer is (4) —
The standard equation of a circle is (

*x*–*h*)^{2}+ (*y*–*k*)^{2}=*r*^{2}, where (*h*,*k*) is the center of the circle and*r*is the radius. Simply rearrange this equation to look like the standard equation of a circle by adding 7 to both sides to get*x*^{2}+*y*^{2}= 16. Therefore,*r*^{2}= 16 and*r*= 4.## 37.

The Correct Answer is (117) —
First, plug the initial values for

*P*and*V*into the equation*PV*=*k*to find that*k*= 78 kPa × 30 cm^{3}= 2,340. Using the same equation, plug in the new volume (20 cubic centimeters) and the*k*value to solve for*P*:*P*× 20 = 2,340

*P*= 117 kPa

## 38.

The Correct Answer is (80) —
First, calculate the

*k*value for the gas in each syringe by using the formula*PV*=*k*. For syringe A,*k*= 50 × 15.6 = 780. For syringe B,*k*= 40 × 15.6 = 624.Now, plug the new volume (30 cubic centimeters) for each syringe into their respective equations to solve for pressure:

Therefore, the new pressure in B as a percent of A is^{20.8}⁄_{26}× 100% = 80%.