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## Question Explanations For

##
*PRACTICE TEST 2 (Calculator Math Test)*

## 1.

The Correct Answer is (C) —
If the length of a list of numbers is even, the median is the mean of the middle two numbers – in this case, 3 and 6.

## 2.

The Correct Answer is (C) —
There are 12 boys in a group of 12 + 18 = 30. 12/30 reduces to 2/5.

## 3.

The Correct Answer is (B) —
To drive 210 miles, Ralph would need to buy 6 gallons of gas (210 miles/35 miles/gallon = 6 gallons). The price of gas in 2008 was $2.50, according to the chart. 6 gallons × $2.50/gallon = $15.

## 4.

The Correct Answer is (C) —
We can write this as 24 = 2(

*a*+*b*) and*a*= 2*b*. Substituting gives 24 = 2(2*b*+*b*) = 6*b*, so*b*= 4.## 5.

The Correct Answer is (D) —
If the trend continues, the number of faculty will increase again in 2005; 26 is the only number that fits this trend.

## 6.

The Correct Answer is (B) —
Plugging in for

*x*and*y*gives 1 + 2(2) + 5(-1)2 and 4(2) + 3(-1), or 10 and 5, respectively. The difference between 10 and 5 is 5.## 7.

The Correct Answer is (B) —
If the average of two numbers is 40, their sum is 80. So the sum of the other 10 numbers is 300 – 80 = 220, and their average is 22.

## 8.

The Correct Answer is (B) —
Setting the equations equal to each other gives 3

*x*+ 1 =*x*– 3; rearranging gives 2*x*= -4, so*x*= -2.## 9.

The Correct Answer is (C) —
The company spent $28,000/700 = $40 per employee, so with 850 employees, they will spend $40 × 850 = $34,000.

## 10.

The Correct Answer is (A) —
The total of all numbers in all sets must add up to 90. Numbers in the spots where the circles overlap are not counted in the spots where the circles do not overlap, so we must subtract the sum of all the given numbers to find

*x*.## 11.

The Correct Answer is (D) —
Set A contains all the numbers that are only in set A (18), the numbers that are in sets A and B (6), the numbers that are in sets A and C (9), and the numbers that are in sets A, B, and C (12). The sum of these numbers is 45, which is 50% of 90.

## 12.

The Correct Answer is (B) —
If set D has 50 integers and set A has 45, their ratio is 50:45, reduced to 10:9.

## 13.

The Correct Answer is (D) —
60% of ¼ of 320 is the same as 0.6 × 0.25 × 320 = 48. 30% of 300 is 90.

## 14.

The Correct Answer is (C) —
The part of the graph between 1950 and 1960 fits a pattern of exponential growth. The population did not grow most quickly between 1980 and 1990. Based on the graph, there were approximately 8 times more people in 1960 than in 1950.

## 15.

The Correct Answer is (C) —
The graph is linear, so the slope is constant. If

*V*=*IR*, and*V*and*I*are both increasing linearly,*R*must be constant, so the slope represents resistance.## 16.

The Correct Answer is (A) —
If the mass decreases by half every 20 years, the function should describe exponential decay. Only (A) and (B) describe exponential decay, but (B) describes the mass being reduced by half 20 times every year instead of once every 20 years.

## 17.

The Correct Answer is (C) —
The greatest possible value for

*a*is 6, since if it were 7 we would get -16 < 5 – 3(7), or -16 < -16, which is false. The least possible value for*a*is 1.## 18.

The Correct Answer is (C) —
Setting the equation equal to 0 and factoring gives us 40

*t*(*t*– 5) = 0, so*t*= 5.## 19.

The Correct Answer is (C) —
The slope of the line between the two points is -4 (rise/run = -24/6 = -4). To find the

*y*-intercept, plug in either ordered pair for*x*and*y*in the equation*y*= (-4)*x*+*b*.## 20.

The Correct Answer is (C) —
To transform

*f*(*x*) into 5*f*(*x*^{2}), all we have to do is square the*x*term in*f*(*x*) and multiply the equation by 5, so we can plug the first equation into the second as follows: 5(*x*^{2}– 3) = 2*x*– 8. Setting this equation equal to zero gives 5*x*^{2}– 2*x*– 7 = 0, and factoring gives (*x*+ 1)(5*x*– 7). This means that*x*can be either -1 or 7/5, but we know that*x*is a positive value. 7/5 is the same as .## 21.

The Correct Answer is (D) —
This describes exponential decay; graph (D) is the only graph that shows this.

## 22.

The Correct Answer is (C) —
The numbers cannot have a range any smaller than 4 (for example, five numbers in a sequence: 1, 2, 3, 4, 5); the smallest median would be for the set (1, 2, 3, 4, 5), which has a median of 3; the highest mean is 17, for the set (15, 16, 17, 18, 19).

## 23.

The Correct Answer is (A) —
Half the weight of the sculpture (

*w*/2) will be zinc ($3), and the other half (*w*/2) will be copper (6*w*). The cost for the zinc half is $3*w*/2, and the copper half is $6*w*/2, for a total of $9*w*/2.## 24.

The Correct Answer is (C) —
If we take

*y*– 4 = 4*x*+ 3*y*and solve for*y*, we find that*y*= -2 – 2*x*. Taking 4*x*+ 3*y*= -(9 –*x*), rearranging and simplifying gives us*x*+*y*= -3. Plugging in for*y*gives*x*– 2 – 2*x*= -3, and simplifying gives*x*= 1. Plugging this in to*y*= -2 – 2*x*, we find that*y*= -4.## 25.

The Correct Answer is (C) —
Bees are in the larval stage for 1 out of the 2 weeks of their development (we don’t count the two weeks of adulthood as “developing”), so selecting 200 at random should give ½ × 200 = 100 bees in the larval stage.

## 26.

The Correct Answer is (C) —
We can write , or

*c*+*d*+*e*= 3*a*. The average of*b*,*c*,*d*, and*e*is , which we can rewrite as .## 27.

The Correct Answer is (D) —
If we draw a line perpendicular to the side of either of the pieces of wood, we can form a triangle (where the hypotenuse is one side of the parallelogram and the short side is 4 cm, the width of the piece of wood). By using opposite angles, we can deduce that this is a 30-60-90 triangle, meaning that the hypotenuse is twice the length of the short side. We can repeat this with the other piece of wood to confirm that all four sides of the parallelogram are 8 cm long. We now have two sides (

*a*and*b*) and an internal angle (*θ*) of the parallelogram. Given this information, the formula for the area of a parallelogram is*A*=*ab*sin*θ*. Plugging in, we get*A*= 8 × 8 × 0.5, or 32.## 28.

The Correct Answer is (65) —
To simplify this expression, we can take the natural logarithm of both sides so that the exponents become coefficients. Using the identity log(

*a*) =^{b}*b*log(*a*), we can write 3^{2x + 2}as log(3)(2*x*+ 2) and 27^{x – 2}as 3log(3)(*x*– 2), since 27 = 3^{3}. Plugging these values back in, we have log(3)(2*x*+ 2) = 3log(3)(*x*– 2). Dividing out log(3) leaves us with 2*x*+ 2 = 3(*x*– 2); solving for*x*gives us*x*= 8. 8^{2}+ 1 = 65.## 29.

The Correct Answer is (5) —
The price of the tickets can be written as 5

*x*+ 30*y*= 185, where*x*is the number of children and*y*is the number of adults. The number of family members can then be written as*x*+*y*= 12. So*x*= 12 –*y*, and plugging this in to the first equation gives 5(12 –*y*) + 30*y*= 185. Solving for*y*gives*y*= 5. (To check our answer, we can write 5(7) + 30(5) = 185, which is correct).## 30.

The Correct Answer is (3.4) —
If the sonar wave moves at 340 meters per second, and is detected by the submarine 20 seconds after it was emitted, the wave traveled 340 m/s × 20 s = 6800 m round-trip. However, this includes both the distance to and the distance from the lighthouse, so we divide by 2 to get 3400 m, or 3.4 km.

## 31.

The Correct Answer is (11.8) —
The sonar signal takes to reach the island. In 6 s, the Wayfarer has traveled 6 m/s × 6 s = 36 m, putting it 2004 m away from the island. If we take the island as our origin, we can represent the position of the Wayfarer as

*y*= -6*x*+ 2004, where*x*is the time in seconds since the signal reflected off the island (-6 is the speed of the submarine with respect to the island and 2004 is its distance from the island). We can similarly represent the position of the returning sonar wave as*y*= 340*x*. Setting these equations equal to each other will give us the time at which the signal reaches the submarine: -6*x*+ 2004 = 340*x*. Solving for*x*and rounding to the nearest tenth gives us 5.8. We then add this to the 6 seconds elapsed from the time the signal was emitted to the time it reflected off the island.