Theatex as higher education assessment_4 pot

■ Integers include the whole numbers and their opposites. Remember, the opposite of zero is zero: –3,–2,–1,0,1,2,3, ■ Rational numbers are all numbers that can be written as fractions, where the numerator and denomina- tor are both integers, but the denominator is not zero. For example, ᎏ 2 3 ᎏ is a rational number, as is ᎏ Ϫ 5 6 ᎏ .The decimal form of these numbers is either a terminating (ending) decimal, such as the decimal form of ᎏ 3 4 ᎏ which is 0.75; or a repeating decimal, such as the decimal form of ᎏ 1 3 ᎏ which is 0.3333333 . . . ■ Irrational numbers are numbers that cannot be expressed as terminating or repeating decimals (i.e. non- repeating, non-terminating decimals such as π, ͙2 ෆ , ͙12 ෆ ). The number line is a graphical representation of the order of numbers. As you move to the right, the value increases. As you move to the left, the value decreases. If we need a number line to reflect certain rational or irrational numbers, we can estimate where they should be. COMPARISON SYMBOLS The following table will illustrate some comparison symbols: = is equal to 5 = 5 ≠ is not equal to 4 ≠ 3 > is greater than 5 > 3 ≥ is greater than or equal to x ≥ 5 (x can be 5 or any number > 5) < is less than 4 < 6 ≤ is less than or equal to x ≤ 3 (x can be 3 or any number < 3) –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 Greater Than Less Than – THEA MATH REVIEW– 84 – ᎏ 3 4 ᎏ ͙2 ෆ ␲ SYMBOLS OF ADDITION In addition, the numbers being added are called addends. The result is called a sum. The symbol for addition is called a plus sign. In the following example, 4 and 5 are addends and 9 is the sum: 4 + 5 = 9 SYMBOLS OF SUBTRACTION In subtraction, the number being subtracted is called the subtrahend. The number being subtracted FROM is called the minuend. The answer to a subtraction problem is called a difference. The symbol for subtraction is called a minus sign. In the following example, 15 is the minuend, 4 is the subtrahend, and 11 is the difference: 15 – 4 = 11 SYMBOLS OF MULTIPLICATION When two or more numbers are being multiplied, they are called factors. The answer that results is called the prod- uct. In the following example, 5 and 6 are factors and 30 is their product: 5 ϫ 6 = 30 There are several ways to represent multiplication in the above mathematical statement. ■ A dot between factors indicates multiplication: 5 • 6 = 30 ■ Parentheses around any one or more factors indicate multiplication: (5)6 = 30, 5(6) = 30, and (5)(6) = 30. ■ Multiplication is also indicated when a number is placed next to a variable: 5a = 30. In this equation, 5 is being multiplied by a. S YMBOLS OF DIVISION In division, the number being divided BY is called the divisor. The number being divided INTO is called the div- idend. The answer to a division problem is called the quotient. There are a few different ways to represent division with symbols. In each of the following equivalent expressions, 3 is the divisor and 8 is the dividend: 8 ÷ 3, 8/3, ᎏ 8 3 ᎏ ,3ͤ8 ෆ – THEA MATH REVIEW– 85 PRIME AND COMPOSITE NUMBERS A positive integer that is greater than the number 1 is either prime or composite, but not both. ■ A prime number is a number that has exactly two factors: 1 and itself. Examples 2, 3, 5, 7, 11, 13, 17, 19, 23 . . . ■ A composite number is a number that has more than two factors. Examples 4, 6, 8, 9, 10, 12, 14, 15, 16 . . . ■ The number 1 is neither prime nor composite since it has only one factor. Operations ADDITION Addition is used when it is necessary to combine amounts. It is easiest to add when the addends are stacked in a column with the place values aligned. Work from right to left, starting with the ones column. Example Add 40 + 129 + 24. 1. Align the addends in the ones column. Since it is necessary to work from right to left, begin to add start- ing with the ones column. Since the ones column totals 13, and 13 equals 1 ten and 3 ones, write the 3 in the ones column of the answer, and regroup or “carry” the 1 ten to the next column as a 1 over the tens column so it gets added with the other tens: 1 40 129 + 24 3 – THEA MATH REVIEW– 86 2. Add the tens column, including the regrouped 1. 1 40 129 + 24 93 3. Then add the hundreds column. Since there is only one value, write the 1 in the answer. 1 40 129 + 24 193 SUBTRACTION Subtraction is used to find the difference between amounts. It is easiest to subtract when the minuend and sub- trahend are in a column with the place values aligned. Again, just as in addition, work from right to left. It may be necessary to regroup. Example If Becky has 52 clients, and Claire has 36, how many more clients does Becky have? 1. Find the difference between their client numbers by subtracting. Start with the ones column. Since 2 is less than the number being subtracted (6), regroup or “borrow” a ten from the tens column. Add the regrouped amount to the ones column. Now subtract 12 – 6 in the ones column. 5 4 ΋ 2 1 – 36 6 2. Regrouping 1 ten from the tens column left 4 tens. Subtract 4 – 3 and write the result in the tens column of the answer. Becky has 16 more clients than Claire. Check by addition: 16 + 36 = 52. 5 4 ΋ 2 1 – 36 16 – THEA MATH REVIEW– 87 MULTIPLICATION In multiplication, the same amount is combined multiple times. For example, instead of adding 30 three times, 30 + 30 + 30, it is easier to simply multiply 30 by 3. If a problem asks for the product of two or more numbers, the numbers should be multiplied to arrive at the answer. Example A school auditorium contains 54 rows, each containing 34 seats. How many seats are there in total? 1. In order to solve this problem, you could add 34 to itself 54 times, but we can solve this problem easier with multiplication. Line up the place values vertically, writing the problem in columns. Multiply the number in the ones place of the top factor (4) by the number in the ones place of the bottom factor (4): 4 ϫ 4 = 16. Since 16 = 1 ten and 6 ones, write the 6 in the ones place in the first partial product. Regroup or carry the ten by writing a 1 above the tens place of the top factor. 1 34 ϫ 54 6 2. Multiply the number in the tens place in the top factor (3) by the number in the ones place of the bottom factor (4); 4 ϫ 3 = 12. Then add the regrouped amount 12 + 1 = 13. Write the 3 in the tens column and the one in the hundreds column of the partial product. 1 34 ϫ 54 136 3. The last calculations to be done require multiplying by the tens place of the bottom factor. Multiply 5 (tens from bottom factor) by 4 (ones from top factor); 5 ϫ 4 = 20, but since the 5 really represents a number of tens, the actual value of the answer is 200 (50 ϫ 4 = 200). Therefore, write the two zeros under the ones and tens columns of the second partial product and regroup or carry the 2 hundreds by writing a 2 above the tens place of the top factor. 2 34 ϫ 54 136 00 – THEA MATH REVIEW– 88 4. Multiply 5 (tens from bottom factor) by 3 (tens from top factor); 5 ϫ 3 = 15, but since the 5 and the 3 each represent a number of tens, the actual value of the answer is 1,500 (50 ϫ 30 = 1,500). Add the two additional hundreds carried over from the last multiplication: 15 + 2 = 17 (hundreds). Write the 17 in front of the zeros in the second partial product. 2 34 ϫ 54 136 1,700 5. Add the partial products to find the total product: 2 34 ϫ 54 136 + 1,700 1,836 Note: It is easier to perform multiplication if you write the factor with the greater number of digits in the top row. In this example, both factors have an equal number of digits, so it does not matter which is written on top. DIVISION In division, the same amount is subtracted multiple times. For example, instead of subtracting 5 from 25 as many times as possible, 25 – 5 – 5 – 5 – 5 – 5, it is easier to simply divide, asking how many 5s are in 25; 25 ÷ 5. Example At a road show, three artists sold their beads for a total of $54. If they share the money equally, how much money should each artist receive? 1. Divide the total amount ($54) by the number of ways the money is to be split (3). Work from left to right. How many times does 3 divide 5? Write the answer, 1, directly above the 5 in the dividend, since both the 5 and the 1 represent a number of tens. Now multiply: since 1(ten) ϫ 3(ones) = 3(tens), write the 3 under the 5, and subtract; 5(tens) – 3(tens) = 2(tens). 1 3ͤ54 ෆ –3 2 – THEA MATH REVIEW– 89 2. Continue dividing. Bring down the 4 from the ones place in the dividend. How many times does 3 divide 24? Write the answer, 8, directly above the 4 in the dividend. Since 3 ϫ 8 = 24, write 24 below the other 24 and subtract 24 – 24 = 0. 18 3ͤ54 ෆ –3↓ 24 –24 0 REMAINDERS If you get a number other than zero after your last subtraction, this number is your remainder. Example 9 divided by 4. 2 4ͤ9 ෆ – 8 1 1 is the remainder. The answer is 2 r1. This answer can also be written as 2 ᎏ 1 4 ᎏ since there was one part left over out of the four parts needed to make a whole. Working with Integers Remember, an integer is a whole number or its opposite. Here are some rules for working with integers: ADDING Adding numbers with the same sign results in a sum of the same sign: (positive) + (positive) = positive and (negative) + (negative) = negative When adding numbers of different signs, follow this two-step process: 1. Subtract the positive values of the numbers. Positive values are the values of the numbers without any signs. 2. Keep the sign of the number with the larger positive value. – THEA MATH REVIEW– 90 Example –2 + 3 = 1. Subtract the positive values of the numbers: 3 – 2 = 1. 2. The number 3 is the larger of the two positive values. Its sign in the original example was positive, so the sign of the answer is positive. The answer is positive 1. Example 8 + –11 = 1. Subtract the positive values of the numbers: 11 – 8 = 3. 2. The number 11 is the larger of the two positive values. Its sign in the original example was negative, so the sign of the answer is negative. The answer is negative 3. SUBTRACTING When subtracting integers, change all subtraction signs to addition signs and change the sign of the number being subtracted to its opposite. Then follow the rules for addition. Examples (+10) – (+12) = (+10) + (–12) = –2 (–5) – (–7) = (–5) + (+7) = +2 MULTIPLYING AND DIVIDING A simple method for remembering the rules of multiplying and dividing is that if the signs are the same when mul- tiplying or dividing two quantities, the answer will be positive. If the signs are different, the answer will be nega- tive. (positive) ϫ (positive) = positive = positive (positive) ϫ (negative) = negative = negative (negative) ϫ (negative) = positive = positive Examples (10)( – 12) = – 120 – 5 ϫ – 7 = 35 – ᎏ 1 3 2 ᎏ = –4 ᎏ 1 3 5 ᎏ = 5 (negative) ᎏᎏ (negative) (positive) ᎏᎏ (negative) (positive) ᎏᎏ (positive) – THEA MATH REVIEW– 91 Sequence of Mathematical Operations There is an order in which a sequence of mathematical operations must be performed: P: Parentheses/Grouping Symbols. Perform all operations within parentheses first. If there is more than one set of parentheses, begin to work with the innermost set and work toward the outside. If more than one operation is present within the parentheses, use the remaining rules of order to determine which operation to perform first. E: Exponents. Evaluate exponents. M/D: Multiply/Divide. Work from left to right in the expression. A/S: Add/Subtract. Work from left to right in the expression. This order is illustrated by the following acronym PEMDAS, which can be remembered by using the first let- ter of each of the words in the phrase: Please Excuse My Dear Aunt Sally. Example + 27 = + 27 = + 27 = 16 + 27 = 43 Properties of Arithmetic Listed below are several properties of mathematics: ■ Commutative Property: This property states that the result of an arithmetic operation is not affected by reversing the order of the numbers. Multiplication and addition are operations that satisfy the commuta- tive property. Examples 5 ϫ 2 = 2 ϫ 5 5a = a5 b + 3 = 3 + b However, neither subtraction nor division is commutative, because reversing the order of the numbers does not yield the same result. Examples 5 – 2 ≠ 2 – 5 6 ÷ 3 ≠ 3 ÷ 6 64 ᎏ 4 (8) 2 ᎏ 4 (5 + 3) 2 ᎏ 4 – THEA MATH REVIEW– 92 ■ Associative Property: If parentheses can be moved to group different numbers in an arithmetic problem without changing the result, then the operation is associative. Addition and multiplication are associative. Examples 2 + (3 + 4) = (2 + 3) + 4 2(ab) = (2a)b ■ Distributive Property: When a value is being multiplied by a sum or difference, multiply that value by each quantity within the parentheses. Then, take the sum or difference to yield an equivalent result. Examples 5(a + b) = 5a + 5b 5(100 – 6) = (5 ϫ 100) – (5 ϫ 6) This second example can be proved by performing the calculations: 5(94) = 5(100 – 6) = 500 – 30 470 = 470 ADDITIVE AND MULTIPLICATIVE IDENTITIES AND INVERSES ■ The additive identity is the value which, when added to a number, does not change the number. For all of the sets of numbers defined above (counting numbers, integers, rational numbers, etc.), the additive identity is 0. Examples 5 + 0 = 5 –3 + 0 = –3 Adding 0 does not change the values of 5 and –3, so 0 is the additive identity. ■ The additive inverse of a number is the number which, when added to the number, gives you the addi- tive identity. Example What is the additive inverse of –3? – THEA MATH REVIEW– 93 [...]... frequently Measurement This section will review the basics of measurement systems used in the United States (sometimes called customary measurement) and other countries, methods of performing mathematical operations with units of measurement, and the process of converting between different units The use of measurement enables a connection to be made between mathematics and the real world To measure any... base is used as a factor to attain a product Example Evaluate 25 2 is the base and 5 is the exponent Therefore, 2 should be used as a factor 5 times to attain a product: 25 = 2 ϫ 2 ϫ 2 ϫ 2 ϫ 2 = 32 Z ERO E XPONENT Any non-zero number raised to the zero power equals 1 Examples 50 =1 700 = 1 29,8740 = 1 N EGATIVE E XPONENTS A base raised to a negative exponent is equivalent to the reciprocal of the base... division apply It is easier to divide if the divisor does not have any decimals In order to accomplish that, simply move the decimal place to the right as many places as necessary to make the divisor a whole number If the decimal point is moved in the divisor, it must also be moved in the dividend in order to keep the answer the same as the original problem; 4 ÷ 2 has the same solution as its multiples... between mathematics and the real world To measure any object, assign a number and a unit of measure For instance, when a fish is caught, it is often weighed in ounces and its length measured in inches The following lesson will help you become more familiar with the types, conversions, and units of measurement Types of Measurements The types of measurements used most frequently in the United States are listed... pounds 32 ounces ᎏᎏᎏ 16 ounces per pound = 2 pounds Therefore, 32 ounces equals two pounds Basic Operations with Measurement You may need to add, subtract, multiply, and divide with measurement The mathematical rules needed for each of these operations with measurement follow A DDITION WITH M EASUREMENTS To add measurements, follow these two steps: 1 Add like units 2 Simplify the answer by converting... the fraction is the same number of 9s as digits Example ៮ Convert 3 to a fraction 103 – THEA MATH REVIEW – ៮ You may already recognize 3 as ᎏ3ᎏ The repeating pattern, in this case 3, becomes our numerator There is one digit in the pattern, so 9 is our denominator 1 ៮ 3 3Ϭ3 1 3 = ᎏ9ᎏ = ᎏᎏ = ᎏ3ᎏ 9÷3 Example ៮ Convert 36 to a fraction The repeating pattern, in this case 36, becomes our numerator There are... the number 2 is the cube root of the number 8 The symbol for cube root is the same as the square root symbol, except for a small three 3 ͙34 It is read as “cube root.” The number inside of the radical is still called the radicand, and the three is called ෆ the index (In a square root, the index is not written, but it has an index of 2.) Example 53 3 = 125, therefore ͙125 = 5 ෆ Like square roots, the... except one must make note of the total number of decimal places in the factors Example What is the product of 0.14 and 4.3? First, multiply as usual (do not line up the decimal points): 4.3 ϫ.14 172 + 430 602 Now, to figure out the answer, 4.3 has one decimal place and 14 has two decimal places Add in order to determine the total number of decimal places the answer must have to the right of the decimal point... common multiples are 12, 24, and 36 From the above it can also be determined that the least common multiple of the numbers 4 and 6 is 12, since this number is the smallest number that appeared in both lists The least common multiple, or LCM, is used when performing addition and subtraction of fractions to find the least common denominator Example (using denominators 4 and 6 and LCM of 12) 1 5 1(3) 5(2)... days = 1 week 52 weeks = 1 year (yr) 12 months = 1 year 365 days = 1 year *Notice that ounces are used to measure the dimensions of both volume and weight Converting Units When performing mathematical operations, it may be necessary to convert units of measure to simplify a problem Units of measure are converted by using either multiplication or division ■ To convert from a larger unit into a smaller . dividend. Since 3 ϫ 8 = 24, write 24 below the other 24 and subtract 24 – 24 = 0. 18 3ͤ 54 ෆ –3↓ 24 – 24 0 REMAINDERS If you get a number other than zero after your last subtraction, this number. multiply as usual (do not line up the decimal points): 4. 3 ϫ . 14 172 + 43 0 602 Now, to figure out the answer, 4. 3 has one decimal place and . 14 has two decimal places. Add in order to deter- mine. REVIEW– 102 Examples Which is larger: ᎏ 1 7 1 ᎏ or ᎏ 4 9 ᎏ ? Cross-multiply. 7 ϫ 9 = 63 4 ϫ 11 = 44 63 > 44 , therefore, ᎏ 1 7 1 ᎏ > ᎏ 4 9 ᎏ Compare ᎏ 1 6 8 ᎏ and ᎏ 2 6 ᎏ . Cross-multiply. 6