The gre anlytycal writting section 4 pptx

( + ) ϫ ( – ) = – (+) Ϭ (–) = – ( – ) ϫ ( – ) = + (–) Ϭ (–) = + A simple rule for remembering these patterns is that if the signs are the same when multiplying or divid- ing, the answer will be positive. If the signs are different, the answer will be negative. ADDING Adding two numbers with the same sign results in a sum of the same sign: ( + ) + ( + ) = + and ( – ) + (– ) = – When adding numbers of different signs, follow this two-step process: 1. Subtract the absolute values of the numbers. 2. Keep the sign of the number with the larger absolute value. Examples: –2 + 3 = Subtract the absolute values of the numbers: 3 – 2 = 1. The sign of the number with the larger absolute value (3) was originally positive, so the answer is positive. 8 + –11 = Subtract the absolute values of the numbers: 11 – 8 = 3 The sign of the number with the larger absolute value (11) was originally negative, so the answer is –3. SUBTRACTING When subtracting integers, change the subtraction sign to addition and change the sign of the number being subtracted to its opposite. Then follow the rules for addition. Examples: (+10) – (+12) = (+10) + (–12) = –2 (–5) – (–7) = (–5) + (+7) = +2 REMAINDERS Dividing one integer by another results in a remainder of either zero or a positive integer. For example: 1 R1 – THE GRE QUANTITATIVE SECTION– 155 4ͤ5 ෆ –4 1 If there is no remainder, the integer is said to be “divided evenly,” or divisible by the number. When it is said that an integer n is divided evenly by an integer x, it is meant that n divided by x results in an answer with a remainder of zero. In other words, there is nothing left over. ODD AND EVEN NUMBERS An even number is a number divisible by the number 2, for example, 2,4, 6, 8, 10, 12,14, and so on.An odd num- ber is not divisible by the number 2, for example, 1, 3, 5, 7, 9, 11, 13, and so on. The even and odd numbers are also examples of consecutive even numbers and consecutive odd numbers because they differ by two. Here are some helpful rules for how even and odd numbers behave when added or multiplied: even + even = even and even ϫ even = even odd + odd = even and odd ϫ odd = odd odd + even = odd and even ϫ odd = even F ACTORS AND MULTIPLES Factors are numbers that can be divided into a larger number without a remainder. Example: 12 ϫ 3 = 4 The number 3 is, therefore, a factor of the number 12. Other factors of 12 are 1, 2, 4, 6, and 12. The common factors of two numbers are the factors that are the same for both numbers. Example: The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 18 = 1, 2, 3, 6, 9, 18. From the previous example, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6. This list also shows that we can determine that the greatest common factor of 24 and 18 is 6. Determining the greatest com- mon factor is useful for reducing fractions. Any number that can be obtained by multiplying a number x by a positive integer is called a multiple of x. Example: Some multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40 . . . Some multiples of 7 are: 7, 14, 21, 28, 35, 42, 49, 56 . . . – THE GRE QUANTITATIVE SECTION– 156 PRIME AND COMPOSITE NUMBERS A positive integer that is greater than the number 1 is either prime or composite, but not both. ■ A prime number has exactly two factors: 1 and itself. Example: 2,3,5,7,11,13,17,19,23, ■ A composite number is a number that has more than two factors. Example: 4,6,9,10,12,14,15,16, The number 1 is neither prime nor composite. Variables In a mathematical sentence,a variable is a letter that represents a number. Consider this sentence:x+ 4 = 10.It is easy to determine that xrepresents 6.However, problems with variables on the GRE will become much more complex than that, and there are many rules and procedures that you need to learn. Before you learn to solve equations with vari- ables,you must learn how they operate in formulas.The next section on fractions will give you some examples. Fractions A fraction is a number of the form ᎏ a b ᎏ ,where a and b are integers and b  0.In ᎏ a b ᎏ , the a is called the numerator and the b is called the denominator. Since the fraction ᎏ a b ᎏ means a Ϭ b, b cannot be equal to zero. To do well when work- ing with fractions, it is necessary to understand some basic concepts. The following are math rules for fractions with variables: ᎏ b a ᎏ ϫ ᎏ d c ᎏ = ᎏ b a ᎏ + ᎏ b c ᎏ = ᎏ a b ᎏ Ϭ ᎏ d c ᎏ = ᎏ a b ᎏ ϫ ᎏ d c ᎏ = ᎏ b a ᎏ + ᎏ d c ᎏ = Dividing by Zero Dividing by zero is not possible. This is important when solving for a variable in the denominator of a fraction. Example: ᎏ a – 6 3 ᎏ a – 3  0 a  3 In this problem, we know that a cannot be equal to 3 because that would yield a zero in the denominator. ab + bc ᎏ bd a ϫ d ᎏ b ϫ c a + c ᎏ b a ϫ c ᎏ b ϫ d – THE GRE QUANTITATIVE SECTION– 157 Multiplication of Fractions Multiplying fractions is one of the easiest operations to perform. To multiply fractions, simply multiply the numerators and the denominators, writing each in the respective place over or under the fraction bar. Example: ᎏ 4 5 ᎏ ϫ ᎏ 6 7 ᎏ = ᎏ 2 3 4 5 ᎏ Division of Fractions Dividing by a fraction is the same thing as multiplying by the reciprocal of the fraction. To find the recipro- cal of any number, switch its numerator and denominator. For example, the reciprocals of the following numbers are: ᎏ 1 3 ᎏ ⇒ ᎏ 3 1 ᎏ = 3 x ⇒ ᎏ 1 x ᎏᎏ 4 5 ᎏ ⇒ ᎏ 5 4 ᎏ 5 ⇒ ᎏ 1 5 ᎏ ᎏ – 2 1 ᎏ ⇒ ᎏ – 1 2 ᎏ = –2 When dividing fractions, simply multiply the dividend by the divisor’s reciprocal to get the answer. For example: ᎏ 1 2 2 1 ᎏ Ϭ ᎏ 3 4 ᎏ = ᎏ 1 2 2 1 ᎏ ϫ ᎏ 4 3 ᎏ = ᎏ 4 6 8 3 ᎏ = ᎏ 1 2 6 1 ᎏ Adding and Subtracting Fractions ■ To add or subtract fractions with like denominators, just add or subtract the numerators and leave the denominator as it is. For example: ᎏ 1 7 ᎏ + ᎏ 5 7 ᎏ = ᎏ 6 7 ᎏ and ᎏ 5 8 ᎏ – ᎏ 2 8 ᎏ = ᎏ 3 8 ᎏ ■ To add or subtract fractions with unlike denominators, you must find the least common denominator,or LCD.In other words, if the given denominators are 8 and 12, 24 would be the LCD because 8 ϫ 3 = 24, and 12 ϫ 2 = 24. So, the LCD is the smallest number divisible by each of the original denominators. Once you know the LCD, convert each fraction to its new form by multiplying both the numerator and denominator by the necessary number to get the LCD, and then add or subtract the new numerators. For example: ᎏ 1 3 ᎏ + ᎏ 2 5 ᎏ = ᎏ 5 5 ( ( 1 3 ) ) ᎏ + ᎏ 3 3 ( ( 2 5 ) ) ᎏ = ᎏ 1 5 5 ᎏ + ᎏ 1 6 5 ᎏ = ᎏ 1 1 1 5 ᎏ Mixed Numbers and Improper Fractions A mixed number is a fraction that contains both a whole number and a fraction. For example, 4 ᎏ 1 2 ᎏ is a mixed number. To multiply or divide a mixed number, simply convert it to an improper fraction. An improper frac- tion has a numerator greater than or equal to its denominator. The mixed number 4 ᎏ 1 2 ᎏ can be expressed as the improper fraction ᎏ 9 2 ᎏ . This is done by multiplying the denominator by the whole number and then adding the numerator. The denominator remains the same in the improper fraction. – THE GRE QUANTITATIVE SECTION– 158 For example, convert 5 ᎏ 1 3 ᎏ to an improper fraction. 1. First, multiply the denominator by the whole number: 5 ϫ 3 = 15. 2. Now add the numerator to the product: 15 + 1 = 16. 3. Write the sum over the denominator (which stays the same): ᎏ 1 3 6 ᎏ . Therefore, 5 ᎏ 1 3 ᎏ can be converted to the improper fraction ᎏ 1 3 6 ᎏ . Decimals The most important thing to remember about decimals is that the first place value to the right is tenths. The place values are as follows: In expanded form, this number can also be expressed as: 1268.3457 = (1 ϫ 1,000) + (2 ϫ 100) + (6 ϫ 10) + (8 ϫ 1) + (3 ϫ .1) + (4 ϫ .01) + (5 ϫ .001) + (7 ϫ .0001) Comparing Decimals Comparing decimals is actually quite simple. Just line up the decimal points and fill in any zeroes needed to have an equal number of digits. Example: Compare .5 and .005 Line up decimal points and add zeroes: .500 .005 Then ignore the decimal point and ask, which is bigger: 500 or 5? 500 is definitely bigger than 5, so .5 is larger than .005 1 T H O U S A N D S 2 H U N D R E D S 6 T E N S 8 O N E S 3 T E N T H S 4 H U N D R E D T H S 5 T H O U S A N D T H S 7 T E N T H O U S A N D T H S . D E C I M A L P O I N T – THE GRE QUANTITATIVE SECTION– 159 Operations with Decimals To add and subtract decimals, you must always remember to line up the decimal points: 356.7 3.456 8.9347 + 34.9854 + .333 – 0.24 391.6854 3.789 8.6947 To multiply decimals, it is not necessary to align decimal points. Simply perform the multiplication as if there were no decimal point. Then, to determine the placement of the decimal point in the answer, count the numbers located to the right of the decimal point in the decimals being multiplied. The total numbers to the right of the decimal point in the original problem is the number of places the decimal point is moved in the product. For example: To divide a decimal by another, such as 13.916 Ϭ 2.45 or 2.45ͤ13 ෆ .9 ෆ 16 ෆ , move the decimal point in the divisor to the right until the divisor becomes a whole number. Next, move the decimal point in the dividend the same number of places: This process results in the correct position of the decimal point in the quotient. The problem can now be solved by performing simple long division: Percents A percent is a measure of a part to a whole, with the whole being equal to 100. ■ To change a decimal to a percentage, move the decimal point two units to the right and add a percent- age symbol. 245 1391.6 5.68 –1225 166 6 –1470 1960 1391.6 245 1 2.3 4 2 2 x .5 6 1 2 3 4 7 4 0 4 6 1 7 0 0 6.9 1 0 4 1 2 3 4 = TOTAL #'s TO THE RIGHT OF THE DECIMAL POINT = 4 – THE GRE QUANTITATIVE SECTION– 160 . move the decimal point two units to the right and add a percent- age symbol. 245 1391.6 5.68 –1225 166 6 – 147 0 1960 1391.6 245 1 2.3 4 2 2 x .5 6 1 2 3 4 7 4 0 4 6 1 7 0 0 6.9 1 0 4 1 2 3 4 =. one integer by another results in a remainder of either zero or a positive integer. For example: 1 R1 – THE GRE QUANTITATIVE SECTION 155 4 5 ෆ 4 1 If there is no remainder, the integer is said. remainder. Example: 12 ϫ 3 = 4 The number 3 is, therefore, a factor of the number 12. Other factors of 12 are 1, 2, 4, 6, and 12. The common factors of two numbers are the factors that are the same for both