Chapter 2: Rational Numbers
2.1 Express 0.013 as a percentage.
Decimal numbers, like percentages, express a value in terms of a whole. This whole value can be expressed in decimal form, as 1 (or 1.0), and as a percentage, 100%. Transforming a decimal into a percent is as simple as moving the decimal point exactly two digits to the right. Here, 0.013 = 1.3%.
2.38 Describe the relationship between multiplying and dividing fractions.
Dividing by a fraction is equivalent to multiplying by the reciprocal of that fraction. a/b / c/d = a/b x d/c (side note: Some students call this the KFC rule (it has nothing to do with chicken), which stands for keep, flip, change. When you divide something by a fraction, KEEP the first fraction the same, FLIP the fraction you're dividing by (take the reciprocal), and CHANGE the division sign to a multiplication sign.)
2.22 Explain what is meant by a least common denominator.
Equivalent fractions might have different denominators. For instance, Problem 2.13 demonstrated that 8/24 and 1/3 have the same value, as 1/3 is 8/24 expressed in lowest terms. It is often useful to rewrite one or more fractions so that their denominators are equal. Usually, there are numerous options from which you can choose a common denominator, and the least common denominator is the smallest of those options.
2.41 Simplify the complex fraction: 3/ 9/7.
A complex fraction contains a fraction in its numerator, denominator, or both. This fraction is complex because its denominator is the fraction . All fractions, including complex fractions, can be rewritten as a quotient. Convert the quotient into a product, calculate the product, and reduce the result to lowest terms. (skipped writing the numbers)
2.11 Explain the process used to reduce a fraction to its lowest terms.
A fraction is in lowest terms only when its numerator and denominator no longer share any common factors. In other words, the fraction is reduced if no number divides evenly into both the numerator and denominator. One effective way to reduce a fraction is to identify the greatest common factor (GCF) of both numbers and then divide each by that GCF. (side note: Except 1, because 1 can divide evenly into anything.)
Note: Problems 2.23-2.25 refer to the fractions 1/2 and 3/10. 2.23 Identify the least common denominator of the fractions.
Begin by identifying the largest of the given denominators; here, the largest denominator is 10. Because the other denominator (2) is a factor of 10, then 10 is the least common denominator (LCD). The LCD is never smaller than the largest denominator. (side note: In other words, 10 / 2 has no remainder.)
2.8 Express the improper fraction 65/4 as a mixed number.
Four divides into 65 a total of 16 times with a remainder of 1. Therefore 65/4 = 16 1/4. The fractional part of the mixed number consists of the remainder divided by the original denominator. (side note: Here's a quick way to figure this out. Either long divide 65 / 4 or use a calculator. You get 16.25. The number left of the decimal (16) is the nonfraction part of the mixed number. To figure out the remainder, multiply that whole number by the denominator (16 x 4 = 64) and subtract what you get from the numerator: 65 - 64 = 1.)
2.32 Use the prime factorization technique described in Problem 2.30 to simplify the expression: 116/945 + 612/1575 .
Identify the prime factorizations of 945 and 1,575. 945 = 3^5 x 5 x 7 1575 = 3^2 x 5^2 x 7 Both factorizations include the same factors (3, 5, and 7), so the LCD includes those factors as well. The highest power of 3 is 33 (from the factorization of 945), and the highest power of 5 is 52 (from the factorization of 1,575), so apply those powers to the factors in the LCD. The remaining factor, 7, is raised to the same power in both factorizations (1). LCD = 3^3 x 5^2 x 7^1 = 27 x 25 x 7 = 4725 Divide the LCD by each denominator to identify the value by which you should multiply to generate the equivalent fractions: 4725 / 945 = 5 and 4725 / 1575 = 3 116/945 + 612/1575 = 116 x 5 over 945 x 5 ll + ll 612 x 3 over 1575 x 3 = 580/4725 + 1836/4725 = 580 + 1838 over 4725 = 2416/4725 (side note: Both numbers are divisible by 5: 945 = (5)(189) and 1,575 = (5)(315). Additionally, 189 and 315 are divisible by 9: 945 = (5)(9)(21) and 1,575 = (5)(9)(35). Finally, 21 and 35 are both divisible by 7.)
Note: Problems 2.34-2.35 refer to the expression (- 7/36)(18/14). 2.35 Reduce the numerators and denominators of the expression before calculating the product.
If a number in any numerator shares a common factor with a number in any denominator, it is beneficial to express the numbers as products that include the common factor. Here, the left numerator and the right denominator share the common factor 7; the left denominator and the right numerator share the common factor 18. (-7/36)(18/14) = (-7 over 2 x 18)(18 over 2 x 7) Multiply the fractions together but do not calculate the individual products. (-7 over 2 x 18)(18 over 2 x 7) = - 7 x 18 over 2 x 18 x 2 x 7 Eliminate the matching pairs of factors in the numerator and denominator. 7 crossed x 18 crossed over 2 x 18 crossed x 2 x 7 crossed = 1 over 2 x 2 = -1/4 No factors remain in the numerator other than the (usually) unwritten factor 1. Notice that this answer and the answer to Problem 2.34 are equal. The method you use to multiply the fraction has no affect on the product. (side note 1: This is a great trick because your answers won't need to be reduced at the end of the problem.)(side note 2: In other words, leave it as a big string of numbers multiplied together.(side note 3: If you cross out a 7 in the top of the fraction, you can cross out a 7 in the denominator. It's a one-for-one exchange—one number on the top for one matching number on the bottom.)
2.37 Simplify the expression: 6(-2/9).
Integers are rational numbers, as any number divided by 1 is equal to itself: . Multiply the fractions. Note that 6 and 9 have common factor 3. (I'm not writing all of the numbers. suck it.)
Note: Problems 2.12-2.13 refer to the rational number 8/24. 2.12 Identify the greatest common factor of the numerator and denominator.
List all the factors of the numerator (numbers that divide evenly into 8) and the denominator (numbers that divide evenly into 24). Factors of 8: 1, 2, 4, 8 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 The largest factor common to both lists is 8, so the greatest common factor of 8 and 24 is 8.
2.16 Reduce the fraction -28/52 to lowest terms.
List the factors of the numerator and denominator. Although this fraction is negative, the technique you use to reduce it to lowest terms remains unchanged. Factors of 28: 1, 2, 4, 7, 14, 28 Factors of 52: 1, 2, 4, 13, 26, 52 Divide the numerator and denominator by 4, the greatest common factor. -28/52 = 28 / 4 over 52 / 4 = -7/13
Note: Problems 2.14-2.15 refer to the rational number 27/63. 2.14 Identify the greatest common factor of the numerator and denominator.
List the factors of the numerator and denominator. Factors of 27: 1, 3, 9, 27 Factors of 63: 1, 3, 7, 9, 21, 63 The largest factor common to both lists is 9, so the greatest common factor of 27 and 63 is 9. (side note: There's no magic trick for generating the list of factors. Start by trying to divide the number by 2, then by 3, and so on, until you identify all the numbers that divide in evenly. Here's one tip: All the numbers in the list are paired up. For instance, after you figure out that 3 is a factor of 63, then 63 / 3 = 21 also has to be in the list of factors.)
Note: Problems 2.34-2.35 refer to the expression (- 7/36)(18/14). 2.34 Simplify the expression using the method described in Problem 2.33.
Multiply the numerators and divide by the product of the denominators. Note that the product of numbers with different signs is negative. (-7/36)(18/14)= -126/504 Divide 126 and 504 by their greatest common factor (126) to reduce the fraction. -126 / 126 over 504 / 126 = -1/4
2.33 Simplify the expression: 3/8(2/5).
Multiplying and dividing fractions does not require a common denominator. Multiply the numerators of the fractions together and divide by the product of the denominators. 3/8(2/5) = 3 x 2 over 8 x 5 = 6/40 To reduce the fraction to lowest terms, divide the numerator and denominator by 2. 6 / 2 over 40 / 2 = 3/20
2.43 Simplify the complex fraction: 20/9 / 16/27.
Rewrite the complex fraction as a quotient and then as a product. Calculate the product and ensure the result is reduced to lowest terms. Notice that 20 and 16 have common factor 4; 27 and 9 have common factor 9. (skipped writing the numbers)
2.40 Simplify the expression: (-5/12)/(-1/6).
Rewrite the quotient as a product, multiply the fractions, and write the result in lowest terms. Recall that the quotient (and product) of two negative numbers is positive (still not writing all of the numbers. suck it.) (side note: If you don't like looking for common factors and canceling them out, you don't have to. You can always wait until the very last step and divide by the greatest common factor. However, it's usually easier to spot common factors when the numbers are smaller.)
2.18 Express 0.45 as a fraction in lowest terms.
The decimal 0.45 is read "forty-five hundredths," because it extends two digits to the right of the decimal point. Forty-five hundredths literally translates into the fraction 45/100. Reduce the fraction to lowest terms. 45 / 5 over 100 / 5 = 9/20 (side note 1: 0.5 is five tenths, 0.05 is five hundredths, 0.005 is five thousandths, 0.0005 is five ten thousandths, etc.) (side note 2: Each digit right of the decimal point is a power of 10. Because 0.45 ends two digits right of the decimal, you divide it by 102 . To make 0.12986 into a fraction, you divide it by 105 , because it has five digits left of the decimal.)
2.20 Express 0.5 as a fraction.
The decimal 0.5 repeats infinitely (0.5 0.5555555... = ); therefore, you cannot use the method described in Problems 2.18-2.19 to convert this number into a fraction. If the digits of a repeating decimal begin repeating immediately after the decimal point (that is, the repeated string begins in the tenths place of the decimal), then you can apply a shortcut to rewrite the decimal as a fraction: Divide the repeated string by as many 9s as there are digits in the repeated string. In this case, the repeated string is consists of one digit (5). To convert into a fraction, divide the repeated string by 9. 0.5 bar over 5 = 5/9 The rationale behind this shortcut is omitted here, as it is based on skills not discussed until Chapter 4. This technique, in its more rigorous form, is explained in greater detail in Problems 4.26-4.28. (side note 1: In Problems 2.18-2.19, the number of digits after the decimal told you what power of 10 to divide by. However, this decimal never ends, so you can't get the power of 10 you need.) (side note 2: If the repeated string is two digits long, divide by 99. If it's three digits long, divide by 999.)
2.28 Simplify the expression: 1/3-5/12+7/16.
The largest denominator (16) is not evenly divisible by both of the other denominators (3 and 12) so multiply it by 2: 16 x 2 = 32. However, 32 is not divisible by all of the denominators either, so multiply the largest denominator by 3: 16 x 3 = 48. Because 48 is divisible by 3, 12, and 16, it is the least common denominator. Rewrite the expression using the method described by Problem 2.27: Divide each denominator into the LCD and multiply the numerator and denominator of each fraction by the corresponding result. 1 x 16 over 3 x 16 - 5 x 4 over 12 x 4 + 7 x 3 over 16 x 3 = 16/48 - 20/48 + 21/48 = 16 - 20 +21 over 48 = -4 + 21 over 48 = 17/48
Note: Problems 2.23-2.25 refer to the fractions 1/2 and 3/10. 2.25 Calculate the sum of the fractions.
To calculate the sum or difference of fractions, those fractions must have a common denominator. According to Problem 2.24, 1/2 = 5/10. 1/2 + 3/10 = 5/10 + 3/10 Add the numerators of the fractions, but not the denominators. 5/10 + 3/10 = 5+3 over 10 = 8/10 Unless otherwise directed, you should always reduce answers to lowest terms. 8/10 = 4/5 (side note: In other words, if you want to ADD or SUBTRACT fractions ....)
2.9 Express 9 3/7 as an improper fraction.
To convert a mixed number into a fraction, multiply the denominator and the whole number and then add the numerator. Divide by the denominator of the mixed number: a b/c = (ca)+b over c. 9 3/7 = (7x9) +3 over 7 = 63 + 3 over 7 = 66/7
Note: Problems 2.23-2.25 refer to the fractions 1/2 and 3/10 . 2.24 Generate equivalent fractions using the least common denominator.
To rewrite 1/2 using the least common denominator, divide the LCD by the current denominator: 10 / 2 = 5 . Multiply the numerator and denominator of 1/2 by that result. 1 x 5 over 2 x 5 = 5/10 Because 3/10 already contains the least common denominator, it does not need to be rewritten. (side note: Multiplying the top and bottom of the fraction by 5 is like multiplying the entire fraction by 5/5. You're allowed to do that because 5 / 5 = 1, and multiplying any number by 1 doesn't change it, according to the multiplicative identity property in Problem 1.36.)
2.42 Simplify the complex fraction: 2/3 / 8/5.
Use the method described in Problem 2.41: Rewrite the complex fraction as a quotient and then as a product. Calculate the product and ensure the result is reduced to lowest terms. (skipped writing the numbers. It's too time consuming.)
2.10 Express 4 5/12 as an improper fraction.
According to Problem 2.9, a b/c = (ca)+b over c. Substitute a = 4, b = 5, and c = 12 into the formula. 4 5/12 = (12 x 4) + 5 over 12 = 48 + 5 over 12 = 53/12
Note: Problems 2.12-2.13 refer to the rational number 8/24. 2.13 Reduce the fraction to lowest terms.
According to Problem 2.12, the greatest common factor of 8 and 24 is 8, so divide both the numerator and the denominator by 8 to reduce the fraction. 8/24 = 8 / 8 over 24 / 8 = 1/3 Though 1/3 and 8/24 are different fractions, they have equivalent values: 1 / 3 = 8 / 24 = 0.3 bar over 3. (side note: If you have two dozen eggs, most people would say, "I have one-third of the eggs" rather than "I have 8 of the 24 eggs." Both are correct, but the first is easier to visualize.)
Note: Problems 2.14-2.15 refer to the rational number 27/63. 2.15 Express the fraction in lowest terms.
According to Problem 2.14, the greatest common factor of 27 and 63 is 9, so divide the numerator and denominator by 9 to reduce the fraction. 27/63 = 27 / 9 over 63 / 9 = 3/7
Note: Problems 2.3-2.4 refer to the rational number 1/4. 2.4 Express the fraction as a percentage.
According to Problem 2.3, 1.4=0.25 . To transform a decimal into a percentage, move the decimal point two digits to the right: 0.25 = 25.0%, or simply 25%.
2.2 Express 0.25% as a decimal
According to Problem 2.1, converting from a decimal to a percentage requires you to move the decimal point two digits to the right. It comes as no surprise, then, that performing the opposite conversion, from percentage to decimal, requires you to move the decimal exactly two digits to the left. In this problem, only one digit, 0, appears to the left of the decimal. The second, unwritten, digit is also 0. Therefore, 0.25% = 00.25% = 0.0025%. (side note: You can put as many zeroes as you want at the beginning of a decimal: 0.25, 00.25, 000.25, and 0000000000.25 all mean the same thing. You can also add zeroes at the end of a decimal: 1.5, 1.50, 1.500, and so on.)
Note: Problems 2.30-2.31 refer to the fractions 11/30 and 19/64. 2.31 Simplify the expression:11/30-19/64
According to problem 2.30, the least common denominator is 960. Multiply the numerator and denominator of the first fraction by 32 (because 960 / 30= 32). Similarly, the numbers in the second fraction should be multiplied by 960 / 64 = 15. These operations rewrite the expression using equivalent fractions with the least common denominator. 11/30 - 19/64 = 11 x 32 over 30 x 32 ll - ll 19x15 over 64 x 15 = 352/960 - 285/960 = 352 - 285 over 960 = 67/960 (side note: 67 is a prime number, so you can only reduce the fraction if 960 is divisible by 67, and it's not.)
Note: Problems 2.5-2.7 refer to the rational number 11/6. 2.6 Express the fraction as a percentage.
According to problem 2.5, 22/6=1.833333.... Move the decimal two places to the right to convert the decimal into a percentage: 1.833333+183.3% (bar over 3).
2.39 Simplify the expression: 4/5 / 2.
Express the integer as a rational number and rewrite the quotient as a product, using the method described in Problem 2.38. Multiply the fractions and reduce the product to lowest terms. (again, not writing all of the numbers. suck it.) (side note: So dividing by 2 is the same as multiplying by 1/2. That makes sense— dividing a number by 2 and taking half of the number mean the same thing.)
2.19 Express 1.843 as an improper fraction.
The decimal 1.843 is read "one and eight hundred forty-three thousandths." Therefore, you should divide 843 by one thousand to convert the decimal into a mixed number: 1 843/1,000. The number left of the decimal, 1, becomes the whole part of the mixed number. That fraction cannot be reduced, because the greatest common factor of 843 and 1,000 is 1. Use the method described in Problem 2.7 to convert the mixed number into an improper fraction. 1 843/1000 = (1000)(1) + 843 over 1000 = 1000 +843 over 1000 = 1843/1000
Note: Problems 2.5-2.7 refer to the rational number 11/6. 2.7 Express the fraction as a mixed number.
The fraction 11/6 is considered improper because its numerator is greater than its denominator. To express it as a mixed number, divide 11 by 6. There is no need to use long division—it is sufficient to conclude that 6 divides into 11 one time with a remainder of 5. Thus 11/6 = 1 5/6 . The whole number portion of the mixed number is the number of times the divisor divides into the dividend; the remainder is the numerator of the fraction. The denominator of the fraction matches the denominator of the original fraction. In other words, given the improper fraction y/x, if x divides into y a total of w times with a remainder of r, then 7/x = w y/x. (side note: Improper doesn't mean unacceptable. It's fine to have fractions with bigger numbers on top than on bottom, and most teachers would rather you leave fractions in improper form, rather than write them as mixed numbers.)
2.29 Simplify the expression: 1/9+7/4-11/12.
The largest denominator (12) is not divisible by both of the remaining denominators, and neither is 12 x 2 = 24. However, 12 x 3 = 36 is divisible by 4, 9, and 12, so 36 is the least common denominator. Rewrite the expression using equivalent fractions with the LCD. 1/9 + 7/4 - 11/12 = 1 x 4 over 9 x 4 + 7 x 9 over 4 x 9 - 11 x 3 over 12 x 3 = 4/36 + 63/36 - 33/36 = 4+63-33 over 36 = 67-33 over 36 = 34/36 = 17/18 (side note: Divide the numerator and denominator by 2 to reduce the fraction.)
Note: Problems 2.26-2.27 refer to the fractions 2/3, 5/6, and 7/9. 2.26 Identify the least common denominator.
The largest denominator of the three fractions is 9. However, both of the remaining denominators are not factors of 9, so 9 is not the LCD. To identify another potential LCD candidate, multiply the largest denominator by 2: 9 x 2 = 18. All the denominators (3, 6, and 9) are factors of 18, so it is the LCD. (side note: If 18 didn't work, you'd test to see if 9 x 3 = 27 were the LCD. If 27 didn't work, you'd multiply 9 by 4, then 5, then 6, and so on, until finally all the denominators divided in evenly.)
Note: Problems 2.3-2.4 refer to the rational number 1/4. 2.3 Express the fraction as a decimal.
The rational number 1/4 represents the quotient 1 / 4. To express the fraction as a decimal, use long division to divide 4 by 1. Set up the long division problem, writing an additional zero at the end of the dividend. Copy the decimal point above the division symbol. For the moment, ignore the decimal point within the dividend and imagine that 1.0 is equal to 10. Because 4 divides into 10 two times, place a 2 above the rightmost digit of 10. Because 4 does not divide evenly into 10, a remainder will exist. Multiply 2 in the quotient by the divisor (4) and write the result (2x4=8) below the dividend (10). Draw a horizontal line beneath 8.Subtract 8 from 10 and write the result below the horizontal line. The difference (10 - 8 = 2) is not 0, so the quotient has not yet terminated. Place another 0 on the end of the dividend and on the end of the number below the horizontal line. This time, dividing the bottommost number by the divisor produces no remainder; 4 divides evenly into 20. Write the result above the division symbol next to the 2 already there. Multiply the newest digit in the quotient (5) by the divisor (4) and write the result beneath 20. Subtract the bottom two numbers. Because the remainder is 0, the division problem is complete: 1/4=1 / 4= 0.25 (side note 1: The number you're dividing BY is called the "divisor" and the number you're dividing INTO is called the "dividend." The answer you get once you're done dividing is called the "quotient.") (side note 2: You can add as many zeroes as you want, and you can do it at any time during the problem. Here's your goal: you want the answer to have either terminated or begun to repeat. If it hasn't done either, pop some more zeroes up there and keep going.) (side note 3: Rational numbers either repeat or terminate. When you get a remainder of 0, you stop dividing, and the decimal terminates.)
2.21 Express 0.72 (bar over 7 and 2) as a fraction
The repeated string of 0.727272 ... consists of two digits, so divide the repeated string (72) by two nines (99) and reduce the fraction to lowest terms. 0.72 bar over 7 and 2 = 72/99 = 72 / 9 over 99 / 9 = 8/11 Remember that this technique applies only when the repeated string of digits begins immediately to the left of the decimal point. If the repeated string begins farther left in the decimal, you should apply the technique described in Problems 4.26-4.28.
2.17 Reduce the fraction 2,024/8,448 to lowest terms and identify the greatest common factor of 2,024 and 8,448.
The technique demonstrated in Problems 2.11-2.16 is not convenient when the numbers in the numerator and denominator are very large. Because 2,024 and 8,448 each have many factors, rather than list all of them, identify one common factor (it does not matter which) and use it to reduce the fraction. In this case, both numbers are even, so you can divide each by 2. 2,024 / 2 over 8448 / 2 = 1,012/4224 The fraction is not yet reduced to lowest terms. Notice that the numerator and denominator of the resulting fraction are again even, so divide both by 2. 1,012 / 2 over 4,224 / 2 = 506/2,112 Once again the fraction consists of even numbers. Divide by 2. 506 / 2 over 2,112 / 2 = 253/1,056 Continue to look for common factors. Notice that 253 and 1,056 are evenly divisible by 11. 253 / 11 over 1,056 / 11 = 23/96 Now that the fraction is reduced to lowest terms, calculate the greatest common factor by multiplying each of the numbers that were eliminated from the fraction. GCF = 2 x 2 x 2 x 11 = 88 (side note 1: If it would take too long to write all the factors, find something that divides evenly into the top and the bottom of the fraction and reduce it. Keep doing that until you can't find any common factors.) (side note 2: When you're looking for factors, don't stop checking at 10, or you'll miss this one. You don't have to try every number in the world, though—it depends on the smaller of the two numbers in the numerator and denominator. If that number is less than 225, you can stop looking for factors at 15. If it's less than 400, you can stop looking for factors at 20. Basically, if the smaller number is less than a number N, then stop checking at N x N .)
Note: Problems 2.30-2.31 refer to the fractions 11/30 and 19/64. 2.30 Identify the least common denominator.
The technique described in Problem 2.23, calculating the least common denominator using consecutive multiples of the largest denominator, are not well suited to these fractions; 64 is not divisibly by 30, and neither are 64 x 2 = 128, 64 x 3 = 192, 64 x 4 = 256, or 64 x 5 = 320. Rather than continue to follow this pattern, you can use the prime factorizations of the denominators to determine the LCD. The first step, appropriately, is to determine the prime factorization of each denominator, 30 and 64; to accomplish this, write each number as a product. There is no single correct product, so it is equally valid to express 30 as 2 x 15, 3 x 10, or 5 x 6. Similarly, you could write 64 as 2 x 32, 4 x 16, or 8 x 8. 30 = 5 x 6 64 = 8 x 8 Factor any composite numbers in the products. For instance, 6 can be expressed as 2 x 3 and 8 can be expressed as 4 x 2. 30 = 5 x (2 x 3) 64 = (4 x 2) x (4 x 2) Continue factoring until only prime numbers remain. The only remaining composite number is 4, so express 4 as 2 x 2. 30 = 5 x 2 x 3 64 = (2 x 2) x 2 x (2 x 2) x 2 Rewrite the factorizations by listing the factors in order from least to greatest, using exponents when possible. 30 = 2 x 3 x 5 64 = 2 ^6 Now that the prime factorizations have been identified, you can begin to construct the least common denominator. Start by listing each unique factor that appears in either factorization. For instance, 30 has prime factors 2, 3, and 5; 64 has a single prime factor, 2, which is already included in the prime factorization of 30, so it isn't included a second time. LCD = 2 x 3 x 5 Each factor in the LCD should be raised to the highest power of that factor in either factorization. Both 3 and 5 only appear in the prime factorization of 30 and are raised to the first power, so 3 and 5 are also raised to the first power in the LCD. However, 2 appears in both factorizations, once to the first power and once to the sixth power. Choose the higher of the two powers, 2^6 , for the LCD. LCD = 2^6 x 3^1 x 5^1 = 2^6 x 3 x 5 Multiply the factors of the LCD together. LCD = 64 x 3 x 5 = 192 x 5 = 960 The least common denominator is 960. (side note 1: Every number unravels into a unique product of prime numbers, according to something called the Fundamental Theorem of Algebra. A prime factorization is like a number's unique fingerprint.) (side note 2: In other words, leave all prime numbers (like 5) alone. However, if you can rewrite a number as a multiplication problem containing something besides 1, you need to do that.) (side note 3: Exponents are covered more in depth in Problems 3.8-3.18. For now it's enough to know that 2 x 2 x 2 x 2 x 2 x 2 = 2^6. Count up how many 2s are multiplied together and turn that number into a power.)
Note: Problems 2.26-2.27 refer to the fractions 2/3, 5/6, and 7/9. 2.27 Simplify the expression:2/3+5/6-7/9 .
To rewrite the fraction using the least common denominator, divide each denominator into 18: 18 / 3 = 6, 18 / 6 = 3, and 18 / 9 = 2. Multiply the numerator and denominator of each fraction by the corresponding result. In other words, multiply the first fraction by 6/6, the second fraction by 3/3, and the third fraction by 2/2. 2/3 + 5/6 - 7/9 = 2 x 6 over 3 x 6 + 5x3 over 6 x 3 - 7 x 2 over 9 x 2 = 12/18 + 15/18 - 14/18 Combine the numerators into a single numerator divided by the least common denominator and simplify. 12/18 + 15/18 - 14/18 = 12 + 15 - 14 over 18 = 27-14 over 18 = 13/18 The result is already in lowest terms: 13/18.
Note: Problems 2.5-2.7 refer to the rational number 11/6. 2.5 Express the fraction as a decimal.
Use the method described in Problem 2.3 to rewrite the fraction as a long division problem. Here, however, there is no immediate need to place a 0 in the dividend. No decimal point is written explicitly, so write one at the end of the dividend and copy it above the division symbol. Six divides into eleven one time, so write 1 above the rightmost digit of 11. Multiply the divisor by the digit just written above the dividend , write the result below the dividend and subtract. The divisor (6) cannot divide into the result (5) a whole number of times. As Problem 2.3 directed, change 5 into 50 and also add a zero to the end of the dividend. Six divides into 50 eight times, so place 8 at the right end of the quotient. Multiply the new digit by the divisor (6), write the result below 50, and subtract. Six does not divide into 2 a whole number of times, so once again insert zeroes after 2 and 11.0. Six divides into 20 three times. Take the appropriate actions in the long division problem. Once again, 2 is the bottommost number in the long division problem. If you place zeros after it and after the dividend, it produces another 3 in the quotient. Once again, 2 is the bottommost number. Repeating this process is futile—each time a zero is added, it produces another 3 in the quotient and a difference of 2, the same number with which you started. Because the division problem has turned into an infinite loop producing the same pattern of digits, you can conclude that 11/6=1.83 (bar over the three). (side note 1:In Problem 2.3, you had to add a zero because 4 divides into 10 but it really doesn't divide into 1 very well. In Problem 2.5, however, 6 does divide into 11, so you don't need to write 11 as 110.) (side note 2: When you're long dividing, what you're dividing INTO has to be bigger than what you're dividing BY. If it's not, add a zero to the dividend and the bottom number.) (side note 3: Put a 3 above the division symbol, multiply it by the divisor (6), write that multiplication result (18) below 20, and then subtract it from 20.) (side note 4: If the decimal form of a rational number doesn't terminate, one or more digits will repeat infinitely. Write the decimal with a little bar over the 3 to indicate that it's an infinitely repeating digit in the decimal.)
2.36 Simplify the expression: (6/35)(14/15).
Write the product as a single fraction. (6/35)(14/15) = 6 x 14 over 35 x 15 Identify common factors of the numerator and denominator: 6 and 15 have common factor 3; 14 and 35 have common factor 7. Rewrite the fraction so that the common factors are explicitly identified. 6 x 14 over 35 x 15 = (2 x 3) x (2 x 7) over (5 x 7) x (3 x 5) = 2 x 2 x 3 x 7 over 3 x 5 x 5 x 7 Eliminate pairs of common factors from the numerator and denominator to reduce the fraction. 2 x 2 x 3 crossed x 7 crossed over 3 crossed x 5 x 5 x 7 crossed = 2 x 2 over 5 x 5 = 4/25
