College Entrance Exam Review Kippap Review Center Math Module Order of Operations. Solving series of operations, we follow the GEMDAS Rule: ● G – Groups ● E – Exponents ● M/D – Multiplication/Division ● A/S – Addition/Subtraction Divisibility Rules. A number is divisible by 2: if the number is even or ends with 0, 2, 4, 6, or 8 Ex. the number “46” ends with “6” so it is even. 3: if the sum of the digits is divisible by 3 Ex. The sum of the digits of the number “63” is 6+3=9 which is divisible by 3. Therefore, 63 is divisible by 3 4: if the last two digits is divisible by 4 Ex. The last two digits of the number “124” is 24 which is divisible by 4. Therefore, 124 is divisible by 4. 5: if the last digit is 0 or 5 Ex. The last digit of the number “420” is “0”. Therefore, 420 is divisible by 5. 6: if the number is divisible by 3 and by 2 Ex. The number “96” is divisible by “3”, because 9+6=15 which is divisible by 3, and it is also divisible by 2 because its last digit is “6”. 7: *There is no simple divisibility rule for the number 7. So, it is recommended do go straight to division and find out if there are remainders. 8: if last three digits is divisible by 8 Ex. The last three digits of the number “1464” is “464” which is divisible by 8. 9: if the sum of the digits is divisible by 9 Ex. The sum of the digits of the number “261” is equal to 9 which is divisible by 9. Therefore, the number is divisible by 9. 10: if the last digit is 0 Ex. The number “12390” ends with “0”. Therefore, it is divisible by 10. 11: *There is no simple divisibility rule for the number 11. So, it is recommended to go straight to division and find out if there are remainders. 12: if the number is divisible by 3 and 4 Ex. The number “624” is (a) divisible by 3 because the sum of its digits is equal to 12 which is divisible by 3 and (b) by 4 because the number’s last two digits, “24”, is divisible by 4. EXPONENTS An exponent refers to the number of times a number is multiplied by itself. Properties of Exponents: ● A number (except for 0) that has an exponent of 0 is equal to 1 Ex.73480 0 = 1 ● A number that has an exponent of 1 is equal to that number Ex.54 1 = 54 ● A number that has a negative exponent is equal to the reciprocal of that number without the negative sign in its exponent Ex.23 −4 = 1 23 4 ● When multiplying two numbers with the same base, add the exponents and retain the base Ex.72 3 * 72 5 = 72 8 ● When dividing two numbers with the same base, subtract the exponents and retain the base Ex. 72 3 ÷ 72 5 = 72 −2 ● When multiplying two numbers with the same exponent, multiply the bases and retain the exponent Ex. 3 5 * 4 5 = 12 5 ● When dividing two numbers with the same exponent, divide the bases and retain the exponent Ex. 12 5 ÷ 4 5 = (12÷4) 5 = 3 5 ● Multiply exponents inside and outside the parentheses Ex. (3 5 ) 3 = 3 15 ● Distribute the exponent outside the parenthesis to the factors inside it Ex. (3x) 3 = 3 3 x 3 Contact us: +63 917 129 0044
[email protected] College Entrance Exam Review Kippap Review Center Math Module .( 3 4 ) 3 = ( 3 3 4 3 ) The root of a number x is another number, which when multiplied by itself a given number of times, equals x. ● A number with an exponent with the form 1/x means the xth root of that number Ex. 29 1/3 = 3 29 FRACTIONS Fractions. A fraction is an expression that represents the part of a whole. Comparing Fractions ● With same denominator o Compare the numerators. The higher the numerator the larger the fraction Ex. is greater than because 7 is 7 13 4 13 greater than 4 ● With different denominators using Cross multiplication o Cross multiply the numerators with the denominators of the other fraction. The fraction with the numerator with the bigger the product is the bigger fraction Ex. , is greater 2 11 3 13 3 13 because 3 * 11 = 33 is greater than 2 * 13 = 26 ● With same numerator o Compare the denominators. The lesser the denominator the greater the fraction. Ex. is greater than because 13 23 13 23 15 is less than 15. Operations on Fractions Adding and Subtracting Fractions ● When adding and subtracting like fractions, just add or subtract the numerators Ex. 3 5 + 1 5 = 4 5 ● When adding and subtracting unlike fractions, first convert to like fractions by looking for the LCD and then just add or subtract the numerators Ex. . The LCD of 4 and 5 is 20. Thus, 3 4 + 1 5 change the fractions to their equivalent fractions with a denominator of 20. . 3 4 + 1 5 = 3*5 4*5 + 1*4 5*4 = 15 20 + 4 20 = 19 20 ● When adding and subtracting mixed fractions, o Separate the whole numbers and the fractions o Add or subtract the whole numbers o Add or subtract the fraction o Add or subtract the fraction o or from the whole number Ex. 6 1 4 − 3 3 5 = (6 − 3) + 1 4 − 3 ( 5 ) = 3 + 5 20 − 12 ( 20 ) = 3 + − 7 ( 20 ) = 3 1 − 7 20 Multiplying Fractions ● To multiply fractions, just multiply the numerators and the denominators. Ex. 3 4 * 1 5 = 3*1 4*5 = 3 20 Dividing Fractions ● To divide fractions, first transform the divisor to it’s reciprocal, then multiply the numerators and denominators. Ex. 3 4 ÷ 1 5 = 3 4 * 5 1 = 15 4 = 3 3 4 DECIMALS Place Value ● In writing decimals, the position of the digits relative to the decimal point is very important. Contact us: +63 917 129 0044
[email protected] College Entrance Exam Review Kippap Review Center Math Module Decimals, Fractions Conversion ● To convert a fraction to a decimal, just divide the numerator by the denominator Ex. 4 5 = 4÷5 = 0. 2 ● To convert a decimal to a fraction, . The number of decimal places that you move to make the numerator whole is the number of zeroes the denominator has. Ex. 0. 078 = 78 1000 = 39 500 Scientific Notation 1056000 → 1. 056 × 10 6 RATIO AND PROPORTION Direct Proportion ● In direct proportions, an increase in one causes an increase in the other. Ex. More people to feed, the more food to be provided; the longer the travel distance, the more gas is needed. Solving for Unknowns Involving Direct Proportion Ex. During a fun runs, 3L of water should be provided for 2 persons. If there are 500 participants at this year’s fun run, how many liters of water should be provided? *In direct proportions, same units are on the same level of the fraction (both numerator or both denominator). The fractions should contain corresponding values. In this case, 3 is to 2 as x is to 500. 3 liters of water 2 persons = x liters of water 500 persons *Isolate the unknown 3 liters of water*500 persons 2 persons = x liters of water 3*500 2 = x liters of water 750 = x liters of water ∴750L of water is needed for 500 participants. Inverse Proportion ● In inverse proportions, an increase in one translates to a decrease in the other factor. Ex. More workers, less time to finish the job; More faucets, less time to fill the tank Solving for Unknowns Involving Inverse Proportion Ex. Five students can finish a large box of pizza in 4 hrs. How long would it take to finish the large box of pizza if there are 7 students? *In inverse proportions, the corresponding values are multiplied to each other 5 students * 4 hrs = 7 students * x hrs *Isolate the unknown 5 students*4 hrs 7 students = x hrs 5*4 7 = x hrs 20 7 = 2 6 7 hrs ∴7 students can finish a large box of pizza in 2 6 7 hrs. PERCENT Percent = per + cent (hundred) o 50% means 50 per 100 or 50/100 o 100% means 100/100 or one whole o 500% means 500/100 or 5 ● To transform a percent to a decimal, move the decimal two places to the left then remove the percent sign Ex. 35. 0% = 0. 35 Contact us: +63 917 129 0044
[email protected] College Entrance Exam Review Kippap Review Center Math Module ● To transform a decimal to a percent, move the decimal two places to the right and append a percent sign Ex. 0. 678 = 67. 8% ● To transform a percent to a fraction, use 100 as the denominator and remove the percent sign Ex. 40% = 40 100 = 2 5 ● To transform a fraction to a percent, convert it to a decimal first, then convert to percent Ex. 3 4 = 0. 75 = 75% MULTIPLICATION AND DIVISION OF SIGNED NUMBERS ● (+) * (+) = (+) ● (+) * (−) = (−) ● (−) * (−) = (+) ● (+) ÷ (+) = (+) ● (+) ÷ (−) = (−) ● (−) ÷ (−) = (+) THE METRIC SYSTEM ● Length: meters (m) ● Mass: kilograms (kg) ● Volume: liters (L) ALGEBRA Algebraic Expression – is an expression that consist of integer constants, variables, and operations Variables – is a quantity represented by letters whose mathematical values may change Coefficients – numbers used to multiply a variable Ex. Consider the following algebraic expression 4x + 3y ● x and y are the variables ● 4 and 3 are their corresponding coefficients Term – is either a single number or variable, or a combination of both (multiplied together); terms are separated by + or – signs Ex. In the previous example, 4x and 3y are two terms separated by a + sign. Like Terms – terms that have the same variable raised to the same exponent Ex. 4x 2 + 7y + 7x 2 ● 4x are like terms because they have 2 and 7x 2 the same variable (x) with the same exponent (2) Monomial – algebraic expression with only one term Ex. 4x is a monomial 2 y 3 Polynomial – algebraic expression with two or more terms Ex. 4x is a polynomial with three 2 + 7y + 7x 2 terms Operations on Algebraic Expressions ● Addition and Subtraction o You may only add or subtract like terms Ex. 4x 2 + 7y + 7x 2 = 4x 2 + 7x 2 ( ) + 7y = 11x 2 + 7y ● Multiplication and Division of Two Monomials o Multiply/divide the coefficients and multiply/divide the variable Ex. 12x 2 ×3x 2 y = (12×3) × x 2 × x 2 ( )×y = 36x 4 y ● Multiplication of Monomial to Polynomial o Multiply the monomial to each term of the polynomial Ex. 4x× 10x 2 ( − 2x + 7) = 40x 3 − 8x 2 + 28x ● Division of Polynomial by a Monomial o Divide each term of the polynomial by the monomial Ex. 40x 2−8x 2+28x 2x Contact us: +63 917 129 0044
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