Cronasia GE Math Topics PDF

SELECTED TOPICS IN GENERAL EDUCATION: MATHEMATICS _________________________________________________________________________________________________________ I. Order of Operations P – Parenthesis E – Exponents MD – Multiply and Divide from left to right AS – Add and Subtract from left...

SELECTED TOPICS IN GENERAL EDUCATION: MATHEMATICS _________________________________________________________________________________________________________ I. Order of Operations P – Parenthesis E – Exponents MD – Multiply and Divide from left to right AS – Add and Subtract from left to right Examples: 𝟑 + (𝟓 × 𝟐) ÷ 𝟐 + 𝟏𝟎 = 3 + 10 ÷ 2 + 10 = 3 + 5 + 10 = 8 + 10 = 𝟏𝟖 𝟖𝟐 − 𝟑𝟐 × (𝟏𝟓 − 𝟏𝟐)𝟐 = 82 − 32 × 32 = 82 − 9 × 9 = 82 − 81 =𝟏 𝟓 + 𝟑 × 𝟒𝟗 − 𝟕 × 𝟐𝟓𝟖 × 𝟎 = 5 + 3 × 49 − 7 × 258 × 0 = 5 + 147 − 7 × 258 × 0 = 5 + 147 − 0 = 𝟏𝟓𝟐 II. Prime Factorization Prime Factorization - It is the process of breaking down a number into factors until all the factors are prime numbers. Prime Number - It is a counting number greater than 1 whose factors are only 1 and itself. Examples: 40 4 10 2 X 2 X 2 X 5 Therefore, the prime factors are 2, 2, 2 and 5. 24 4 6 2 X 2 X 2 X 3 Therefore, the prime factors are 2, 2, 2 and 3. Trial-and-Error Method Two conditions must be met: 1. Multiply the numbers in each option, then check if the product is equal to the given number. 2. Check if the numbers multiplied resulting to the given number are prime numbers.  SELECTED TOPICS IN GENERAL EDUCATION: MATHEMATICS Examples: What are the prime factors of 225? A. 5 x 5 x 3 x 3 C. 3 x 3 x 5 B. 5 x 3 x 4 x 2 D. 9 x 5 x 5 Rationalization: A. 5 x 5 x 3 x 3 = 225 so this is the correct answer. B. 5 x 3 x 4 x 2  4 is not a prime number. C. 3 x 3 x 5 = 45 which is not 225 D. 9 x 5 x 5  9 is not a prime number. What are the prime factors of 90? A. 2 x 15 x 3 C. 2 x 2 x 2 x 3 B. 3 x 3 x 3 x 2 D. 3 x 3 x 5 x 2 Rationalization: A. 2 x 15 x 3  15 is not a prime number. B. 3 x 3 x 3 x 2 = 54 which is not 90 C. 2 x 2 x 2 x 3 = 24 which is not 24 D. 3 x 3 x 5 x 2 = 90 so this is the correct answer. III. Least Common Multiple (LCM) It is the LEAST number that can be divided by all of the numbers given in a set. Technique (BOTTOM TO TOP): Find the least number in the options (bottom) that is divisible by all of the numbers in the stem (top). Clues:  Least number  Same  Lowest number  Together  Smallest number  Both Examples: What is the least common multiple of 24 and 80? A. 360 C. 240 B. 60 D. 480 Ronald and Tim both did their laundry today. Ronald does laundry every 6 days and Tim does laundry every 9 days. How many days will it be until Ronald and Tim both do laundry on the same day again? A. 12 C. 36 B. 18 D. 72 IV. Greatest Common Factor (GCF) It is the GREATEST number that divides all of the numbers given in a set. Technique (TOP TO BOTTOM): Divide the numbers in the stem (top) by the greatest number in the options (bottom). Clues:  Greatest number  Highest number  Biggest number  Largest number  SELECTED TOPICS IN GENERAL EDUCATION: MATHEMATICS Examples: What is the greatest common factor of 18, 54, and 90? A. 2 C. 9 B. 3 D. 18 Madame Irene has 18 boys and 27 girls in choir. She wants them to stand in equal rows. Only boys or girls will be in each row. What is the greatest number of students that can stand in each row? A. 18 C. 3 B. 6 D. 9 V. Ratio and Proportion A ratio is a comparison of two quantities by division. A is to B – in words A:B – in colon form A/B – in fraction form To reduce the ratio to its lowest term is to divide the quantities by their GCF. Examples: In Velois store, a certain handkerchief is sold at 8 dollars while a belt is sold at 2 dollars. What is the ratio of belt to handkerchief? Answer should be in a simplified form. A. 8/2 C. 4/1 B. 2/8 D. 1/4 Solution: belt 2 2÷2 𝟏 = = = 𝐨𝐫 𝟏: 𝟒(𝐃) handkerchief 8 8 ÷ 2 𝟒 There are 15 boys and 33 girls in the class of III-Chrysoberyl. What is the ratio of the number of girls to the number of boys? A. 5:11 C. 11:16 B. 11:5 D. 16:11 Solution: girls 33 33 ÷ 3 𝟏𝟏 = = = 𝐨𝐫 𝟏𝟏: 𝟓 (𝐁) boys 15 15 ÷ 3 𝟓 A proportion is an equality of two ratios. 6 8 = 9 12 A. Direct Proportion - As one quantity increases, the other quantity also increases. - As one quantity decreases, the other quantity also decreases. 𝐱𝟏 𝐱𝟐 Formula: = 𝐲𝟏 𝐲𝟐 Example: A syrup is made by dissolving 2 cups of sugar in 3 cups of boiling water. How many cups of sugar should be used for 6 cups of boiling water? A. 3 C. 5 B. 4 D. 6  SELECTED TOPICS IN GENERAL EDUCATION: MATHEMATICS Solution: x1 x2 = y1 y2 2 x = Cross-multiply. 3 6 2(6) = 3x 12 = 3x 12 3x = Divide both sides by 3. 3 3 𝐱=𝟒 B. Inverse Proportion - As one quantity increases, the other quantity decreases and vice versa. Formula: (𝐱 𝟏 )(𝐲𝟏 ) = (𝐱 𝟐 )(𝐲𝟐 ) Example: Four pipes can fill a tank in 20 minutes. How long will it take to fill the tank with 8 pipes open? A. 10 mins. C. 30 mins. B. 20 mins. D. 40 mins. Solution: (𝐱 𝟏 )(𝐲𝟏 ) = (𝐱 𝟐 )(𝐲𝟐 ) 4(20) = 8y Multiply. 80 = 8y 80 8y = Divide both sides by 8. 8 8 𝐲 = 𝟏𝟎 C. Partitive proportion - One quantity is being divided into unequal parts. Example: The ratio of the three sides of a triangle is 1:2:3. What are the measurements of each side if the perimeter of the triangle is 120cm? A. 30, 30, 60 C. 20, 40, 60 B. 60, 60, 30 D. 10, 50, 60 Given: Ratio: 1:2:3 1x = 1st side; 2x = 2nd side; 3x = 3rd side Total: 120 cm Solution: 1x + 2x + 3x = 120 6x = 120 6x 120 = 6 6 x = 20  Not the final answer First side Second side Third side = 1x = 2x = 3x = 1(20) = 2(20) = 3(20) = 20 = 40 = 60  SELECTED TOPICS IN GENERAL EDUCATION: MATHEMATICS VI. Percent - meaning “per hundred” Conversion Techniques Percent to Decimal Decimal to Percent A. Percent to Decimal To convert a percent to decimal, we remove the percent symbol and move the decimal point two places to the left. 55%  55  0.55 13.2%  13.2  0.132 B. Decimal to Percent To convert decimal to percent, we move the decimal point two places to the right and affix the percent symbol. 0.345  34.5  34.5% 25.5  2550  2550% VII. Percentage, Rate and Base Definitions: Percentage (P) – a part of the whole – “is” – “what is” Base (B) – the whole – “of” – “of what number” Rate (R) – the number in percent (%) – “what percent” Formulas: 𝐏 = 𝐁 × 𝐑 𝐏 𝐁= 𝐑 𝐏 𝐑= 𝐁 Sample Problem on Percentage What is the 20% of 700? A. 140 C. 162 B. 217 D. 210 Required: P=?  SELECTED TOPICS IN GENERAL EDUCATION: MATHEMATICS Given: R = 20% B = 700 Solution: P=BxR P = 700 x 20% P = 14,000% P = 140 A student earned a grade of 90% on a math test that had 30 problems. How many problems on this test did the student answer correctly? A. 24 C. 26 B. 25 D. 27 Sample Problem on Rate What percent of 30 is 15? A. 50% C. 40% B. 47% D. 52% Required: R=? Given: B = 30 P = 15 Solution: P R= B 15 R= 30 1500% R= (Add two zeros in the numerator. ) 30 𝐑 = 𝟓𝟎% There are 40 carpenters in a crew. On a certain day, 8 were present. What percent showed up for work? A. 10% C. 30% B. 20% D. 40% Sample Problem on Base 30 is 10% of what number? A. 350 C. 450 B. 300 D. 500 Given: P = 30 R = 10% Required: B=? Solution: P B= R 30 B= 10% 3000% B= (Add two zeros in the numerator. ) 10% 𝐁 = 𝟑𝟎𝟎  SELECTED TOPICS IN GENERAL EDUCATION: MATHEMATICS A student answered 40 problems on a test correctly and received a grade 40%. How many problems were on the test, if all the problems were worth the same number of points? A. 20 C. 100 B. 50 D. 150 VIII. Algebraic Expression - These are expressions that contain numbers, variables, and operations to state a relationship. - Examples 9, 89x, 2t+3, y2 + 3y + 10 Translating Algebraic Expressions Examples Addition Verbal Expression: a number increased by 5 Numerical Expression: x + 5 Subtraction Verbal Expression: a number decreased by 8 Numerical Expression: x – 8 Multiplication Verbal Expression: a number multiplied by 5 Numerical Expression: 5x Division Verbal Expression: a number divided by 7 x Numerical Expression: x ÷ 7 or 7 Combination of Operations Verbal Expression: eight times a number decreased by 2 Numerical Expression: 8x – 2 B. Algebraic Equation - It consists of two algebraic expressions set equal to each other. - Steps: 1. Transpose. 2. Combine like terms. 3. Divide both sides.  SELECTED TOPICS IN GENERAL EDUCATION: MATHEMATICS Examples: Solve for x: 3x + 2x = 30 A. 3 C. 5 B. 4 D. 6 Solution: 3x + 2x = 30 5x = 30 Combine like terms. 5x 30 = 5 Divide both sides of the equation by 5. 5 𝐱 = 𝟔 (𝐃) Solve for x: 5x + 3x + 2 = 42 A. 3 C. 5 B. 4 D. 6 Solution: 5x + 3x + 2 = 42 5x + 3x = 42 − 2 Transpose +2. 8x = 40 Combine like terms. 8x 40 = 8 Divide both sides of the equation by 8. 8 𝐱 = 𝟓 (𝐂) Solve for x: 5x – 3 – x + 2x = 45 A. 5 C. 7 B. 6 D. 8 Solution: 5x – 3 – x + 2x = 45 5x – x + 2x = 45 + 3 Transpose -3. 6x = 48 Combine like terms. 6x 48 = Divide both sides of the equation by 6. 6 6 x=8