Symbols of addition 2 pps

Factors and Multiples FACTORS Factors are numbers that can be divided into a larger number without a remainder. Example 12 ÷ 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 com- mon factors of two numbers are the factors that both numbers have in common. Examples The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 18 = 1, 2, 3, 6, 9, and 18. From the examples above, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6. From this list it can also be determined that the greatest common factor of 24 and 18 is 6. Determining the greatest common factor (GCF) is useful for simplifying fractions. Example Simplify ᎏ 1 2 6 0 ᎏ . The factors of 16 are 1, 2, 4, 8, and 16. The factors of 20 are 1, 2, 4, 5, and 20. The common factors of 16 and 20 are 1, 2, and 4. The greatest of these, the GCF, is 4. Therefore, to simplify the fraction, both numerator and denominator should be divided by 4. ᎏ 1 2 6 0 ÷ ÷ 4 4 ᎏ = ᎏ 4 5 ᎏ MULTIPLES Multiples are numbers that can be obtained by multiplying a number x by a positive integer. Example 5 ϫ 7 = 35 The number 35 is, therefore, a multiple of the number 5 and of the number 7. Other multiples of 5 are 5, 10, 15, 20, etc. Other multiples of 7 are 7, 14, 21, 28, etc. – THEA MATH REVIEW– 95 The common multiples of two numbers are the multiples that both numbers share. Example Some multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36 . . . Some multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48 . . . Some common multiples are 12, 24, and 36. From the above it can also be determined that the least com- mon multiple of the numbers 4 and 6 is 12, since this number is the smallest number that appeared in both lists. The least common multiple, or LCM, is used when performing addition and subtraction of fractions to find the least common denominator. Example (using denominators 4 and 6 and LCM of 12) ᎏ 1 4 ᎏ + ᎏ 5 6 ᎏ = ᎏ 1 4 ( ( 3 3 ) ) ᎏ + ᎏ 5 6 ( ( 2 2 ) ) ᎏ = ᎏ 1 3 2 ᎏ + ᎏ 1 1 0 2 ᎏ = ᎏ 1 1 3 2 ᎏ = 1 ᎏ 1 1 2 ᎏ Decimals The most important thing to remember about decimals is that the first place value to the right of the decimal point is the tenths place. The place values are as follows: In expanded form, this number can also be expressed as: 1,268.3457 = (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) 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 • D E C I M A L 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 POINT – THEA MATH REVIEW– 96 ADDING AND SUBTRACTING DECIMALS Adding and subtracting decimals is very similar to adding and subtracting whole numbers. The most important thing to remember is to line up the decimal points. Zeros may be filled in as placeholders when all numbers do not have the same number of decimal places. Example What is the sum of 0.45, 0.8, and 1.36? 11 0.45 0.80 + 1.36 2.61 Take away 0.35 from 1.06. 1 0 ΋ .0 1 6 –0.35 0.71 MULTIPLICATION OF DECIMALS Multiplication of decimals is exactly the same as multiplication of integers, except one must make note of the total number of decimal places in the factors. Example What is the product of 0.14 and 4.3? First, multiply as usual (do not line up the decimal points): 4.3 ϫ .14 172 + 430 602 Now, to figure out the answer, 4.3 has one decimal place and .14 has two decimal places. Add in order to deter- mine the total number of decimal places the answer must have to the right of the decimal point. In this problem, there are a total of 3 (1 + 2) decimal places. When finished multiplying, start from the right side of the answer, and move to the left the number of decimal places previously calculated. .602 – THEA MATH REVIEW– 97 ۍۍۍ In this example, 602 turns into .602 since there have to be 3 decimal places in the answer. If there are not enough digits in the answer, add zeros in front of the answer until there are enough. Example Multiply 0.03 ϫ 0.2. .03 ϫ .2 6 There are three total decimal places in the problem; therefore, the answer must contain three decimal places. Starting to the right of 6, move left three places. The answer becomes 0.006. D IVIDING DECIMALS Dividing decimals is a little different from integers for the set-up, and then the regular rules of division apply. It is easier to divide if the divisor does not have any decimals. In order to accomplish that, simply move the decimal place to the right as many places as necessary to make the divisor a whole number. If the decimal point is moved in the divisor, it must also be moved in the dividend in order to keep the answer the same as the original prob- lem; 4 ÷ 2 has the same solution as its multiples 8 ÷ 4 and 28 ÷ 14, etc. Moving a decimal point in a division prob- lem is equivalent to multiplying a numerator and denominator of a fraction by the same quantity, which is the reason the answer will remain the same. If there are not enough decimal places in the answer to accommodate the required move, simply add zeros until the desired placement is achieved. Add zeros after the decimal point to continue the division until the dec- imal terminates, or until a repeating pattern is recognized. The decimal point in the quotient belongs directly above the decimal point in the dividend. Example What is .425ͤ1. ෆ 53 ෆ ? First, to make .425 a whole number, move the decimal point 3 places to the right: 425. Now move the decimal point 3 places to the right for 1.53: 1,530. The problem is now a simple long division problem. 3.6 425.ͤ1, ෆ 53 ෆ 0. ෆ 0 ෆ –1,275↓ 2,550 –2,550 0 – THEA MATH REVIEW– 98 COMPARING DECIMALS Comparing decimals is actually quite simple. Just line up the decimal points and then fill in zeros at the end of the numbers until each one has an equal number of digits. Example Compare .5 and .005. Line up decimal points. .5 .005 Add zeros. .500 .005 Now, ignore the decimal point and consider, which is bigger: 500 or 5? 500 is definitely bigger than 5, so .5 is larger than .005. ROUNDING DECIMALS It is often inconvenient to work with very long decimals. Often it is much more convenient to have an approxi- mation for a decimal that contains fewer digits than the entire decimal. In this case, we round decimals to a cer- tain number of decimal places. There are numerous options for rounding: To the nearest integer: zero digits to the right of the decimal point To the nearest tenth: one digit to the right of the decimal point (tenths unit) To the nearest hundredth: two digits to the right of the decimal point (hundredths unit) In order to round, we look at two digits of the decimal: the digit we are rounding to, and the digit to the imme- diate right. If the digit to the immediate right is less than 5, we leave the digit we are rounding to alone, and omit all the digits to the right of it. If the digit to the immediate right is five or greater, we increase the digit we are round- ing by one, and omit all the digits to the right of it. Example Round ᎏ 3 7 ᎏ to the nearest tenth and the nearest hundredth. Dividing 3 by 7 gives us the repeating decimal .428571428571 Ifwe are rounding to the nearest tenth, we need to look at the digit in the tenths position (4) and the digit to the immediate right (2). Since 2 is less than 5, we leave the digit in the tenths position alone, and drop everything to the right of it. So, ᎏ 3 7 ᎏ to the nearest tenth is .4. To round to the nearest hundredth, we need to look at the digit in the hundredths position (2) and the digit to the immediate right (8). Since 8 is more than 5, we increase the digit in the hun- dredths position by 1, giving us 3, and drop everything to the right of it. So, ᎏ 3 7 ᎏ to the nearest hun- dredth is .43. – THEA MATH REVIEW– 99 Fractions To work well with fractions, it is necessary to understand some basic concepts. SIMPLIFYING FRACTIONS Rule: ᎏ a bc c ᎏ = ᎏ a b ᎏ ■ To simplify fractions, identify the Greatest Common Factor (GCF) of the numerator and denominator and divide both the numerator and denominator by this number. Example Simplify ᎏ 6 7 3 2 ᎏ . The GCF of 63 and 72 is 9 so divide 63 and 72 each by 9 to simplify the fraction: ᎏ 6 7 3 2 ÷ ÷ 9 9 = = 7 8 ᎏ ᎏ 6 7 3 2 ᎏ = ᎏ 7 8 ᎏ ADDING AND SUBTRACTING FRACTIONS Rules: To add or subtract fractions with the same denominator: ᎏ a b ᎏ ± ᎏ b c ᎏ = ᎏ a ± b c ᎏ To add or subtract fractions with different denominators: ᎏ a b ᎏ ± ᎏ d c ᎏ = ᎏ ad b ± d cb ᎏ ■ To add or subtract fractions with like denominators, just add or subtract the numerators and keep the denominator. Examples 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 – THEA MATH REVIEW– 100 . share. Example Some multiples of 4 are: 4, 8, 12, 16, 20 , 24 , 28 , 32, 36 . . . Some multiples of 6 are: 6, 12, 18, 24 , 30, 36, 42, 48 . . . Some common multiples are 12, 24 , and 36. From the above. common. Examples The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24 . The factors of 18 = 1, 2, 3, 6, 9, and 18. From the examples above, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6. From. without a remainder. Example 12 ÷ 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 com- mon factors of two numbers are the factors