Answers for Practice Test 1, Mathematical Reasoning

Answers and Explanations

C. Sign D offers the worst deal. This problem tests your understanding of numbers and their equivalents (integers, fractions, decimals, and percents) in a real-world situation. Signs A, B, and C give customers 50% off. Sign D gives them . Sign D offers the least discount.
A. ab must be perpendicular to ad. This problem involves measurement and geometry and tests your understanding of perpendicular and parallel lines in a geometrical figure. Frames are rectangles. Each pair of opposite sides must be parallel and intersecting sides (ab and ad) must be perpendicular.
$1,710.00. This problem tests your knowledge and mastery of number operations and number sense. Use a calculator, because numerous conversions are involved, including the following:
- Area of the deck is square feet
- 9 square feet = 1 square yard
- 37.33 square yards
Because you have to buy carpet at a full square yard at a time, you’d have to buy 38 square yards.

One square yard of decking costs $45.00, and

or 38 square yards of decking (rounded to the nearest square yard) costs $1,710.00.
B. grommets. This problem tests your data-analysis skills. You’re asked to interpret and draw inferences from the bar graph and include additional data from the presentation. Because the profit per unit is the same for grommets and gadgets but differs from the profit on widgets, which had twice the profitability, to make a fair comparison, you’d have to double the sales of widgets. In this case, grommets seem to be less profitable than the other two lines but not by much. Because grommets sold the fewest numbers and were the least profitable product, they’re recommended as the one to drop.
average. Quan’s measurements are equal to or higher than those of his class, except for his average. His average is 94 while the class’s average is 94.27. This problem tests your skills in calculations of statistical measurement. You’re asked to use the presented data to calculate the measures of Quan’s performance and compare them to those of his classmates. This is a good question on which to use the calculator because it involves a series of calculations.

You find the average by adding the marks and dividing by the total number of marks. The median is the middle value; in Quan’s case, the middle value is the fourth value 97. Quan’s average was 94, his median was 97, and the range of his marks was 13. The mode is the most often, or common, value in the list, and the range is the difference between the largest and smallest numbers. This question is a good example of why some familiarity with mathematical vocabulary is an asset.
A. June. Alice has converted her story into a graph, and you’re being asked to interpret the line graph in conjunction with her story. Because her average daily time had been increasing until May, dropped in June, and recovered in July and August, you can assume that the twisted ankle slowed her down. It likely happened in June.
D. 19:18. A number of operations are involved in solving this problem. You’re asked to average a set of grades for each person and compare them by using a ratio. You can simplify this question, using a calculator.
- The total of Paula’s marks is .
- The total of Dominic’s marks is .
Because you divide each total by 5 to get the average marks for Paula and Dominic, you can simply use the ratio of the totals to get the answer because it will equal the ratio of the averages. (Note that if one of the students had six grades and the other had five, for example, you’d have to use the ratio of the averages, not of the totals.)

The ratio of Paula’s marks to Dominic’s marks is 380:360, which you can simplify by dividing each number by 20 to get 19:18.
66. This problem involves algebra, functions, and patterns. The numbers 4, 6, 10, and 18 form a pattern (also called a series). After looking carefully at the series, you see that the second term is formed by subtracting 1 from the first term and multiplying by 2. Try this on the third number: . You’ve found your pattern. Continuing the series: 4, 6, 10, 18, 34, 66, … , the first term you come to that is a multiple of 11 is 66.

You could also simply double the difference between the previous two numbers and add it to the second number to create the next one. For example, the difference between 4 and 6 is 2. Double that and add it to the to get the next number. The difference between 6 and 10 is 4. Double that and add it to the to get the next number. Continue with this pattern until you find the number you need.
B. $16.13. This problem involves data analysis and manipulation of numbers and is best done using a calculator. Most of the information given is irrelevant, except to decide that Simone may have bought at a high point. The important price to consider is $15.19. In addition to this price per share, Simone has to pay her broker 3% commission.

Therefore, her final price per share on September 24 is . Because you’re dealing with money, you have to round the number to two decimal places, making her final price per share $15.65. This amount of money came out of her bank account for each share she bought.

If Simone decides to sell the shares at this price, $15.65, she has to pay her broker another 3% commission, or . Rounded to two decimals, she has to pay a commission of $0.47 per share. She then receives the value of the shares, $15.65, minus the commission of $0.47, for a total of $15.18 per share — that is, for each share she sells, the broker deposits $15.18 into her account. Notice that this amount is less than the amount she paid for each share.

To break even, Simone has to receive $15.65 per share — after the commission. Set the equation up this way:
- , where x is the selling price
Now divide both sides by 0.97 to get .
D. –4. This question involves algebra. You have to solve a linear equation, as follows:

Cross-multiply and write this equation as . Then, getting rid of the parentheses, the equation looks like this: . Next, bring all the a’s to the left and the numbers to the right, so you have . Then, combine the a’s to get . Finally, divide both sides by 56 to get one a on the left: .
2/3. This question tests your skills in measurement and geometry. You’re asked to find the slope of a line drawn for you.

The x-axis runs horizontally across the grid. The y-axis runs vertically, up and down the grid. The origin is where the two axes (that’s the plural of axis) intersect. Points to the left of the y-axis have negative x-values. Points below the x-axis have negative y-values. The x-intercept of a line is the point where the line cuts the x-axis. The y-intercept of a line is the point where the line cuts the y-axis. All lines parallel to the x-axis have slopes of 0.

The slope of a line is the rise over the run. The rise is 4, and the run is 6. This means that the slope is 4/6 or 2/3 (divide by 2 to simplify).
. This question tests your skills in measurement and geometry. You’re asked to identify the x-intercept and the y-intercept and to draw a line with a slope of 2/3 on the graph.

If you draw a line through the point on the y-axis having the same slope, it crosses the x-axis at . Simply count over 3 points to the left (the run), down 2 (the rise), and you’re at . But you’re asked for the x-intercept, so repeat this process. Go over 3 more points to the left and down 2 more, and you’re at .
D. not enough information given. This question doesn’t provide enough information for you to give an accurate answer. If the fire were rectangular in shape, the answer would be different from a circular fire or an irregularly shaped fire. The question provides information only about the shape of the barbeque pit.
C. $2.19. Consider the price per square foot at each store:
- Carnie’s Carpets: $21.50 per square yard per square foot
- Flora’s Flooring: $2.45 per square foot
- Dora’s Deep Discount: The area of an -foot rug is square feet. The cost for 96 square feet is $210.24 or per square foot.
A. 3rd quarter. In this question, you’re asked to analyze graphs to identify patterns in a workplace situation.

In the 2013 graph, the third quarter of 2013 produces a little more than 30% of the output. The best answer for this question is the 3rd quarter.
1,140. This problem involves measurement, specifically, area and money. Assuming that the estimate for renovation is accurate, the number of square feet of renovation that the Ngs can afford for $18,000 is square feet square feet. Round this number to 1,140 because you usually don’t add part of a square foot.
(6, 6). This problem involves data analysis, statistics, and probability. You’re being asked to graph a point representing the medians of two sets of data. First, find the median (the middle number, when put in order) of the first set of numbers. The median is 6. Then find the median of the second set of numbers. Again, it’s 6.

B. sedan. This problem is based on measurement, using uniform rates, and it asks you to make a decision based on factual information. To figure the cost of gasoline over the five years, set up the problem this way:

To help you decide which car LeeAnne should buy, create a chart like the following:

Vehicle Type	Miles/Gallon	Total Gas Costs
SUV	12.8	$24,609.38
Sedan	19.6	$16,071.43
2-door	19.5	$16,153.83
All-wheel drive	17.2	$18,313.94
Sports car	18.6	$16,935.47

From these figures, you can see that the sedan is the best buy. It gets the most miles per gallon, so LeeAnne wouldn’t have to fill the tank as frequently.

C. divide then add. This problem involves number operations. Instead of asking you for the answer, which is pretty simple, you’re asked to provide the operations that are required to solve the problem. First, you divide (miles to site by miles per hour), and then you add (the amount of time Tom wants to arrive early). Remember to keep the units consistent.
D. 11. This question involves data analysis. You’re asked to apply measures of central tendency (the mean) and analyze the effect of changes in data on this measure. If Leonora’s present average is 77.8% and she wants to get an average of 80%, she needs enough marks to get an additional 2.2% .

Because Leonora is taking five subjects, she requires 5 extra points for each percent increase. Thus, she requires additional points. The problem says that English is her best subject, so she would need the 11 extra points in English.
A. 7 cups of soup and cup of lentils. This question tests your ability to figure out how a change in the amount of rice used results in changes to the amount of soup and lentils needed. Because each cup of rice requires 2 cups of soup, cups of rice require cups of soup. Because each cup of rice requires cup of lentils, cups of rice require cup of lentils.
C. 1:46. This question is a test in probability. You’re asked to figure out the probability of an event occurring. If you had an entire deck of 52 cards, the probability of drawing an ace of hearts would be 1:52. If you remove 6 cards and none of them is the ace of hearts, you may as well have a 46-card deck . The probability of drawing an ace of hearts from a 46-card deck is 1:46.
B. the larger rug. This problem tests your measurement skills. You’re asked to predict the impact of changes in the linear dimensions of the rug on its area and cost. Choice (C) seems logical, but the question never mentions the cost of the paneling or the skylight, so you can’t consider it as an answer.

Draw a sketch of the room with the larger rug. It will have a tiled area around it. You have to figure out how many square feet of tile and carpet you need for this floor treatment, as follows:
- The area of the room is square feet.
- The larger rug will cover square feet of the floor. This leaves 56 square feet to be covered with tile. The cost of the rug is . The cost of the tile is . The total cost is .
- The smaller rug will cover square feet of the floor. This leaves square feet to be covered with tile. The cost of the rug is . The cost of the tile is . The total cost is . The smaller rug will cost more for the entire floor treatment.
- Tile costs more per square foot than carpeting, so you know without doing any figuring that having more tile will result in higher costs.

C. loveseat. This question is an exercise in data analysis. You’re asked to compare sets of data based on the mean (average) prices of four other stores. You can summarize the average prices on a sketch table like this one:

Item	Store A	Store B	Store C	Store D	Average Price	Friendly Furniture
Couch	$1,729.00	$1,749.00	$1,729.00	$1,699.00	$1,726.50	$1,719.00
Dining room set	$4,999.00	$4,899.00	$5,019.00	$4,829.00	$4,936.50	$4,899.00
Loveseat	$1,259.00	$1,199.00	$1,279.00	$1,149.00	$1,221.50	$1,229.00
Coffee table	$459.00	$449.00	$479.00	$429.00	$454.00	$449.00
Reclining chair	$759.00	$799.00	$739.00	$699.00	$749.00	$739.00

You can see that the only item Friendly Furniture sells for over the average price is the loveseat, which is the answer to the question.

320. This question tests your knowledge of number operations by asking you to solve a problem involving calculations. Sarah ate 48/18 pistachios per minute. In 2 hours or 120 minutes, she could eat .
multiplication. This question is about number operations; it asks you to select the appropriate operation to solve a problem. Because the first operation performed is to find the volume of the room, and the formula for volume is , the first operation you use to solve the problem is multiplication.
D. not enough information given. This question tests your knowledge of measurement and geometry. You’re asked to visualize and describe geometrical figures under a 90-degree rotation. Each of the figures is changed by the rotation. Try drawing each of these shapes, picking a point on the perimeter and rotating it 90 degrees. Because this is a timed test, try drawing one or two, noticing that they change quite a bit. Use your imagination to check the rest. After discovering that none of the four shapes has the same relationship to the horizontal after a 90-degree rotation about a point on its perimeter, you have your answer — not enough information given.
7:1. This question tests your data-analysis skills by asking you to interpret a chart and answer a question involving calculation.
The largest budget is the Operations budget, while the smallest budget is Human Resources. The ratio between these two budgets is 14.7 to 2.1 or 7:1 (dividing both sides by 2.1).

If you wanted to do this in your head, notice that 14:2 (the approximate ratio between the Operations budget and the Human Resources budget) is double 7:1.
5. This question tests your knowledge of patterns by asking you to compare information from different types of graphs to extract information. Graph 5 has the first and third quarters in the required ratio and is the correct answer.
D. not enough information. This problem involves measurement and geometry, and it asks you to use the Pythagorean theorem. You can’t actually solve this problem, however. Because the rangefinder is measuring the distance from the forester’s eye and you don’t know how high his eye is above the ground, you can’t calculate the height of the tree. You can calculate the distance from the forester’s eye to the top of the tree by using the Pythagorean theorem, but the question asks for the height of the tree (which is the distance from the ground — not the forester’s eye — to the top of the tree). Thus, you don’t have enough information.
$78.50. This question tests your knowledge of number operations by asking you to perform several operations to calculate an answer. After the fourth week, Lawrie would’ve deposited . There would’ve been two withdrawals totaling . Her balance after the fourth week would be .
$3.20. This question tests your skills in using percentages and discounts. Store A offers Sarah 1/3 off or off the original price. Store B offers her 30% off; 30% is 0.30, so she’ll get off the original price. By buying at Store A, she’d get the chair for less. Thus, she’d save $3.20.
D. 100. This question tests your skills by asking you to use information from a graph to solve a problem. From the graph, you can figure out that the volume in decibels is the square of the volume setting. For a volume setting of 4, the volume is 16 decibels. Therefore, for a setting of 10, the volume is 100 decibels .
D. 12. This question tests your skills in algebra by asking you to solve equations. The equation given is . If , the square root of 144 is 12. Thus, the answer is 12.
A. 36. If the volume decreases by half for every 10 feet away from the stage you get and the volume at the stage is 144 decibels, a person sitting 10 feet from the stage would hear at a volume of 72 decibels (144/2), and a person sitting 20 feet from the stage would hear at a volume of 36 decibels (72/2).
12. This question involves number operations. You’re asked to calculate the average miles per gallon for a vehicle. Rather than provide you with the number of gallons used, you’re given the cost of gasoline and the cost of the 240-mile trip. To calculate the amount of fuel used, you divide $54.00 by $2.70 to get 20 gallons. You can do this operation mentally to speed things up. Next, you divide the miles, 240, by the fuel used, 20 gallons, to get the mileage, 12 miles per gallon .
D. circle. This question tests your skills in measurement and geometry. To remain at a constant temperature, you have to remain at a constant distance from the fire.

The path of a point that travels a constant distance from a point is a circle.
4.48. This problem tests your ability to do calculations and use a formula: Volume = length × width × depth. Thus, . The average depth = (the answer is rounded).
D. 2,949. This question tests your ability to make a decision based on data presented in a table and then to use that information to answer a question. The least economical car costs $1,823 to drive for a year, while the most economical car costs $840 for the same time under the same conditions. The difference in cost for one year is . The cost for three years is .

You could also estimate an answer. The difference annually is just under $1,000. For three years, that’s just under $3,000. So Choice (D) is the best answer.
B. . This question tests your ability to analyze data, using the mean and median to answer a question about the data given. The mean of the city mileages is the sum of the mileages divided by 10 (the number of entries), which equals 16.8. The median of the mileages is the one midway between the two in the middle, or 16.5. The difference between the two numbers is 0.3 or .
(0, 5). This question tests your ability to analyze data by representing data graphically.

For Vehicle A, the difference between the city and highway mileage is 5 miles per gallon . The point you want on the y-axis is (0, 5), which you need to mark on the graph.
D. . This question tests your skill in algebra by asking you to solve a system of linear equations:

A linear equation is one in which the powers of the variables are all equal to 1. To solve this system, you have to eliminate x by multiplying each equation by a number that allows you to subtract one from the other and end up with just y’s. Multiply the first equation by 5 and the second equation by 2:

Subtract the second equation from the first, and you get . (Note that you can also multiply the first equation by and add the two equations together. Either way gets you the same answer.)
B. 4 times as many. This question asks you to analyze a situation presented in a table. The table tells you that the country with the highest participation rate is the United States, with a participation rate of 66.4. The country with the lowest participation rate is Portugal, with a participation rate of 15.5. Because you’re asked for an approximation, you can say that the participation rate in the United States is 60 and in Portugal, it’s 15, which means that 4 times as many adults participate in adult education in the United States than in Portugal.
D. $18.00. This problem involves number operations. The total amount of Gordon’s bills is . If Gordon allocates only $250.00 to pay these bills, he ends up short. Be wary of Choice (B), which is a special trap for people who don’t read the question carefully.
3 hours. At the end of the first month, Georgette will owe . The second month’s interest will be . At $11 an hour, Georgette would have to work an additional 2.76 hours, or practically 3 hours because no one would hire someone to work 2 hours, 45 minutes, and 36 seconds.
D. approximately 5,300. This problem tests your knowledge of measurement and geometry by asking you to solve a problem involving volume and weight. You can do this problem in your head, but we take you through the steps using calculations first.

The formula for volume of a cylinder (the cylinder is the circular inside of the pool to a height of 9 inches) is , where approximately 3.14, radius, and height. If the diameter is 12 feet, the radius is 6 feet. If the height is 9 inches, it’s 9/12 feet, which can be simplified to 3/4 feet.

In a formula, don’t forget that all units must be the same — that is, feet and feet or inches and inches.
- The volume is cubic feet.
- Because 1 cubic foot weighs 62.42 pounds, the weight of 84.78 cubic feet is or 5,300 rounded to the nearest hundred.
To do this problem in your head, multiply 6 by 6 to get 36. Multiply 36 by 3/4 to get 27, and multiply 27 by 3 to get 81. The approximate volume of the pool is 81 cubic feet, which isn’t bad for an approximation. For your purposes, say the volume is 80 cubic feet, which is still close. The weight of a cubic foot of water is 62.42 pounds, so round it to 60 pounds. Now, multiply 80 by 60 to get 4,800, which is closest to Choice (D). You can go with that approximation because it’s very close to one of the answers.
A. 6.4%. This question tests your ability to evaluate an answer by using a formula. This formula, , isn’t in the format you want because you want to calculate the rate, which means solving for r. You can change the equation to , which allows you to calculate the rate from the information given. Substituting into this equation, you get .

(Remember that 1 year and 3 months is , or 1.25 of a year.)

Then .
C. $4.90. This question involves number operations. You’re asked to calculate — in your head — the answer to a problem.

To use mental math to solve this problem, round everything. Consider the apples at $0.80 a pound, bananas at $0.20 each, milk at $1.30, and a loaf of bread at $1.00. The total for this approximation is . Looking at the answer choices, Choice (C) is the only one close to this approximation.
A. 28. This question tests your knowledge of patterns by asking you to figure out the next number in a series. By looking at the series, it looks like each number is the square of the placement of the number in the list, plus 3. That is, the first number is plus 3, or 4. The second number is plus 3, or 7. The third term is plus 3, or 12. The fifth term would be plus 3, which is 28.
C. . This question tests your skills in geometry by asking you to visualize a graph of an object. Because the object is a rectangle, the opposite sides are equal in length and are parallel, the fourth corner will be 2 units to the left of the y-axis, giving it an x-coordinate of , and 2 units below the x-axis, giving it a y-coordinate of . Therefore, the point would be .

The x-coordinate is the distance from the y-axis, and the y-coordinate is the distance from the x-axis.

Answers for Practice Test 1, Mathematical Reasoning

Answers and Explanations

Answer Key