cbest: FRACTIONS

prime numbers

a number that can be evenly divided by only itself and one. for example, 19 is a prime number because it can be evenly divided by 19 and 1, but 21 is not a prime number because 21 can be evenly divided by other numbers (3 and 7). the only even prime number is 2; thereafter any even number can be divided evenly by 2. zero and 1 are not prime number. the first ten numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

adding mixed numbers

the first rule (find the LCD) applies, but make sure that you always add the whole numbers to get your final answer. ex: 2 1/2 =2 2/4 because we multiplied 1/2 by 2 to have the same denominator as 3 1/4. so we have 2 2/4 + 3 1/4 = 5 3/4

order of operation

there is an order in which the operations on numbers must be done so that everyone doing a problem involving several operations and parentheses will get the same results. the order of operations is: step 1: parentheses- changing signs to their opposites. simplify (if possible) all expressions in parentheses. step 2: exponents - apply exponents to their appropriate bases. step 3. multiplication or division- do the multiplication or division in the order that it appears as you read the problem left to right. step 4. addition or subtraction - do the addition/subtraction in the order it appears as you read the problem left to right. easy way to remember is PEMDAS.

adding and subtracting decimals pg. 165-166

to add or subtract, just line up the decimal points and then add or subtract in the same manner you would add or subtract regular numbers. adding zeros can make the problem easier when you line them up. for example 23.6 can be 23.600 to evenly line up with 1.750 this can only be done to the right of the decimal. you can also add decimal places to a number that doesn't have a decimal. Ex: 17 can be 17.00 and then if you need to borrow to subtract 17.00 - 8.43=8.57. you can do so by turning one of the zeros from 17.00 into a 10 which would then turn the 0 to the left into a 9 because we are borrowing and then the 7 would turn into a 6.

changing decimals to fractions

to change a decimal to a fraction: 1. read it: 0.8 (eight-thenths) 2. write it: 8/10 3. reduce it: 4/5 or 1. read it: 0.028 (twenty-eight thousands) 2. write it: 28/1000 3. reduce it: 7/250

changing fractions to decimals

to change a fraction to a decimal, simply do what the operation says. in other words, 13/20 means 13 divided by 20, so do just that (insert decimal points and zeros accordingly). example on page 167.

changing fractions to percents

to change a fraction to percent: 1. multiply by 100 2. divide by denominator 3. insert a percent sign 1/2=1/2 x 100=100/2=50% 2/5=2/5 x 100=200/5=40%

changing percents to decimals

to change a percent to decimals: 1. eliminate the percent sign. 2. move the decimal point two places to the left (sometimes adding zeros is necessary). 75%=.75 5%=.05 23%=.23 .2%=.002

changing decimals to percents

to change decimals to percents: 1. move the decimal point two places to the right. 2. inset a percent sign. .75=75% .05=5%

changing percents to fractions

to change percents to fractions: 1. eliminate the percent sign. 2. place the number without the percent sign over 100. (i.e., divide the percent by 100). 3. reduce if necessary 60%= 60/100 =3/5 13%= 13/100

reducing fractions

Fraction must be reduced to its lowest terms. do this by dividing the numerator and denominator by the largest number that divides evenly into both. ex: reduce 14/16 by dividing both terms by 2 to get 7/8. likewise, reduce 20/25 to 4/5 by dividing both numerator and denominator by 5.

inequalities

a statement in which the relationships are not equal. instead of using an equal sign as in an equation, an inequality uses > (greater than) and < (less than) or the greater than or equal to symbol. and the less than or equal to symbol. when working with inequalities, treat them exactly like equations, except: if you multiply or divide both side by a negative number, you must reverse the direction of the sign. Ex: solve for x: 2x + 4 > 6 this will equal x>1

equations

an equation is a relationship between numbers and/or symbols. it helps to remember that an equation is like a balance scale, with the equal sign serving as the center. thus, if you have the same thing on the both sides if the equal sign (say , add 5 to each side), the equation stays balanced. to solve the equation x-5=23, you must get x by itself on one side; therefore, add 5 to both sides. sometimes you may have to use more than one step to solve for an unknown.

proportions

are two equivalents ratios or fractions. proportions are quickly solved by cross-multiplication. Ex: solve for x 3/x=5/7 cross multiply and get 5x=21, then 21/5=x or 4 1/5. this problem could have also have been presented in written form as "3 is to x as 5 is to 7; find the value of x": we would have. example #2: solve for x: p/q=x/y xq=py x=py/q

common fractions/improper fractions

common fraction: a fraction whose numerator is smaller than its denominator. all common fractions have a value that is less than one. 3/5 is a common fraction. improper fraction: a fraction whose numerator is the same or more than the denominator. all improper fractions have values equal to one or more than one. ex: 6/6=1, 5/4=1 1/4.

fractions

consist of two numbers: a numerator (number on top) and a denominator (number on the bottom). numerator/denominator.

mixed numbers

contains both a whole number and a fraction. ex: 5 1/4 and 290 3/4. to change an improper fraction to a mixed number, divide the denominator into the numerator. ex: 18/5=3 3/5 with the house division the denominator would go outside the house and the 18 inside the house. to change a mixed number to an improper fraction, multiply the denominator times the whole number, add in the numerator, and put the total over the original denominator. ex: 4 1/2= 9/2 2x4+1= 9

subtracting mixed numbers

find the LCD, except you now subtract the numerators. 7/8 - 1/4 we multiply 1/4 by 2 to get 2/8, so we have 7/8 - 2/8 = 5/8. example #2: 3/4 - 1/3 =we multiply 3/4 by 3 and 1/3 by 4. we get 9/12 - 4/12 = 5/12

adding fractions

first change all denominators to least common multiple (LCM) or lowest common denominator (LCD) - the lowest number that can be divided evenly by all the denominators in the problem. when you make all the denominators the same, you can add fractions by simply adding the numerators (the denominator remains the same). one way to find this value is to make a list of the multiples for the values involved and then find the least common one. ex: find LCM for 24 and 36. multiples of 24: 24, 48, 72, 96, 120, 144. multiples of 36: 36, 72, 108, 144, 180, 216. notice that 72 and 144 are both common multiples, but that 72 is the least common multiple. now apply this to adding of fractions. ex: 5/24 + 7/36 = as we saw above, the (LCD) for 24 and 36 is 72. 5/24 = 15/72, since 24 is multiplied by 3 to get 72, the 5 is also multiplied by 3. 7/36 = 14/72, since 36 is multiplied by 2 to get 72, the 7 is also multiplied by 2. Now that the denominators are the same, add the numerators and keep the denominator 29/72. if the denominators are originally the same then you would simply add the fractions. ex: 6/11 + 3/11 = 9/11

adding and subtracting monomials

follow the rules as with regular signed numbers, provided that the terms are alike. ex: 3x+2x=5x

dividing fractions pg.165

if either numerator or denominator consists of several numbers, combine them into one number. then reduce if necessary. Ex: 28+14/26+17= 42/43 or 1/4+1/2 /1/3+1/4= 1/4+2/4/4/12+3/12= 3/4/7/12= 3/4 x 12/7=36/28 =9/7=1 2/7

numerator

indicates how many of these equal parts are contained in the fraction. thus, if the fraction is 3/5 of a pie, then the denominator 5 indicates that the pie is divided into 5 equal parts, of which 3 (numerator) are in the fraction.

denominator

indicates the number of equal parts into which something is divided.

diving decimals

it is the same as dividing other numbers, except the divisor (the number you're dividing by) has a decimal, you move to the right as many places as necessary until it is a whole number. then move the decimal point in the divided (the number being divided into) the same number of places. sometimes you may have to add zeros to the dividend (the number inside the division sign). example on page 166.

monomials and polynomials

monomial (mono means one) us an algebraic expression that consists of only one term (meaning mathematical expression and it consists of one number and one variable). polynomial consist of two or more terms. meaning that it has a combination of many terms. each series of terms in polynomial must be joined by addition or subtraction. look at page 175 for reference. other terms binomial (means 2) trinomial (means 3), beyond 3 terms it is usually called polynomial at times even when there is just 2. remember if x is alone it counts as 1. if there is a number with no variable imagine there is an x to the power of 0 since it would have dramatic effect we could simply add it to help us solve the problem. review math antics video.

signed numbers (positive numbers and negative numbers)

on a number line, numbers to the right of 0 are positive. numbers to the left of 0 are negative. given any two numbers on a number line, the one on the right is always larger, regardless of its sign (positive or negative).

parentheses

parentheses are used to group numbers. everything inside parentheses must be done before any other operations. Ex: 50(2 + 6) = 50 (8)= 400. when parentheses is preceded by a minus sign, change the minus to a plus by changing all the signs in front of each term inside the parentheses. then remove the parentheses. Ex: 6-(-3 + a - 2b + c) = 6+(+3 -a + 2b - c)= 6 + 3 - a + 2b - c = 9 - a + 2b -c

multiplying fractions

simply multiply the numerator, then multiply the denominators. reduce to lowest terms if necessary. Ex: 2/3 x 5/12 = 10/36 reduce 10/26 to 5/18. this answer had to be reduced as it wasn't in lowest terms. canceling when multiplying fractions: you can first "cancel" the fractions. that eliminates the need to reduce your answer. to cancel, find a number that divides evenly into one numerator and one denominator. in this case, 2 divides evenly into 2 in the numerator (it goes in one time) and into 12 in the denominator (it goes in 6 times). thus: 2/3 x 5/12 = 5/18, after you reduce 2/3 to 1/3 x 5/6 after reducing 5/12. remember, you may cancel only when multiplying fractions.

subtracting mixed numbers

sometimes you may have to "borrow" from the whole number, just like you sometimes borrow from the next column when subtracting numbers. ex: 651-129=522. the 1 on 651 turns to 11 because it borrow 1 from the 5 next to it so 651 turns to 64(11). 4 1/6 - 2 5/6= 4 1/6 turns into 3 7/6 because you borrow one, in the form 6/6, from the 1st column. so the answer is 1 2/6 = 1 1/3. we need to borrow from the whole number of 4 1/6 to subtract to 2 5/6, so we break it down like this 3+1+1 1/6 = 3 +6/6 + 1/6 = 3 + 7/6 so its 3 7/6 - 2 5/6 = 1 2/6 = 1 1/3. another example 6 turns into 5 5/5 - 3 1/5 = 2 4/5.

finding a percent of a number

to determine a percent of a number, change the percent to a fraction or decimal (whichever is easier for you) and multiply. remember, the word "of" means multiply. Ex: what is 20% of 80? 20/100 x 80= 1600/100= 16 or .20 x 80= 16.00=16 what is 12% of 50? 12/100 x 50=600/100=6 or .12 x 50=6.00=6 what is 1/2% of 18? 1/2/100 x 18=1/200 x 18 or 18/200= 9/100 or .005 x 18=.09

evaluating expressions

to evaluate an expression, evaluate, replace, or substitute the numerical value in the variable and simplify the expression using order of operations. Ex: evaluate 2x to the power of 2 +3y+6 if x=2 and y=9 2(2 to the power of 2)+3(9)+6= 2(4)+27+6= 8+27+6=41

finding the percentage increase or percentage decrease

to find the percentage change (increase or decrease), use this formula: change/starting point x 100 = percentage change Ex: what is the percentage decrease of a $500 item on sale for $400? change 500-400= 100 change/starting point x 100= 100/500 x 100= 1/5 x 100=20% decrease. what is the percentage increase of jon's salary if it went from $150 a month to $200 a month? change: 200-150= 50 change/starting point x 100= 50/150 x 100= 1/3 x 100= 33 1/3%n increase

multiplying decimals

to multiply decimals, just multiply as usual. then count the total number of digits above the line and to the right of all the decimal points. place the decimal point in your answer so that there are the same number of digits to the right of it as there are above the line. example on page 166.

multiplying mixed numbers

to multiply mixed numbers, first change any mixed number to an improper fraction. then multiply as shown in the preceding section. to change mixed numbers to improper fractions: 1. multiply the whole number by the denominator of the fraction. 2. add this to the numerator of the fraction. 3. this is now your numerator. 4. the denominator remains the same. Ex: 3 1/3 x 2 1/4 = 10/3 x 9/4 = 90/12 = 7 6/12 = 7 1/2. then change the answer, if in improper fraction form, back to a mixed number and reduced if necessary.

multiplying and dividing signed numbers

to multiply or divide signed numbers, treat them just like regular numbers, but remember, but remember this rule: an odd number of negative signs produces a negative answer. an even number of negative signs produces a positive answer. Ex: (-3)(+8)(-5)(-1)(-2) = +240 divide both side by -5 (-3)(+8)(-1)(-2)=-48 divide both sides by -2 -6/-2=+32 -64/+2= -32

subtraction of signed numbers

to subtract positive or negative numbers, just change the sign of the number being subtracted and then add. still keep the sign of the larger number.

other applications of percent

turns the question word-for-word into a question. for "what" substitute x; for "is" substitute an equal sign; for "of" substitute a multiplication sign. change percents to decimals of fractions, whichever you find easier. then solve the equation. for example: 18 is what percent of 90? 18= x(90) 18/90=x 1/5=x 20%=x 10 is 50% of what number? 10=.50(x) 10/.50=x 20=x what is 15% of 60? x= 15/100 x 60= 90/10= 9 or .15(60)= 9

squares

two square a number, just multiply by itself. Ex: 6 squared is 36. 356 is called a perfect square (the square of a whole number). any exponent means multiply by itself that many times. Ex: 5 squared = 5 x 5= 25 8 squared = 8 x 8-= 64 remember, x to the power of 1 = x and to the power of 0= 1, when x is any number (other than 0). list of perfect squares: 1= 1, 2= 4, 3 =9, 4 =16, 5 =25, 6 =36, 7 =49, 8 =64, 9 =81, 10 =100, 11 = 121, 12 =144

addition of signed numbers

when adding two numbers with the same sign ( either both positive or both negative), add the numbers and keep the same sign. when adding two numbers with different signs (one positive and one negative), subtract the numbers and keep the sign from the larger number.

representation of multiplication

when two or more letter, or a number and letter, are written next to each other, they are understood to be multiplied. thus, 8x means 8 times x. or ab means a times b. or 18ab means 18 times a times b. parentheses or the dot on the air between two numbers also represents a multiplication.

deimals

you can write fractions in decimal forms by using a symbol called a decimal point. all numbers to the left of the decimal point are whole numbers. all numbers to the right of the decimal point are fractions with denominators of only 10, 100, 1,000, 10,000, and so on, and so forth, as follows: .6=6/10=3/5 .7=7/10 .07=7/100 .007=7/1,000 .0007=7/10,000 .00007=7/100,000 .25=25/100=1/4