CLAST Arithmetic by Joyce 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Review Terms A whole number is any non-fractional number from 0 to infinity An integer is represented by all whole numbers and, additionally, includes all non-fractional negative numbers A mixed number is composed of an integer and a fraction The numerator is the top part of a fraction The denominator is the bottom part of a fraction. A variable is a letter or symbol used to represent a number in a mathematical expression. A bar over a number(s) means the number(s) are repeated. For example, means 5.213131313... 3/30/99 CLAST Arithmetic by Joyce

Add & Subtract Fractions Convert all numbers to fractions, including whole numbers and mixed numbers The denominators must be the same - Find the least common denominator and convert to equivalent fractions Combine - add and/or subtract the numerators, keeping the same denominator Reduce and/or convert the answer to a mixed number, if possible Determine the sign of the answer 3/30/99 CLAST Arithmetic by Joyce

Convert a Whole Number to a Fraction To convert a whole number to a fraction, simply place the number over 1 10 converts to 17 converts to 12345678 converts to -11 converts to or or 3/30/99 CLAST Arithmetic by Joyce

Convert a Mixed Number to an Improper Fraction A fraction in which the numerator is larger than the denominator is known as an improper fraction. To convert a mixed number to an improper fraction : Multiply the whole number by the denominator Add the numerator to the product Put the result in the numerator over the original denominator Example: 2 2 x 4 = 8 3 + 8 = 11 Answer: 3/30/99 CLAST Arithmetic by Joyce

Least Common Denominator (LCD) The LCD is a number into which all the denominators will divide with no remainder. To find the LCD, completely factor all of the denominators. Example: Find the LCD of and completely factor 8 as 2 x 2 x 2 (factor 2, power of 3) completely factor 10 as 2 x 5 (factor 2, power of 1) (factor 5, power of 1) The LCD is the product of all the different factors, with each factor raised to the greatest power that it occurs. Answer: LCD is 40 (2 x 2 x 2 x 5) 3/30/99 CLAST Arithmetic by Joyce

Convert to Fractions with Common Denominators Whatever you do to the numerator you must also do to the denominator to maintain the value of the fraction. Example: Before and can be added, they must be converted to fractions with common denominators. The common denominator is 40 (see previous slide) Multiply by Multiplying by is the same as multiplying by one. = Multiply by Multiplying by is the same as 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Add Fractions Convert to fractions with common denominators, then add the numerators. The denominators are not added. Reduce and/or convert the answer to a mixed number, if possible Example 1: Add and + = factor 9 as 3 x 3 factor 40 as 2 x 4 x 5 No common factors. The answer can not be reduced. Example 2: Add 1 and + = factor 10 as 2 x 5 factor 5 as 1 x 5 The 5’s cancel each other. reduces to , written as 2 3/30/99 CLAST Arithmetic by Joyce

Add/Subtract: Determine the Sign To evaluate expressions involving adding negative numbers, first combine the numbers with like signs to get two subtotals. Ignore the signs of the two subtotals and subtract the smaller subtotal from the larger subtotal. The answer takes on the sign of the larger subtotal. To subtract a negative number, change the sign of the negative number to positive, then add. Example 1: Add -5, 3, and 7 Example 2: Subtract -7 from 5 The subtotal of the positive numbers is 10 The problem is 5 - (-7) The subtotal of the negative numbers is -5 Change -7 to 7 The difference of 10 and 5 (ignore all signs) is 5 Then add 5 + 7 The answer is 5, since the larger subtotal is positive The answer is 12 3/30/99 CLAST Arithmetic by Joyce

Absolute Value of a Real Number The absolute value of a number is the distance the number is from zero on the number line Absolute value is designated by two vertical lines | | Absolute value is always positive, since it represents distance Example 1: the absolute value of 4, written |4|, is 4 4 is four units to the right of zero Example 2: the absolute value of - 5, written |-5|, is 5 - 5 is five units to the left of zero 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Multiply Fractions Review Order of operations Review Conversion to fractions Reduce by canceling common factors, if possible Multiply across resulting numerators Multiply across resulting denominators Reduce again and/or convert the answer to a mixed number, if possible Determine the sign of the answer 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Divide Fractions Review order of operations and conversion to fractions Change the division sign to a multiplication sign and invert the fraction to the right of the sign Reduce by canceling common factors, if possible Multiply across resulting numerators Multiply across resulting denominators Reduce again and/or convert the answer to a mixed number, if possible Determine the sign of the answer 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Divide Fractions Example: Divide Change to multiplication and invert Cancel common factors Multiply across resulting numerators Multiply across resulting denominators The answer is (positive) This fraction can not be reduced nor converted to a mixed number 3/30/99 CLAST Arithmetic by Joyce

Multiply/Divide: Determine the Sign of the Answer If both signs are alike (even number of similar signs), the sign of the answer is positive. Examples: 2 x 3 = 6 (two positive signs) (-2) x (-3) = 6 (two negative signs) If the signs are different (odd number of similar signs), the sign of the answer is negative. Example: (-2) x 3 = -6 (one negative sign) 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Order of Operations When evaluating a mathematical expression, perform operations in this order, working from left to right: Parentheses Please Exponents Excuse Multiply/Divide My Dear Add / Subtract Aunt Sally 3/30/99 CLAST Arithmetic by Joyce

Add/Subtract Decimals Arrange vertically by lining up decimal points Insert zeros to right of decimal point, if needed Add or subtract, beginning with the right hand column Determine the sign of the answer 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Multiply Decimals It is not necessary to line up the decimal points. Multiply (ignore decimal points), starting at the right most column or number. Count the total number of digits to the right of the decimals in the multipliers. Count the same number of digits from the right, in your answer, and insert a decimal point. Insert zeros as placeholders when necessary. Determine the sign of the answer. 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Dividing Decimals If shown as or , for example, rewrite as Move the decimal in the divisor (.56) to the right as many places as necessary to make a whole number becomes (two places right) Move the decimal in the dividend (789) to the right an equal number of places. becomes Put the decimal in the quotient (answer) area directly above the decimal in the dividend. Divide, inserting zeros in the quotient or dividend as placeholders, if needed. Determine the sign of the answer. 3/30/99 CLAST Arithmetic by Joyce

Decimals, Percents, & Fractions Percent (%) denotes a whole quantity divided into 100 equal parts % means the fractional part of 100 Example: 4% = = .04 (four hundredths) Percents may be added, subtracted, multiplied, and divided 3/30/99 CLAST Arithmetic by Joyce

Common Percent, Fraction, and Decimal Equivalents % Decimal Fraction Money 25 .25 25 cents 10 .10 10 cents 75 .75 75 cents 20 .20 20 cents 30 .30 30 cents 50 .50 50 cents 3/30/99 CLAST Arithmetic by Joyce

Decimal/Percent Conversion If you do not see a decimal, it is understood to be at the right of the number. Decimal to percent Move the decimal two places to the right Add the % symbol Example: .25 becomes 25% Percent to decimal Move the decimal two places to the left Remove the % symbol Example: 25% becomes .25 3/30/99 CLAST Arithmetic by Joyce

Decimal/Percent/Fraction Conversion Percent to a decimal and a fraction convert percent to a decimal (move decimal left, remove %) divide by 100 reduce if possible Fraction to a decimal and a percent divide the denominator into the numerator convert decimal to a percent (move decimal right, add %) 3/30/99 CLAST Arithmetic by Joyce

Solve for one Variable in a Percentage Sentence Use the mathematical sentence c = a%b, substitute known values for the variables, and solve What is 6% of 200? Answer: 12 This sentence becomes c = 6% (200) where a = 6 and b = 200 Convert % to a decimal (or fraction) and multiply by b 12 is 6% of what? Answer: 200 This sentence becomes 12 = 6%b where c = 12 and a = 6 Convert % to a decimal (or fraction) and divide into c 12 is what percent of 200? Answer: 6% This sentence becomes 12 = a%(200) where c = 12 and b = 200 Divide b into c to get a decimal (or fraction) and convert to a percent 3/30/99 CLAST Arithmetic by Joyce

Calculate Percent Increase or Decrease for a Single Amount Given an original amount and a percent increase or decrease, find the new amount Change the percent to a decimal multiply the decimal by the original amount add the result to the original amount if increasing, subtract if decreasing Example: If you increase 500 by 25 %, what is the result? 25% = .25 .25(500) = 125 500 + 125 = 625 3/30/99 CLAST Arithmetic by Joyce

Calculate Percent Increase or Decrease for More than One Amount Given an original amount and a new amount, find the percent of increase or decrease. Calculate the amount of increase or decrease Divide the resulting difference by the original amount Convert the result to a percentage Example: If 500 is decreased to 375, find the percent of decrease. 500 - 375 = 125 = .25 .25 = 25% 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Round and Estimate You can estimate an answer by rounding the numbers before calculating the result. Try to eliminate unreasonable answers. Round up if the last digit is five or greater 85 becomes 90, rounded to tens 855 becomes 860 (to tens), or 900 (to 100s) Otherwise round down 84 becomes 80, rounded to tens 844 becomes 840 (to tens), or 800 (to 100s) 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Average Average is the sum of all the items divided by the total number of items. The answer must be between the highest and lowest numbers Example: 14+26+9+12+38 = 99 = 19 Example: Round and estimate 14+26+9+12+38 10+30+10+10+40 = 100 = 20 3/30/99 CLAST Arithmetic by Joyce

Number Line & the Order Relationship between Real Numbers A real number is any number between negative infinity and positive infinity any number is either less than (<), equal to (=), or greater than (>) another number Number Line -5 - 0 2 5 a > b if a is to the right of b on the number line Example: 5 > 2 a < b if a is to the left of b on the number line Example: -5 < - < 3/30/99 CLAST Arithmetic by Joyce

Determine the Order Relationship Between two Real Numbers To compare fractions, first find a common denominator, then compare numerators To compare decimals, first add zeros until you have the same number of places in each number, then compare each digit To compare decimals with fractions, first make them either both decimals or both fractions To compare square roots, square both sides and compare the results 3/30/99 CLAST Arithmetic by Joyce

Recognize the Meaning of Exponents An exponent, or power, indicates how many times a number is multiplied by itself. Examples: 22 means 2 x 2 = 4 24 means 2 x 2 x 2 x 2 = 16 Any number, except the number zero, raised to the power of zero, is 1 Examples: (-2) 0 = 1, 120 = 1 Any number raised to the power of one, is the number itself Example: 21 = 2 Any number, except the number zero, raised to a negative power, is the inverse of that number Example: 2-1 = 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Square Roots The square root of n, is the number which, when multiplied by itself, equals n. Every positive rational number has two square roots. One is positive and one is negative Example: Find the square roots of 16 find the factors of 16: 16x1, 8x2, and 4x4 or -16x-1, -8x-2, and -4x-4 4 multiplied by itself is 16, -4 multiplied by itself is 16 so the square roots of 16 are +4 and -4 or ± 4 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Laws of Exponents Let a and b represent any numbers, and m and n represent any integers. Then am x an = am+n (am)n = a mn ( )n = provided b  0 (ab)n = anbn = am-n provided a  0 a 0 = 1 provided a  0 3/30/99 CLAST Arithmetic by Joyce

Identify Place Value and Use Expanded Notation Place value represents the distance from the decimal. Example: For the number 1234.678 1 is in the thousands place 1x103 or 1 thousand 2 is in the hundreds place 2x102 or 2 hundred 3 is in the tens place 3x101 or thirty 4 is in the ones place 4x100 or four 6 is in the tenths place 6x10-1 or 6 tenths 7 is in the hundredths place 7x1/10-2 or 7 hundredths 8 is in the thousandths place 8x1/10-3 or 8 thousandths 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Use Expanded Notation 24 could be written as 2 x 101 + 4 x 100 2 x 10 + 4 x 1 245.6 could be written as 2 x 102 + 4 x 101 + 5 x 100 + 6 x 10-1 3/30/99 CLAST Arithmetic by Joyce

Find Missing Numbers Given a Pattern Look for a common relationship, evidenced by a pattern in the pairs of numbers. Check all pairs for the same pattern. Linear relationships involve +, -, x, / Example: (2,4),(3,6), (7,14) the second number of the pair is two times the first number of the pair Quadratic relationships involve squaring Example: (2,4), (3,9), (7,49) the second number of the pair is the square of the 3/30/99 CLAST Arithmetic by Joyce

Find Missing Numbers Given a Pattern: Progression A progression is a sequence of numbers which follow an established pattern. An arithmetic progression adds or subtracts the same number Example: 1,3,5,7 In this sequence, 2 is added to each successive number. The number 2 is called the common difference. A geometric progression multiplies or divides by the same number Example: 2,4,8,16 In this sequence, each successive number is multiplied by 2. The number 2 is the common ratio. An harmonic progression keeps a pattern Example: , , In this sequence, the numerator stays the same, and 3 is the common difference in the denominator. 3/30/99 CLAST Arithmetic by Joyce

Solve Real-world Problems Involving Arithmetic Logic & Structure Read the problem. Then use reasoning to Determine what you know Determine what you need to find Determine what is extraneous Determine the steps There is only one correct answer! 3/30/99 CLAST Arithmetic by Joyce

CLAST Arithmetic by Joyce Tips for Taking Tests Take all the practice tests You will learn how questions are phrased You will not waste time trying to figure out the meaning of the question during the test You will learn some answers You can work backwards from the answers Know what to memorize and what to learn 3/30/99 CLAST Arithmetic by Joyce