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Fractions, decimals, and integers are not examples of whole numbers, rational numbers, and natural numbers. Numbers divisible by 2 are even numbers. All others are odd numbers. The absolute value of a number is always a positive number since absolute value measures distance. That is, the distance the particular number is from 0 on the number line. Integers are the set of whole numbers and their opposites Dividing a number by a fraction less than one increases the number. Example for solving a simple equation: 3x + 2 = 17 3x + 2-2 = 17-2 3x=15 3x/3 = 15/3 x = 5 (x) (y) is the correct way to write the product of x and y. The average of a group of numbers is called the mean. The middle number of the group is called the median. The number that appears the most often in a listing of numbers is the mode. Consecutive numbers are 1 number apart. Take any number n and add 1 to it and you have the next number n+1. Example: the sum of two consecutive numbers is 39. What are the numbers? Let n = the first number and let n+1 equal the second number. Then n+(n+1)=39. 2n+1 =39 2n+1-1+39-1 2n=38 n=19. The two numbers are 19 and 20. An equation contains terms and has an equal sign. (An expression contains terms but no equal sign.) The area of a rectangle is length times width. A=lw. To find a missing length or width given one side and the area: A/l=w. Area divided by length equals width. A/w=l. Area divided by width equals length. The perimeter of a rectangle is equal to the sum of lengths of the four sides, (or 2 times the length + 2 times the width.) The area of a square is equal to the side squared. A = s 2 o Example 1: given the area of a room equals 208 ft 2 and one side is 13 ft., what is the other side? A=lw 208 ft 2 = 13 ft. X w. 208 ft 2 / 13 ft.= 16 ft. o Example 2: What is the perimeter of the above rectangle? The perimeter, the distance round the rectangle, is 2 times 13 ft. + 2 times 16 ft. 26 ft. + 32 ft. = 58 ft. o Example 3: Given the area of a square is 8,100 ft 2 What us the measure of each side? 8,100 ft 2. = s 2 8100 ft 2 = s 900=s The radius is one half of the diameter or a line that connects the center to any part of the circle. D=2r, or r=d/2 o Example: If the diameter of a circular stadium was 500 yards, what was the radius? ½ of 500 yards is 250 yards. Formulas: d = rate X time or d = rt R = d t t = d r 2
A race car driver can drive at the rate of 120 miles an hour. How far will he get in (a) 3 hours (b) 1 ½ hours (c) 10 minutes. 1 hour = 120 ml; 3 hours = 120 X (3) = 360 mi 1 ½ hrs = 120 + 60 = 180 mi 10 min = 10 (120) or fractional part of an hour times 120 60 1200 60 = 20 mi There are 3 parts to these problems: Principal, Interest, and Amount. I = P x R x T A = P + I I = A P P = A I A man borrowed a small loan of $500 for 1 year. The bank charged him 5%. What is the interest for the use of this money and what will be the total amount repaid? I = P R T = 500.05 I = $25; A = P + I = $500 + $25 = $525; A = $525 What is profit? Profit is the excess between the money you make or earn and the money you spend. You must make more than you spend to make a profit; if not you experience a loss. The formula is P = Earning Spending. If the answer is positive you make a profit; if the answer is negative, you make a loss. In business terms P = Revenue Expenditure or P = R - E Example: You start a lawn mowing business for three months during the summer. Your expenditures of gas and oil for the mower is $25 for each of the 3 months. Your revenue was $50 for the first month $85 total for the last two months. Did you make a profit? Expenditures are 3 X $25 = $75. Revenue is $50 + $85 = $135. P = $135 $75 = $60. Yes, you made a profit. Similarity refers to objects that have the same shape but different size. Similar objects are dilations, that is, they are either smaller or larger than each other and can mathematically be expressed as a ratio. Example Given a set of similar triangles with one being ½ of the other. If one triangle has sides of 10, 20, 30 cm. the dimensions of other are 20, 40 50 cm. False, because all of the dimension of the second triangle are not twice that of the first, It took 20 hours to get from Miami to Beijing. How many seconds was that? Think 1 minute = 60 seconds; 60 minutes = 1 hour. 60 sec/min X 60 min/hr X 20 hrs = 72,000 sec. How many meters are there in 19 kilometers? Think 1km = 1000m, so 19 km = 19 000 km Triangles are congruent if each of the 3 sides is congruent to a corresponding side of another triangle. We say SSS (side, side, side) as the reason for congruency In other words, three sides in one triangle must be identical to three sides in another triangle. SAS (Side-Angle-Side): Two triangles are congruent if a pair of corresponding sides and the included angle are equal Example: An isosceles triangle had two sides 18cm long. The base angles were 50º. 3
Another isosceles had two sides 8 cm long with the angle in between 50º. Are they congruent? No. Although the included angles are congruent the sides are not since one triangle has two sides 18 cm long and the other has two sides 8 cm long. It does not matter how the triangles are labeled. As long as the angles or measures are the same the triangles are the same (a) true (b) false We say SSS (side, side, side) as the reason for congruency. In this case corresponding means similar location. (ASA) means angle, side in between, angle. (SAS) means side-angle between-side. The X-coordinate always comes first followed by the Y-coordinate. Slope = Vertical change (rise) of Y values Y 2 Y 1 Horizontal change (run) of X values X 2 X 1 When the vertical change (Y 2 Y 1 ) is a drop it is a decrease so the slope is negative or downward. In a business graph, when the fraction (slope) is positive, the line is ascending or going up; the trend or change is increasing. Lines 1 and 2 show increasing trends or change. If you graphed a line of business profits and it looked like this, the business is doing well. When the numerator is zero, the line is horizontal, an indication of no change. This is a stagnant business indicator, a bad sign for a business. When the denominator is zero, the line has no slope and therefore is vertical. Remember, if the denominator is zero, a zero denominator is undefined. The equation of a line should always be Y = mx + b not 2 times mx. We can only add or subtract like terms. x, y, z, w are not like terms. x + x + x can be added and will result in 3x. x +x +y +y will result in 2x +2y. (Sometimes this is referred to as collect like terms then add, subtract, multiply, or divide.) 4
First get x (or whatever letter the question uses) on the left hand side of the equals sign, by itself. Then solve the equations by balancing: whatever we do to one side of an equation, we must do the same to the other side. So if we add 4 to the left hand side, we must add 4 to the right hand side as well. If we multiply on the left side by 2, we multiply on the right side by 2 as well. Example: If 5x+5=15 solve for the value of x. o Solution: 5x+5-5=25-5 (Whatever we do to one side of the equation we must do to the other side in order maintain the balance or equality. 5x=105x/5=10/5. x=2. Example: 5z 10=30. Multiply each side of the equation by 10. 10 x 5z=30 x 10 10 5z=300 z=60 Square roots are the reverse of squares. is a square root symbol also called a radical sign. 4 reads square root of 4. Finding the square root means finding the number that was multiplied by itself to get the square. The square root of 36 equals 6. When the square is a small number we can do it mentally. If the square is a big number we can use a prime factor tree to find the root The absolute value of any number is the distance between that number and zero on -7 = 5 + -12 Thus the number line. Since it is distance it is always positive. What is profit? Profit is the excess between the money you make or earn and the money you spend. You must make more than you spend to make a profit; if not you experience a loss. The formula is P = Earning Spending. If the answer is positive you make a profit; if the answer is negative, you make a loss. In business terms P = Revenue Expenditure or P = R - E Example: You start a lawn mowing business for three months during the summer. Your expenditures of gas and oil for the mower is $25 for each of the 3 months. Your revenue was $50 for the first month $85 total for the last two months. Did you make a profit? Expenditures are 3 X $25 = $75. Revenue is $50 + $85 = $135. P = $135 $75 = $60. Yes, you made a profit. Commutative property of multiplication states that ab=ba. The associative property of multiplication states that (ab)c=a(bc). Subtracting consecutive odd numbers always results in 2. Let n =first odd number and n+2 =the next odd number. Then, evaluating (n+2)-n, you end up with 2. 10y +3z +9 +(5y +4z +7) = Solution: collect like terms.(10y+5y +3z+4z +9 +7) = 15y + 7z + 16. (y + 2) (y 4) = y 2-2y -8. Use the FOIL method. FOIL means first outside inside last. Multiply the outside numbers: y times y, and 2 times -4; then multiply the insides 2 by y times 2. We say multiply the extremes, multiply the means (middle numbers), then add the means 5
A spinner had these numbers on it; two 5s 3, 4, 1, 2. What is the probability of getting a 5? The answer would be 2 out of 6. There are 6 total chances and there is 2 chances of getting a 5. We may express the probability as the fraction 2/6 or 1/3; to a decimal as.3333 or as a percent 33.33% Use the same spinner as above. How many chances are there to get a 2? Express the answer as a percent. There are 6 total chances and there is 1 chances of getting a 2. We may express this as 1 chance out 6. The percentage is 1/6 which is equal to 16.67% A deck of cards has 13 each of clubs, hearts, diamonds, and spades; 4 kings (one of each suit); 4 aces (one of each suit); 4 queens (one of each suit), 4 jacks (one of each suit), and 4 of each number from 2 to 10. What is the probability of selecting a king? There are 4 kings (one of each suit) out of 52 cards. The p(4 kings from a deck of 52 cards) = 4/52 or 1/13 Each die (a cube) has 6 faces. On each face are small dots: either one dot, two dots, three dots, 4 dots, five dots, or six dots. Two or more of these cubes are called dice. They are picked up, shaken, and then rolled onto a surface. What is the probability of rolling an odd number on a die? Since a die has 6 numbers with 1 to 6 on each face, there will be just 3 odd numbers on it. The p (odd numbers on the roll of a die) = 3 / 6 or ½ When flipping a coin, there's only one way you can land on heads. There are two possible outcomes, heads or tails, so the probability of getting heads is 1/2, or 0.5, or 50%, depending on how you want to state the probability. To get the probability of two independent events, you multiply. For example, if you want to calculate the probability of flipping a coin twice and it landing on heads twice, you would multiple 1/2 by 1/2, or 50% by 50%. To get the probability of four independent events, you multiply ½ x ½ x ½ x. ½. How many combinations of 4 pizza toppings can I get from a 5 choices? C (5,4) = 5! = 5 4!(5-4)! Example: In a bag of marbles there were 4 red, 3 blue, 5 green, 3 black, and 5 orange What are my chances of taking a red marble out of the bag? p (red marble out of bag) = 4/20 or 1/5 or 20% Example: In a bag of marbles there were 4 red, 3 blue, 5 green, 3 black, and 5 orange 8. What are my chances of picking an orange marble out of the bag? p(orange out of bag) =5/20 or ¼ or 25%, Example: In a bag of marbles there were 4 red, 3 blue, 5 green, 3 black, and 5 orange.9. What are my chances of picking a black marble out of the bag? p(black out of bag) = 3/20 or 15%. 6
How many combinations of 4 things can you make using 3 at a time? This problem can be worked by using combination theory C(n, k) = n! k!(n-k)!. C( 4,3) = 4! = 4 x 3 x 2 x 1 = 4 3!(4-3)! (3 x 2 x 1)(1) 7
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