ACT Applied Mathematics

From: http://www.act.org/workkeys/assess/math/index.html

Applied Mathematics

According to the Bureau of Labor Statistics, 80% of the fastest-growing jobs in the United States require some training after high school. Many of these jobs require a strong basis in math and science. Mathematics engages critical thinking, abstract reasoning, and problem solving. Individuals can use these skills throughout their lifetime for higher learning and many different careers. The WorkKeys Applied Mathematics assessment helps current and potential employees measure the math skills they have against those the workplace requires. This assessment measures the skill people use when they apply mathematical reasoning, critical thinking, and problem-solving techniques to work-related problems. The test questions require the examinee to set up and solve the types of problems and do the types of calculations that actually occur in the workplace. This test is designed to be taken with a calculator. A formula sheet that includes all formulas required for the assessment is provided. While individuals may use calculators and conversion tables to help with the problems, they still need to use math skills to think them through.

Characteristics/Skills There are five levels of difficulty. Level 3 is the least complex and Level 7 is the most complex. The levels build on each other, each incorporating the skills assessed at the previous levels. For example, at Level 5, individuals need the skills from Levels 3, 4, and 5. Examples are included with each level description.

Level Characteristics of Items Skills  Translate easily from a word  Solve problems that require a problem to a math equation single type of mathematics operation 3  All needed information is (addition, subtraction, multiplication, View presented in logical order and division) using whole numbers sample  Add or subtract negative numbers item  No extra information  Change numbers from one form to another using whole numbers, fractions, decimals, or percentages

 Convert simple money and time units (e.g., hours to minutes) Level Characteristics of Items Skills  Information may be presented  Solve problems that require one out of order or two operations 4  May include extra, unnecessary  Multiply negative numbers View information  Calculate averages, simple ratios, sample simple proportions, or rates using whole item  May include a simple chart, numbers and decimals diagram, or graph  Add commonly known fractions, decimals, or percentages (e.g., 1/2, . 75, 25%)  Add up to three fractions that share a common denominator  Multiply a mixed number by a whole number or decimal

 Put the information in the right order before performing calculations Level Characteristics of Items Skills  Problems require several steps of Decide what information, logic and calculation (e.g., problem calculations, or unit conversions to use 5 may involve completing an order form to solve the problem View by totaling the order and then  Look up a formula and perform sample computing tax) single-step conversions within or item between systems of measurement  Calculate using mixed units (e.g., 3.5 hours and 4 hours 30 minutes)  Divide negative numbers  Find the best deal using one- and two-step calculations and then comparing results  Calculate perimeters and areas of basic shapes (rectangles and circles)

 Calculate percent discounts or markups Level Characteristics of Items Skills  May require considerable  Use fractions, negative numbers, translation from verbal form to ratios, percentages, or mixed numbers 6 mathematical expression  Rearrange a formula before View solving a problem sample  Generally require considerable  Use two formulas to change from item setup and involve multiple-step one unit to another within the same calculations system of measurement  Use two formulas to change from one unit in one system of measurement to a unit in another system of measurement  Find mistakes in questions that belong at Levels 3, 4, and 5  Find the best deal and use the result for another calculation  Find areas of basic shapes when it may be necessary to rearrange the formula, convert units of measurement in the calculations, or use the result in further calculations  Find the volume of rectangular solids

 Calculate multiple rates Level Characteristics of Items Skills  Content or format may be  Solve problems that include unusual nonlinear functions and/or that involve 7  Information may be incomplete more than one unknown View or implicit  Find mistakes in Level 6 questions sample  Convert between systems of item  Problems often involve multiple measurement that involve fractions, steps of logic and calculation mixed numbers, decimals, and/or percentages  Calculate multiple areas and volumes of spheres, cylinders, or cones  Set up and manipulate complex ratios or proportions  Find the best deal when there are several choices

 Apply basic statistical concepts Level 3 Applied Mathematics Sample Item

In your job as a cashier, a customer gives you a $20 bill to pay for a can of coffee that costs $3.84. How much change should you give back?

A. $15.26 B. $16.16 C. $16.26 D. $16.84 E. $17.16

Why this is a Level 3 item:

 Examinees must perform a single subtraction operation.  Numbers are presented in the logical order ($20 – $3.84).  Number of dollars must be converted to a decimal (dollars and cents: $20.00). Level 4 Applied Mathematics

Sample Item Over the last 5 days, you made the following number of sales calls: 8, 7, 9, 5, and 7. On the average, how many calls did you make each day?

A. 5.8 B. 7.0 C. 7.2 D. 9.0 E. 36.0

 There is more than one step of logic and calculation.  Examinees must divide using positive numbers.  Examinees must figure out averages. Level 5 Applied Mathematics

Sample Item

Quik Call charges 18¢ per minute for long-distance calls. Econo Phone totals your phone usage each month and rounds the number of minutes up to the nearest 15 minutes. It then charges $7.90 per hour of phone usage, dividing this charge into 15-minute segments if you used less than a full hour. If your office makes 5 hours 3 minutes worth of calls this month using the company with the lower price, how much will these calls cost?

A. $39.50 B. $41.48 C. $41.87 D. $54.00 E. $54.54

 There are several steps of logic and calculation.  Examinees must perform calculations using mixed numbers.  Examinees must compare their answers with two sets of calculations and choose the "best deal." Level 6 Applied Mathematics

Sample Item

You are preparing to tile the floor of a rectangular room that is 15½ feet by 18½ feet in size. The tiles you plan to use are square, measuring 12 inches on each side, and are sold in boxes that contain enough tile to cover 25 square feet. How many boxes of tiles must you order to complete the job?

A. 11 B. 12 C. 34 D. 59 E. 287

 Examinees must do multiple steps of logic, calculations or conversion.  Examinees must use mixed numbers.  Examinees must eliminate unnecessary information.  Examinees must find the area of a basic shape and use the result in further calculations. Level 7 Applied Mathematics Test

Sample Item

The farm where you just started working has a vertical cylindrical oil tank that is 2.5 feet across on the inside. The depth of the oil in the tank is 2 feet. If 1 cubic foot of space holds 7.48 gallons, about how many gallons of oil are left in the tank?

A. 37 B. 59 C. 73 D. 230 E. 294

 There are multiple steps of calculation.  Examinees must look up and use the formula for the volume of a cylinder.  Examinees must convert from cubic feet to gallons. Answer Key: ACT Applied Mathematics

Answer to Level 3 Sample Item

A. $15.26 = $20.00 – $4.74

B. $16.16 = $20.00 – $3.84 Correct C. $16.26 = $20.00 – $3.74 D. $16.84 = $20.00 – $3.16 E. $17.16 = $20.00 – $2.84

To get the average, first add up the numbers in the series, then divide the sum by the quantity of numbers in the series.

A. (8 + 7 + 9 + 5) ÷ 5 = 5.8: Forgot to include the last 7 B. 7.0 is the mode value of the numbers in the series

C. (8 + 7 + 9 + 5 + 7) ÷ 5 = 7.2 Correct D. (8 + 7 + 9 + 5 + 7) ÷ 4 = 9 E. 8 + 7 + 9 + 5 + 7 = 36.0: Forgot to divide by 5

To calculate Quik Call's price, convert 5 hours 3 minutes to minutes (60 x 5 = 300; 300 + 3 = 303 minutes) and then multiply this by the rate of 18¢ ($0.18) per minute, which equals $54.54. To calculate Econo Phone's price, round 5 hours 3 minutes up to 5 hours 15 minutes and then convert the 15 minutes to ¼ or .25 hours. This value of 5.25 hours is multiplied by the rate of $7.90 per hour to obtain $41.48. Compare the two costs, $54.54 and $41.48, to determine the lower cost.

A. 5 hr x $7.90/hr = $39.50

B. 5.25 hr × $7.90/hr = $41.48 Correct C. 5.3 hr x $7.90/hr = $41.87 D. 5 hr x 60 min/hr x 18¢/min = $54.00 E. 5 hr x 60 min/hr = 300 min; 303 min x 18¢/min = $54.54

Answer to Level 6 Sample Item First, convert 15½ to 15.5 and 18½ to 18.5 and multiply these two values to obtain the area of the room in square feet. Next, divide this number (286.75) by the area covered by one box of tiles (25 sq ft). Finally, round 11.47 boxes up to 12 boxes, as it is probably not possible to order a fraction of a box of tiles.

A. 15½ becomes 15.5 and 18½ becomes 18.5; 15.5 x 18.5 = 286.75 sq ft; 286.75 sq ft ÷ 25 = 11.47, then rounded down to 11 boxes B. 15½ becomes 15.5 and 18½ becomes 18.5; 15.5 x 18.5 = 286.75 sq ft; 286.75 sq ft ÷ 25 = 11.47, then rounded up to 12 boxes Correct C. 15.5 + 18.5 = 34: Added the room dimensions D. 15.5 + 18.5 + 25 = 59: Added all the dimensions given in feet E. 15.5 x 18.5 = 286.75, then rounded up to 287: The area of the room in square feet

First, look up the formula for the volume of a cylinder (π r 2h). Next, divide the diameter (2.5 ft) by 2 to find the radius (1.25 ft) and calculate the volume of the cylinder (9.81 cu ft). Finally, the volume must be multiplied by 7.48, the number of gallons per cubic foot, to find the number of gallons remaining in the tank.

A. Multiplied 2.5 x 2 x 7.48 = 37.4, then rounded down to 37 B. Forgot to square 1.25; 3.14 x 1.25 x 2 x 7.48 = 58.72, then rounded up to 59 C. 2.5 ÷ 2 = 1.25; 3.14 x (1.25)2 x 2 = 9.81 cu ft; 9.81 cu ft x 7.48 gal/cu ft = 73.4 gallons, then rounded down to 73 Correct D. (1.25 x 3.14)2 x 2 x 7.48 = 230.47, then rounded down to 230 E. 3.14 x (2.5)2 x 2 x 7.48 gal/cu ft = 293.59 gallons, then rounded up to 294: Used diameter2 instead of the radius