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## Question Explanations For

##
*PRACTICE TEST 3 (Calculator Math Test)*

## 1.

The Correct Answer is (C) —
If we plug in 1 for

*x*, we get (1 – 1)(1 + 4). This simplifies to (0)(5) = 0.## 2.

The Correct Answer is (A) —
When multiplying two exponential terms with the same base (for example, 2

^{5}and 2^{7}), the exponents are added (not multiplied). So (B) is incorrect, since 2^{3}× 2^{4}= 2^{7}. When an exponential term is itself raised to a power (for example, (2^{6})^{6}), the exponents must be multiplied. So (C) is also incorrect because (2^{6})^{6}= 2^{36}. Finally, 2^{6}+ 2^{6}is equal to 2(2^{6}) or 2^{1}× 2^{6}, which is the same as 2^{7}, so (D) is incorrect.## 3.

The Correct Answer is (C) —
If a wren has 5 chicks, she must catch 5 × 8 = 40 insects for them. With the 20 she must catch for herself, she must catch a total of 60 insects. If it takes 1 hour to catch 6 insects, it will take 10 to catch 60.

## 4.

The Correct Answer is (D) —
We can rewrite

*a*# 8 = 0 as*a*^{2}– 2(8) = 0, or*a*^{2}– 16 = 0 (which is the same as expression II). So*a*^{2}= 16 and*a*= 4 (expression III). Since*a*is squared,*a*can also equal -4 (expression IV). Expression I would only be true if*a*= 0.## 5.

The Correct Answer is (C) —

*B*is the set {2, 7, -2, 0, 8} with each number halved, giving us {1, 3.5, -1, 0, 4}. The range is the difference between the smallest and largest elements in the list, in this case -1 and 4. The difference between -1 and 4 is 5.## 6.

The Correct Answer is (B) —
For two hours, the lawyer charges $600. For the next hour, he charges $400, for a total of $1000 for three hours. Since the next hour is free, the lawyer charges $1000 for four hours. $1000/4 = $250.

## 7.

The Correct Answer is (A) —
We can rewrite this relationship as 0.5

*m*+ 0.5*n*= 0.75*m*. Subtracting 0.5*m*from both sides gives 0.5*n*= 0.25*m*. Dividing both sides by 0.25 gives 2*n*=*m*, so*n*is 50% of*m*.## 8.

The Correct Answer is (D) —
If $140,000 is 70% of the budget, we can represent the budget as

*b*and write 140,000 = 0.7*b*. Dividing both sides by 0.7 gives*b*= 200,000.## 9.

The Correct Answer is (D) —
Judging by the slope of the line, the width appears to increase by 4 inches every two months. At 5 months, the graph is at 18 inches, so two months later, it will be at 22 inches.

## 10.

The Correct Answer is (C) —
With the single songs package, Samantha will pay $1.50 × 12 = $18 for 12 songs. With the 10 songs package, she will pay $26 ($13 for the first ten songs and $13 for the next two). With the unlimited package, she will pay $15. If she buys the 10 songs package for $13 and two single songs for $1.50 each, her total will be $16. The cheapest option, therefore, is the unlimited package.

## 11.

The Correct Answer is (A) —
There were 35 sedimentary rocks found below 1000 feet, out of 60 sedimentary rocks total. 35/60 ≈ 58%.

## 12.

The Correct Answer is (D) —
Suppose there are only 3 gorillas of each species. If the ages of the 3 lowland gorillas are, for example, 1, 12, and 14, then the median age is 12 (as stated in the question) but the average age is 9, which means (A) and (C) may not be true – and since the question asks which statements

*must*be true, we can eliminate them. Similarly, if the ages of the Cross River gorillas are 1, 2, and 33, the average age is 12 (as stated in the question), but the median age is 2, so we can eliminate (B).## 13.

The Correct Answer is (D) —
Set the equations equal to each other: 2

^{a}= 2*a*+ 8. Plugging in 1 for a gives 2 = 10; plugging in 2 gives 4 = 12; plugging in 4 gives 16 = 16.## 14.

The Correct Answer is (A) —
The range of a group is the difference between its largest and smallest numbers – in this case, 13 and 60. To increase the range by 5, we must either add 5 to 60 or subtract 5 from 13. Thus, our options are 65 and 8; only 65 is given as an option.

## 15.

The Correct Answer is (A) —
Each point on the graph represents one student, and there are 8 points at and above 3 hours.

## 16.

The Correct Answer is (B) —
There are 13 students, so the median is the 7th highest heart rate – in this case, 80.

## 17.

The Correct Answer is (B) —
The average heart rate is about 77. The median is 80 (as we found in the previous question). The mode is the value that appears most often, in this case, 80, which appears four times. Given this information, (B) is true: the average heart rate (77) is less than the mode (80). (The range is only 100 – 60 = 40, so (D) is not true).

## 18.

The Correct Answer is (A) —
If an aquarium has 12 dolphins, and each tank can hold three dolphins, the aquarium has 4 tanks. If each tank costs $250 per day, the cost of all the tanks is $250 × 4 = $1000. If each of the 12 dolphins eats $20 worth of food, the cost to feed the dolphins is $20 × 12 = $240. The sum of these numbers is $1240.

## 19.

The Correct Answer is (D) —
If 60% of cones can detect red, as shown in the graph, and we have 200 cones in our sample, we should expect to find 200 × 0.6 = 120 cones that detect the color red.

## 20.

The Correct Answer is (C) —
We can rewrite this equation as

*x*^{2}+*x*– 20 = 0. Factoring gives us (*x*+ 5)(*x*– 4). This means that*x*may equal -5 or 4. Plugging these into the second equation gives us (-5^{2}) – (-5) = 30 and (4^{2}) – (4) = 12. Of these two, only 30 is given as an option.## 21.

The Correct Answer is (B) —
If the company made no profit,

*r*–*c*= 0. This means that (-*x*^{2}+ 17*x*) – (7*x*+ 25) = 0. We can simplify this equation by combining terms to get –*x*^{2}+ 10*x*– 25 = 0. Factoring gives us (-*x*+ 5)(*x*– 5) = 0, so*x*= 5.## 22.

The Correct Answer is (C) —
Suppose the group consists of two numbers,

*x*and*y*. If*a*=*r*, then (*x*+*y*)/2 =*y*–*x*, since that is how average and range are defined. If a new number,*z*, is added to the group, where*z*is between*x*and*y*, the relationship*a*=*r*only holds if (*x*+*y*+*z*)/3 = (*x*+*y*)/2. So we can see that for any group of unique integers,*a*=*r*is only true if (*x*+*y*)/2 = (*x*+*y*+*z*)/3, where*x*and*y*are the largest and smallest numbers and*z*is the average of all the other numbers. This equation is the same as the written-out statement in answer choice C). We can eliminate choice A) by creating a group that contradicts it: for example, {1, 2, 9, 10, 11} has a median of 9 and an average of 6.6, so*m*>*a*, but there are as many numbers greater than the median as there are numbers that are smaller, so A) is not*always*true and cannot be the correct answer. Similarly, for B), we can find a group such as {1, 2, 3, 10, 11}, in which*m*<*a*(in this case, 3 < 5.4) but there are again as many numbers greater than the median as there are numbers that are smaller, meaning that B) is not always true and therefore cannot be the correct answer. Finally, answer D) is not always true because we can find a set such as {5, 7, 9, 10, 14, 15} in which*a*=*r*(10 = 10 in this case), but half of the integers are*not*larger than*a*(we could also find another set in which half the integers are not smaller than*a*).## 23.

The Correct Answer is (A) —
Since the function is linear, we know it can be written in the form

*y*=*mx*+*b*. We can find the slope,*m*, by calculating rise (the difference in*y*values: 5 – 1 = 4) over run (the difference in*x*values: 3 – 1 = 2), or 4/2 = 2. Since we have sample*x*and*y*values, we can find*b*by plugging them into the standard form: 1 = 2(1) +*b*or 5 = 2(3) +*b*, so*b*= -1. So*y*= 2*x*- 1, and*f*(-4) = 2(-4) – 1 = -9.## 24.

The Correct Answer is (A) —
We can rewrite the first equation as

*y*= 2*x*+ 13 and the second as 3*y*= -2*x*– 1 or . Setting them equal to each other gives , and multiplying both sides by 3 gives 6*x*+ 39 = -2*x*– 1. Adding 2*x*to both sides gives 8*x*+ 39 = -1, and subtracting 39 gives 8*x*= -40. Dividing by 8 gives*x*= -5. Plugging this back into either equation gives*y*= 3. This point, (-5, 3), is the intersection of the two equations, and -5 × 3 = -15.## 25.

The Correct Answer is (C) —
If Computer A can do 375 calculations in 25 seconds, it can make 375/25 = 15 calculations in one second. That means that to make 100 calculations, Computer A would take 100/15 = 6.67 seconds. In this time, Computer B can make 140 calculations, meaning it can make 140/6.67 = 21 calculations in one second. To make 6,300 calculations, Computer B would need 6,300/21 = 300 seconds, or 5 minutes.

## 26.

The Correct Answer is (B) —
We can simplify the second equation by factoring:

*y*^{2}= (*x*+ 3)(*x*+ 3) or*y*^{2}= (*x*+ 3)^{2}. Taking the square root of both sides gives*y*=*x*+ 3. We can plug this into the first equation to get*x*+ 3 = 2*x*^{2}. Setting this equation equal to zero, we get 2*x*^{2}–*x*– 3 = 0. Factoring gives us (2*x*– 3)(*x*+ 1), so*x*can be either 3/2 or -1. Of these two, only -1 is given as an option.## 27.

The Correct Answer is (B) —
The average of the set before

*a*is added is 7 and the sum of the five numbers is 35. To get a new average of 5, the sum would have to be 30 (since there are now 6 numbers). Since negative numbers are not allowed, option I. is not possible. To get a new average of 7, the sum would have to be 42. Adding 7 to the set would bring the sum to 42 and the average to 7, so option II. is valid. To get a new average of 8, the sum would have to be 48. This would require adding 13 to the set, but*a*must be less than 10, so option III. is not possible.## 28.

The Correct Answer is (250) —
We can write this relationship as 3

*x*– 0.5*x*, which is equal to 2.5*x*, or 250% of*x*.## 29.

The Correct Answer is (138) —
If the area of the shaded region is 198 and there are 22 identical squares, each square must have area 198/22 = 9. This means that each square has side length 3. There are 46 such sides that form the perimeter (defined as the edges forming the boundary of the figure), and 46 × 3 = 138.

## 30.

The Correct Answer is (70) —
In Iowa, gasoline is $2 per gallon, meaning that $100 dollars will buy 50 gallons of gasoline. In Ohio, gasoline is $2.50 per gallon, meaning that $100 dollars will buy 40 gallons of gasoline. Since for every gallon, the trucks can travel 7 miles, and the truck in Iowa has 10 more gallons than the truck in Ohio, it can travel 10 × 7 = 70 more miles than the truck in Ohio.

## 31.

The Correct Answer is (1.22) —
20 gallons of gas in Iowa would cost 20 × $2 = $40. 50 gallons of gas in Illinois would cost 50 × $2.25 = $112.50. 10 gallons in Indiana would cost 10 × $3 = $30, so the truck driver’s total cost is $40 + $112.50 + $30 = $182.50. For 150 crates, his cost to break even is $182.50/150 = $1.22 per crate.