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00 Key Math Concepts for the CT Number Properties. UNDEFINED On the CT, undefined almost always means division by zero. The epression is undefined if bc a either b or c equals 0.. REL/IMGINRY real number
00 Key Math Concepts for the CT Number Properties. UNDEFINED On the CT, undefined almost always means division by zero. The epression is undefined if bc a either b or c equals 0.. REL/IMGINRY real number is a number that has a location on the number line. On the CT, imaginary numbers are numbers that involve the square root of a negative number. 4 is an imaginary number.. INTEGER/NONINTEGER Integers are whole numbers; they include negative whole numbers and zero. 4. RTIONL/IRRTIONL rational number is a number that can be epressed as a ratio of two integers. Irrational numbers are real numbers they have locations on the number line they just can t be epressed precisely as a fraction or decimal.for the purposes of the CT, the most important irrational numbers are,, and π.. DDING/SUBTRCTING SIGNED NUMBERS To add a positive and a negative, first ignore the signs and find the positive difference between the number parts. Then attach the sign of the original number with the larger number part. For eample, to add and 4, first we ignore the minus sign and find the positive difference between and 4 that s. Then we attach the sign of the number with the larger number part in this case it s the minus sign from the 4. So, + ( 4) =. Make subtraction situations simpler by turning them into addition. For eample, think of 7 ( ) as 7 + (+). To add or subtract a string of positives and negatives, first turn everything into addition. Then combine the positives and negatives so that the string is reduced to the sum of a single positive number and a single negative number. 6. MULTIPLYING/DIVIDING SIGNED NUMBERS To multiply and/or divide positives and negatives, treat the number parts as usual and attach a negative sign if there were originally an odd number of negatives. To multiply,, and, first multiply the number parts: = 0.Then go back and note that there were three an odd number negatives, so the product is negative: ( ) ( ) ( ) = 0. CT STUDY IDS 00 Key Math Concepts for the CT 7. PEMDS When performing multiple operations, remember PEMDS, which means Parentheses first, then Eponents, then Multiplication and Division (left to right), then ddition and Subtraction (left to right). In the epression 9 ( ) + 6, begin with the parentheses: ( ) =. Then do the eponent: = 4. Now the epression is: Net do the multiplication and division to get 9 8 +, which equals. 8. BSOLUTE VLUE Treat absolute value signs a lot like parentheses. Do what s inside them first and then take the absolute value of the result. Don t take the absolute value of each piece between the bars before calculating. In order to calculate ( ) + ( 4) + ( 0), first do what s inside the bars to get:,which is, or. 9. COUNTING CONSECUTIVE INTEGERS To count consecutive integers, subtract the smallest from the largest and add. To count the integers from through, subtract: = 8. Then add : 8 + = 9. Divisibility 0. FCTOR/MULTIPLE The factors of integer n are the positive integers that divide into n with no remainder. The multiples of n are the integers that n divides into with no remainder. 6 is a factor of, and 4 is a multiple of. is both a factor and a multiple of itself.. PRIME FCTORIZTION prime number is a positive integer that has eactly two positive integer factors: and the integer itself. The first eight prime numbers are,,, 7,,, 7, and 9. To find the prime factorization of an integer, just keep breaking it up into factors until all the factors are prime. To find the prime factorization of 6, for eample, you could begin by breaking it into 4 9: 6 = 4 9 =. RELTIVE PRIMES To determine whether two integers are relative primes, break them both down to their prime factorizations. For eample: = 7, and 4 =. They have no prime factors in common, so and 4 are relative primes.. COMMON MULTIPLE You can always get a common multiple of two numbers by multiplying them, but, unless the two numbers are relative primes, the product will not be the least common multiple. For eample, to find a common multiple for and, you could just multiply: = LEST COMMON MULTIPLE (LCM) To find the least common multiple, check out the multiples of the larger number until you find one that s also a multiple of the smaller. To find the LCM of and, begin by taking the multiples of : is not divisible by ; 0 s not; nor is 4. But the net multiple of, 60, is divisible by, so it s the LCM.. GRETEST COMMON FCTOR (GCF) To find the greatest common factor, break down both numbers into their prime factorizations and take all the prime factors they have in common. 6 =, and 48 =. What they have in common is two s and one, so the GCF is = =. 00 Key Math Concepts for the CT CT STUDY IDS 6. EVEN/ODD To predict whether a sum, difference, or product will be even or odd, just take simple numbers like and and see what happens. There are rules odd times even is even, for eample but there s no need to memorize them. What happens with one set of numbers generally happens with all similar sets. 7. MULTIPLES OF ND 4 n integer is divisible by if the last digit is even. n integer is divisible by 4 if the last two digits form a multiple of 4. The last digit of 6 is, which is even, so 6 is a multiple of. The last two digits make 6, which is not divisible by 4, so 6 is not a multiple of MULTIPLES OF ND 9 n integer is divisible by if the sum of its digits is divisible by.n integer is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits in 97 is, which is divisible by but not by 9, so 97 is divisible by but not MULTIPLES OF ND 0 n integer is divisible by if the last digit is or 0. n integer is divisible by 0 if the last digit is 0.The last digit of 66 is, so 66 is a multiple but not a multiple of REMINDERS The remainder is the whole number left over after division. 487 is more than 48, which is a multiple of, so when 487 is divided by, the remainder will be. Fractions and Decimals. REDUCING FRCTIONS To reduce a fraction to lowest terms, factor out and cancel all factors the numerator and denominator have in common. 8 6 = = 7 9. DDING/SUBTRCTING FRCTIONS To add or subtract fractions, first find a common denominator, and then add or subtract the numerators = = = 0. MULTIPLYING FRCTIONS To multiply fractions, multiply the numerators and multiply the denominators. 7 4 = = DIVIDING FRCTIONS To divide fractions, invert the second one and multiply. = = = 6. CONVERTING MIXED NUMBER TO N IMPROPER FRCTION To convert a mied number to an improper fraction, multiply the whole number part by the denominator, then add the numerator. The result is the new numerator (over the same denominator). To convert 7, first multiply 7 by, then add, to get the new numerator of. Put that over the same denominator,, to get. 6. CONVERTING N IMPROPER FRCTION TO MIXED NUMBER To convert an improper fraction to a mied number, divide the denominator into the numerator to get a whole number quotient with a remainder. The quotient becomes the whole number part of the mied number, and the remainder becomes the new numerator with the same denominator. For eample, to convert 0 8, first divide into 08, which yields with a remainder of. Therefore, 8 =. 0 CT STUDY IDS 00 Key Math Concepts for the CT 7. RECIPROCL To find the reciprocal of a fraction, switch the numerator and the denominator. The reciprocal of 7 is 7.The reciprocal of is.the product of reciprocals is. 8. COMPRING FRCTIONS One way to compare fractions is to re-epress them with a common denominator. 4 8 and is greater than, 8 so 4 is greater than 7. nother way to compare fractions is to convert them both to decimals. 4 converts to.7, and 7 converts to approimately CONVERTING FRCTIONS TO DECIMLS To convert a fraction to a decimal, divide the bottom into the top. To convert 8,divide 8 into, yielding REPETING DECIML To find a particular digit in a repeating decimal, note the number of digits in the cluster that repeats. Ifthere are digits in that cluster, then every nd digit is the same. If there are digits in that cluster, then every rd digit is the same. nd so on. For eample, the decimal equivalent of 7 is , which is best written.07. There are digits in the repeating cluster, so every rd digit is the same: 7. To find the 0th digit, look for the multiple of just less than 0 that s 48. The 48th digit is 7, and with the 49th digit the pattern repeats with 0. The 0th digit is.. IDENTIFYING THE PRTS ND THE WHOLE The key to solving most story problems involving fractions and percents is to identify the part and the whole. Usually you ll find the part associated with the verb is/are and the whole associated with the word of. In the sentence, Half of the boys are blonds, the whole is the boys ( of the boys ), and the part is the blonds ( are blonds ). 0 Percents. PERCENT FORMUL Whether you need to find the part, the whole, or the percent, use the same formula: Part = Percent Whole Eample: What is % of? Setup: Part =. Eample: Setup: is % of what number? =.0 Whole Eample: 4 is what percent of 9? Setup: 4 = Percent 9. PERCENT INCRESE ND DECRESE To increase a number by a percent, add the percent to 00%, convert to a decimal, and multiply. To increase 40 by %, add % to 00%, convert % to., and multiply by = FINDING THE ORIGINL WHOLE To find the original whole before a percent increase or decrease,set up an equation. Think of a % increase over as.. Eample: fter a % increase, the population was 9,46. What was the population before the increase? Setup:.0 = 9,46. COMBINED PERCENT INCRESE ND DECRESE To determine the combined effect of multiple percents increase and/or decrease, start with 00 and see what happens. Eample: price went up 0% one year, and the new price went up 0% the net year. What was the combined percent increase? Setup: First year: 00 + (0% of 00) = 0. Second year: 0 + (0% of 0) =. That s a combined % increase. 4 00 Key Math Concepts for the CT CT STUDY IDS Ratios, Proportions, and Rates 6. SETTING UP RTIO To find a ratio, put the number associated with the word of on top and the quantity associated with the word to on the bottom and reduce. The ratio of 0 oranges to apples is 0 which reduces to. 7. PRT-TO-PRT ND PRT-TO-WHOLE RTIOS If the parts add up to the whole, a part-to-part ratio can be turned into two part-to-whole ratios by putting each number in the original ratio over the sum of the numbers. Ifthe ratio of males to females is to, then the males-to-people ratio is + = + and the females-to-people ratio is =.Or, of all the people are female. 8. SOLVING PROPORTION To solve a proportion, cross multiply: 9. RTE = 4 4 = = 4 =.7 To solve a rates problem, use the units to keep things straight. Eample: Setup: If snow is falling at the rate of foot every 4 hours, how many inches of snow will fall in 7 hours? foot 4 hours = inches 7 hours inc 4 ho hes urs = inches 7 hours 4 = 7 = 40. VERGE RTE verage rate is not simply the average of the rates. verage per B = T T verage Speed = To T otal otal B tal distance otal time To find the average speed for 0 miles at 40 mph and 0 miles at 60 mph, don t just average the two speeds. First figure out the total distance and the total time. The total distance is = 40 miles. The times are hours for the first leg and hours for the second leg, or hours total. The average speed, then, is miles per hour. verages 4. VERGE FORMUL To find the average of a set of numbers, add them up and divide by the number of numbers. verage = S um of the terms Number of terms To find the average of the five numbers,,, 40, and 40, first add them: = 0. Then divide the sum by : 0 = VERGE OF EVENLY SPCED NUMBERS To find the average of evenly spaced numbers, just average the smallest and the largest. The average of all the integers from through 77 is the same as the average of and = 9 0 = 4 4. USING THE VERGE TO FIND THE SUM Sum = (verage) (Number of terms) If the average of ten numbers is 0, then they add up to 0 0, or FINDING THE MISSING NUMBER To find a missing number when you re given the average, use the sum. If the average of four numbers is 7, then the sum of those four numbers is 4 7, or 8. Suppose that three of the numbers are,, and 8. These numbers add up to 6 of that 8, which leaves for the fourth number. CT STUDY IDS 00 Key Math Concepts for the CT Possibilities and Probability 4. COUNTING THE POSSIBILITIES The fundamental counting principle: if there are m ways one event can happen and n ways a second event can happen, then there are m n ways for the two events to happen. For eample, with shirts and 7 pairs of pants to choose from, you can put together 7 = different outfits. 46. PROBBILITY Probability = If you have shirts in a drawer and 9 of them are white, the probability of picking a white shirt at 9 random is = 4.This probability can also be epressed as.7 or 7%. Powers and Roots Favorable outcomes Total possible outcomes 47. MULTIPLYING ND DIVIDING POWERS To multiply powers with the same base, add the eponents: To divide powers with the same base, subtract the eponents: y y 8 y 8 y. 48. RISING POWERS TO POWERS To raise a power to an eponent, multiply the eponents.( ) SIMPLIFYING SQURE ROOTS To simplify a square root, factor out the perfect squares under the radical, unsquare them and put the result in front DDING ND SUBTRCTING ROOTS You can add or subtract radical epressions only if the part under the radicals is the same. + =. MULTIPLYING ND DIVIDING ROOTS The product of square roots is equal to the square root of the product:.the quotient of square roots is equal to the square root of the quotient: 6. 6 lgebraic Epressions. EVLUTING N EXPRESSION To evaluate an algebraic epression, plug in the given values for the unknowns and calculate according to PEMDS. To find the value of 6 when =, plug in for : ( ) ( ) DDING ND SUBTRCTING MONOMILS To combine like terms, keep the variable part unchanged while adding or subtracting the coefficients.a a ( )a a 4. DDING ND SUBTRCTING POLYNOMILS To add or subtract polynomials, combine like terms. ( 7) ( ) = ( ) ( 7 ) = 9. MULTIPLYING MONOMILS To multiply monomials, multiply the coefficients and the variables separately. a a ( )(a a)6a. 6. MULTIPLYING BINOMILS FOIL To multiply binomials, use FOIL.To multiply ( + ) by ( + 4), first multiply the First terms:.net the Outer terms: 4 = 4.Then the Inner terms: =.nd finally the Last terms: 4 =. Then add and combine like terms: 00 Key Math Concepts for the CT CT STUDY IDS 7. MULTIPLYING OTHER POLYNOMILS FOIL works only when you want to multiply two binomials. If you want to multiply polynomials with more than two terms, make sure you multiply each term in the first polynomial by each term in the second. ( + + 4)( + ) = ( + ) + ( +) + 4( + ) = = Factoring lgebraic Epressions 8. FCTORING OUT COMMON DIVISOR factor common to all terms of a polynomial can be factored out.ll three terms in the polynomial 6 contain a factor of.pulling out the common factor yields ( 4 ). 9. FCTORING THE DIFFERENCE OF SQURES One of the test maker s favorite factorables is the difference of squares. a b = (a b)(a + b) 9, for eample, factors to ( )( ). 60. FCTORING THE SQURE OF BINOMIL Learn to recognize polynomials that are squares of binomials: a + ab + b = (a + b) a ab + b = (a b) For eample, 4 9 factors to ( ), and n 0n factors to (n ). 6. FCTORING OTHER POLYNOMILS FOIL IN REVERSE To factor a quadratic epression, think about what binomials you could use FOIL on to get that quadratic epression. To factor 6, think about what First terms will produce,what Last terms will produce +6, and what Outer and Inner terms will produce.common sense and trial and error lead you to ( )( ). 6. SIMPLIFYING N LGEBRIC FRCTION Simplifying an algebraic fraction is a lot like simplifying a numerical fraction. The general idea is to find factors common to the numerator and denominator and cancel them. Thus, simplifying an algebraic fraction begins with factoring. To simplify first factor the numerator and 9 4) ( + ) denominator: 9 = ( ( ) ( + ) Canceling + from the numerator and denominator leaves you with 4. Solving Equations 6. SOLVING LINER EQUTION To solve an equation, do whatever is necessary to both sides to isolate the variable.to solve 9, first get all the s on one side by adding to both sides: 7 9. Then add to both sides: 7, then divide both sides by 7 to get: =. 64. SOLVING IN TERMS OF To solve an equation for one variable in terms of another means to isolate the one variable on one side of the equation, leaving an epression containing the other variable on the other side. To solve 0y 6y for in terms of y, isolate : 0y = + 6y + = 6y + 0y 8 = 6y = y 6. TRNSLTING FROM ENGLISH INTO LGEBR To translate from English into algebra, look for the key words and systematically turn phrases into algebraic epressions and sentences into equations. Be careful about order, especially when subtraction is called for. 7 CT STUDY IDS 00 Key Math Concepts for the CT 8 Eample: Setup: The charge for a phone call is r cents for the first minutes and s cents for each minute thereafter. What is the cost, in cents, of a call lasting eactly t minutes? (t ) The charge begins with r, and then something more is added, depending on the length of the call. The amount added is s times the number of minutes past minutes. If the total number of minutes is t, then the number of minutes past is t. So the charge is r + s(t ). Intermediate lgebra 66. SOLVING QUDRTIC EQUTION To solve a quadratic equation, put it in the a b c 0 form, factor the left side (if you can), and set each factor equal to 0 separately to get the two solutions. To solve 7, first rewrite it as 7 0. Then factor the left side: ( )( 4) = 0 = 0 or 4 = 0 = or 4 Sometimes the left side might not be obviously factorable. You can always use the quadratic formula. Just plug in the coefficients a, b, and c from a b c 0 into the formula: b ± b 4 ac a To solve 4 0, plug a =, b = 4, and c = into the formula: = 4 ± 8 = = ± 67. SOLVING SYSTEM OF EQUTIONS You can solve for two variables only if you have two distinct equations. Two forms of the same equation will not be adequate. Combine the equations in such a way that one of the variables cancels out.to solve the two equations 4 y 8 and y, multiply both sides of the second equation by to get: y 9. Now add the equations; the y and the y cancel out, leaving: =. Plug that back into either one of the original equations and you ll find that y = SOLVING N EQUTION THT INCLUDES BSOLUTE VLUE SIGNS To solve an equation that includes absolute value signs, think about the two different cases. For eample, to solve the equation, think of it as two equations: = or = = or SOLVING N INEQULITY To solve an inequality, do whatever is necessary to both sides to isolate the variable. Just remember that when you multiply or divide both sides by a negative number, you must reverse the sign. To solve + 7 , subtract 7 from both sides to get: 0. Now divide both sides by, remembering to reverse the sign: . 70. GRPHING INEQULITIES To graph a range of values, use a thick, black line over the number line, and at the end(s) of the range, use a solid circle if the point is included or an open circle if the point is not included.the figure here shows the graph of . 0 4 Coordinate Geometry 7. FINDING THE DISTNCE BETWEEN TWO POINTS To find the distance between points, use the Pythagorean theorem or special right triangles. The difference between the s is one leg and the difference between the y s is the other leg. 00 Key Math Concepts for the CT CT STUDY IDS y 74. USING N EQUTION TO FIND N INTERCEPT P (, ) O To find the y-intercept, you can either put the equation into y = m + b (slope-intercept) form in which case b is the y-intercept or you can just plug = 0 into the equation and solve for y.to find the -intercept, plug y = 0 into the equation and solve for. In the figure above, PQ is the hypotenuse of a -4- triangle, so PQ =. You can also use the distance formula: d = ( ) + (y y ) To find the distance between R(, 6) and S(, ): d = ( ) = () + ( 6) + ( 8) = 68 = 7 7. USING TWO POINTS TO FIND THE S
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