M.1.2 Perform arithmetic operations with rational numbers.

M.1.2 Perform arithmetic operations with rational numbers. This objective includes, but is not limited to, the following examples of knowledge, skills, and abilities. • Complete computations with integers using the four basic operations. • Complete computations with decimals using the four basic operations. • Complete computations with fractions and mixed numbers using the four basic operations. • Complete computations involving the order of operations, excluding complex fractions. Completing basic computations by hand can at times be quicker than using a calculator. Additionally, calculators don’t always complete the mathematical order of operations in ways you assume they will. For this TEAS task, you need to be competent doing these types of calculations by hand. You will apply the order of operations—including addition, subtraction, multiplication, and division—and be expected to do so using integers, decimals, fractions, and mixed numbers. Practice makes perfect, so it might be helpful to search the Internet for additional practice problems. One mnemonic device for the conventions of the mathematical order of operations for basic calculations is PEMDAS (parentheses, exponents, multiplication and division, addition and subtraction). There is some controversy about the use of PEMDAS (details of which can be easily found by an Internet search), but the TEAS will focus on noncontroversial aspects of the order of operations. For the TEAS, the “E” in PEMDAS (exponents) will not be required. However, parentheses, multiplication and division, and addition and subtraction will be required competencies. To successfully execute these operations, you must (in order) follow these rules. 1. First, perform any calculations inside parentheses. 2. Next, perform all multiplications and divisions, completing the operations as they occur from left to right. 3. Finally, perform all addition and subtraction, completing the operations as they occur from left to right. The following are three basic examples with rationales. Ex. 1: 6 + 7 × 8 = 6 + 7 × 8 Multiply first. = 6 + 56 Then add. = 62 Ex. 2: 16 ÷ 8 - 2 = 16 ÷ 8 - 2 Divide first. = 2 - 2 Then subtract. = 0 Ex. 3: (25 - 11) × 3 = (25 - 11) × 3 Perform operations in parentheses first. = 14 × 3 Then multiply. = 42 Key terms addition. Calculation of a total of two or more numbers. division. Separation of numbers into parts. mixed number. A number formed by an integer and a fraction. multiplication. Addition of a number to itself a specified number of times. order of operations. The sequence of operations that must be followed to simplify an expression. subtraction. Removing one number from another. MATHEMATICS 07/24/15 November 23, 2015 2:14 PM 4_M.1_TEASVI 07/268 Number and algebra ATI TEAS STUDY MANUAL The following are several examples involving multiple steps. Example: 3 + 6 × (5 + 4) ÷ 3 - 7 Solution: Step 1: 3 + 6 × (5 + 4) ÷ 3 - 7 = 3 + 6 × 9 ÷ 3 - 7 Parentheses Step 2: 3 + 6 × 9 ÷ 3 - 7 = 3 + 54 ÷ 3 - 7 Multiplication Step 3: 3 + 54 ÷ 3 - 7 = 3 + 18 - 7 Division Step 4: 3 + 18 - 7 = 21 – 7 Addition Step 5: 21 - 7 = 14 Subtraction Example: 9 - 5 ÷ (8 - 3) × 2 + 6 Solution: Step 1: 9 - 5 ÷ (8 - 3) × 2 + 6 = 9 - 5 ÷ 5 × 2 + 6 Parentheses Step 2: 9 - 5 ÷ 5 × 2 + 6 = 9 - 1 × 2 + 6 Division Step 3: 9 - 1 × 2 + 6 = 9 - 2 + 6 Multiplication Step 4: 9 - 2 + 6 = 7 + 6 Subtraction Step 5: 7 + 6 = 13 Addition In the last two examples, you will notice that multiplication and division were evaluated from left to right according to rule 2. Similarly, addition and subtraction were evaluated from left to right, according to rule 3. When two or more operations occur inside a set of parentheses, these operations should be evaluated according to rules 2 and 3. This is done in the example below. Example: 150 ÷ (6 + 3 × 8) - 5 Solution: Step 1: 150 ÷ (6 + 3 × 8) - 5 = 150 ÷ (6 + 24) - 5 Multiplication inside parentheses Step 2: 150 ÷ (6 + 24) - 5 = 150 ÷ 30 - 5 Addition inside parentheses Step 3: 150 ÷ 30 - 5 = 5 - 5 Division Step 4: 5 - 5 = 0 Subtraction Example: 36 - 6 12 + 3 Solution: This problem includes a fraction bar (also called a vinculum), which means we must divide the numerator by the denominator. However, we must perform all calculations above and below the fraction bar BEFORE dividing. Thus: 36 - 6 = (36 - 6) 12 + 3 (12 + 3) Evaluating this expression, we get: (36 - 6) = 30 = 2 (12 + 3) 15 Extensions of the above example include decimals, fractions, and mixed numbers. With decimals, all processes are identical. With fractions, you might need to find a common denominator or convert fractions to decimals. The same goes for mixed numbers, but you might need to convert to a fraction first. To help prepare for the TEAS, search the Internet for order of operations practice and work on your mastery of these types of problems. M.1.2 Practice problems 1. Solve the following problems: 4 + 3 × (9 - 6) (12 - 2) ÷ (11-6) 6 × 5 + 4 ÷ 2 – 2 × 5 30 - 3 × 2 18 - 24 ÷ 2 2. Which of the following is the correct value of 3 + 2 × 6 – 4? A. 32 B. 10 C. 11 D. 26 3. Which of the following is the correct value of the expression below? 15 + 2 × 5 11 - 24 ÷ 4 A. 21 B. 5 C. ≈ 9.9 07/24/15 November 23, 2015 2:14 PM 4_M.1_TEASVI 07/274 Number and algebra ATI TEAS STUDY MANUAL M.1.6 Solve real-world problems involving percentages. This objective includes, but is not limited to, the following examples of knowledge, skills, and abilities. • Define percent in terms of a real-world context. • Calculate the percent of a number in terms of a real-world context. • Find percent of increase or decrease between two numbers in terms of a real-world context. Percentage is a form of number quantity. For this task on the TEAS, you’ll need to understand what percentage means in real-world contexts, as well as how to work with percentages within these contexts. You will have to find percentage of a number quantity, as well as percent increase or decrease. Utilize the problemsolving skills discussed in the previous chapter and, if needed, search the Internet for additional practice problems of these types. Percent means “per 100,” or a value’s proportional equivalent compared to 100. One percent can be interpreted as one one-hundredth of something. Examples of percent expressions are 45%, 23.8%, 100%, and 0.05%. When a percent is less than or equal to 100%, then you can say “out of” 100. For example, 50% is 50 out of 100. But if a percent is more than 100%, you need to rethink the wording. It doesn’t make sense to say that 150% is 150 out of 100. 150% is 150 for each 100. For example, 150% of 10 is 15. Below are 100 small squares, and 50 have been shaded. One way to describe the shading it to say 50% has been shaded—in other words, 50 out of 100. Likewise, 50 cents is 50% of 100 cents (or 50% of $1.00). Other real-word percentage contexts include: percent off for a sale price; annual percent interest rates at a bank; annual percent gain or loss for a company or business; percent commission for a salesperson; percent depreciation of assets; percentages of ingredients in a mixture or recipe. Another consideration is finding a percent of a number. For instance, you might need to pay sales tax on a purchased item. The cash register and cashier will do it automatically, but you should have at least a rough idea of what it should be. If you buy about $50 worth of groceries and the sales tax rate is 8%, the sales tax in addition to the $50 is $4. To figure a percent of a number, you convert the percent to a decimal, then multiply the percent times the original number. In this case, 8% is 0.08 as a decimal (move the decimal to the left two places in the 8). When you multiply 0.08 times $50, you get $4 for the tax. So your total bill of groceries plus tax will be $54. Another application is percent increase or decrease. In a sense, sales tax is a percent increase. You are essentially increasing the cost of an item by a certain percentage. The 8% sales tax problem described earlier means a percent increase of 8% on the total cost of the purchase. An example of a percent decrease is purchasing an item on sale. If an item is on sale for 20% off, the cost of the item (before sales tax) is decreased by 20%. If an item has a regular price of $70 and is on sale at 20% off, you need to find 20% of the price and subtract. Multiply $70 by 0.2 to get $14