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Middle School Mathematics (0069)

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  1. As x moves from –4 to 0, that is, from left to right on the number line, its value increases.

    The number line shows numbers to the right are increasing up to one-half and those to the left are decreasing to minus 4.

    Similarly, the value of y increases from –2 to 0. Thus, it can be seen that as x increases, y increases. The correct answer is D.

  2. The number of passengers who use the airport each year, 350 thousand, can be written as 350,000; 350 million can be written as 350,000,000. 350,000,000 ÷ 350,000 = 1,000, so the correct answer is C.

  3. If a represents the number of adults, then 5a represents the number of children and 6a represents the total number of people at the show. Since 6a represents a whole number that is a multiple of 6, there cannot be 80 people at the show, for 80 is not a multiple of 6. The correct answer is B.

  4. This question asks you to apply your knowledge of percent increase or decrease to determine a selling price based on cost of a car to the dealer, c.  Since the original price of the car was 25 percent greater than the cost to the dealer, the original price was c + 0.25c = 1.25c.  Since the selling price was 25 percent less than this amount, only 75 percent of this amount will be paid, so the selling price of the car was 0.75(1.25c).  Thus, the correct answer choice is (D).

  5. When triangle ABC is reflected across the y-axis, the figure formed is located in quadrant I and is the mirror image of the given figure. Rotating the triangle about vertex C by 90 degrees clockwise yields choice A.

  6. The length of the large block, 12 centimeters, is 3 times the length of a small block, so each small block is 12 ÷ 3 = 4 centimeters long. Similarly, the width of a small block is 8 ÷ 2 = 4 centimeters, and the height of a small block is 9 ÷ 3 = 3 centimeters. Thus, the correct answer is D.

  7. This question asks you to apply your understanding of angles in a plane and, in particular, properties of angles associated with parallel and transversal lines. You should be able to show, using pairs of alternate interior angles and corresponding angles, that angle measured x degrees and angle measured y degrees are supplementary angles. Recall that the sum of the measures of supplementary angles is 180°. That is, x + y = 180. It is given that y = 3x. Substituting for y, you get 4x = 180. Hence, x = 45. Therefore, the correct answer choice is B.

  8. This question asks you to apply your knowledge of circles, squares, and proportional reasoning to find the ratio of the areas of two squares. There are many ways to approach this problem. One approach is to use the information given and many things that you know about circles, squares, and triangles and do lots of computation. Another is to use your knowledge of what happens to area when you scale up corresponding linear dimensions in a figure. If you like to compute, here is what you might do. First consider circle A. The radius of circle A is 1, and the diameter is 2. This diameter is also the diagonal of the inscribed square and the hypotenuse of a right triangle with side a. By the Pythagorean theorem, a2 + a2 = 22; 2a2 = 2 × 2, a2 = 2, and a, thus the length of a side of square A is 2. So the area of square A is (2)2 = 2. Likewise, the area of square B is (22)2 = 8. Thus the ratio of the area of square A to the area of square B is 2 : 8, which is 1 : 4. The correct answer is D.

    Alternatively, you may recall that when you are comparing two similar figures whose corresponding linear dimensions have a ratio of 1 to 2, as in this problem, the ratio of the areas of the figures is the ratio of the square of the linear dimensions; that is, 12 to 22, which is 1 to 4. Hence, the correct choice is D.

  9. This question asks you to identify a function by applying your understanding of functions to different mathematical statements. To answer questions, such as this, that ask "which of the following," you need to consider only the choices given. There are usually other correct answers to the question, as in this case, that you are not asked to consider. To answer this question, you should recall that if y is a function of x, then each value of x (in the domain of the function) results in only one value of y. In choices A and B, most values of x have two different corresponding values of y. You can see this by solving the equations in A and B for y. In A, y = +4-x or y = -4-x. Similarly, in B, y = ±4-x2. So neither A nor B defines y as a function of x. In choice D, for each value of x, there is more than one value of y that satisfies the inequality. So D does not define y as a function of x. However, in C, for each value of x, there is only one value of y that corresponds to that value of x. Thus, the correct answer choice is C.

  10. This question asks you to apply your knowledge of graphing data in a coordinate plane to a situation involving graduate rate. You should notice that each of the choices given is the graph of a step function. You will need to identify the graph that includes the correct cost for the first step and the correct interval between steps. Since the cost for the first 14 mile or less is $2.50, the cost for the first step (the value on the vertical axis) should be 2.5 over the horizontal interval from 0 to 14 mile, with a solid dot at 14 mile. (There should be no cost at a distance of 0 miles, since there is no charge if there is no ride.) In each of the subsequent horizontal intervals of 14 mile, the cost value on the vertical axis should show an increment of $0.50, with a solid dot at the right endpoint of each interval. Only choice A illustrates this correctly. Choice C has the correct cost values for each step but does not represent the endpoints of each interval correctly. The correct answer choice, therefore, is A.

  11. Since the 29 children have a total of 35 dogs and cats, at least 6 children must have both a dog and a cat. If there are exactly 6 children with both a cat and a dog, then 14 children have only a dog and 9 children have only a cat. On the other hand, all 15 cat owners could also own a dog; then 5 children have only a dog and 9 children have neither a dog nor a cat. Thus, the correct answer is D.

  12. The circle graph shows the distribution of the trash content in percent; the question asks for the weight of the plastics content in tons. From the graph we see that plastics account for 8% of the total weight of the trash. The problem states that 60 tons of the trash consist of paper; the graph shows that this amount equals 40% of the total, so

    60 = 0.4 × (total weight)

    and the total weight is 600.4 = 150 tons.

    The weight of plastics equals 8% of 150 tons, or (0.08)(150) = 12 tons.

    There is another, slightly faster, way to solve this problem. We use the fact that the ratio of plastics to paper in the trash is the same, whether the two amounts are given as percents or in tons. This gives us the proportion

    tons of plasticstons of paper=8%40%=15

    or

    tons of plastics60=15

    tons of plastics=605=12

    The correct answer is D.

  13. The bar graph presents information for eight different years. The vertical scale goes from 0 to 80,000. The zeros are left off the scale because the title tells you to read the numbers in thousands. To find the number of students in any one year, read the height of the corresponding bar from the left-hand scale and multiply that height by 1,000.

    The bar for 1950 has a height of about 27, so the number of students in 1950 was about 27,000. You have to find the number of years in which there were more than twice as many, that is, more than 54,000 students. To do this, count the number of bars that are higher than 54. These are the bars for 1975, 1980, and 1985. Thus, there were three years in which there were more than twice as many students as in 1950. The correct answer is D.

  14. To compute a percent increase, you need the increase in the number of students and the number of students before the increase.

    The graph shows that the number of students in 1970 was 40,000 and the number of students in 1980 was 70,000, an increase of 30,000 students. To find the percent increase, divide this number by the base number, that is, the number of students before the increase, or 40,000.

    30,000 40,000 = 3 4 =0.75=75%

    The correct answer is A.

  15. Given the conditions of the experiment, it is reasonable to assume that the 90 snails captured by the biologist 15 days after the markings were made represent a random sample of all the snails.

    Thus, about 12 90 , or 215, of the population had been marked. Thus, the original 84 snails marked represented approximately 215 of the entire population and the biologist should estimate the snail population to be 84× 15 2 , or 630.

    The correct answer is A.

  16. The probability that the student guesses any one answer correctly is 1/2, and, since the student is randomly guessing, the guesses are independent events. Thus, the probability of guessing all 20 answers correctly is (12)20, and the correct answer is B.

  17. The problem gives a set of test scores and the definition of median. The first part of the definition tells you to order the scores, that is, to arrange them in order from smallest to largest. Here are the numbers ordered from smallest to largest:

    86, 86, 88, 90, 92, 98

    Because there is an even number of scores (6), there are two middle numbers in the set, 88 and 90, and the average of the two middle numbers is

    88+902=1782=89

    Thus, the median of Robin's scores is 89 and the correct answer is A. (Notice that the median of a set of numbers need not be one of the numbers in the set.)