# ALGABRA 1 - Subtracting Fractions with Different Denominators

The following expression contains two fractions which have different denominators:

5/7 - 5/8 = ?

Rewrite each fraction so that both fractions have the same common denominator, and then subtract as indicated.

### Comments for ALGABRA 1 - Subtracting Fractions with Different Denominators

 Sep 30, 2011 Subtracting Fractions with Different Denominators by: Staff ----------------------------------------------------------- Part II When you convert 5/7 (the original fraction) and 40/56 to their decimal equivalents, you can see that both fractions are equivalent. Each fraction has the same decimal value: 5 / 7 = 0.714286 40 / 56 = 0.714286 Therefore, 5 / 7 = 40 / 56 Next, apply the same idea to convert the fraction 5/8 to a fraction with the denominator of 56. “IF” the denominator of 5/8 could by multiplied by 7, the new denominator would be 56: 8*7 = 56 This is exactly what you want. However, as before, 5/8 does not have the same value as 5/56. In order to preserve the value of the original fraction (5/8), both the numerator and denominator must each be multiplied by the same amount: the number 7. Therefore, = 5/8 * 7/7 = (5 * 7) / (8 * 7) = (35) / (56) = 35 / 56 When you convert 5/8 (the original fraction) and 35/56 to their decimal equivalents, you can see that both fractions are equivalent. Each fraction has the same decimal value: 5 / 8 = 0.625 35 / 56 = 0.625 Therefore, 5 / 8 = 35 / 56 Now you can complete the subtraction: 5 / 7 = 40 / 56 5 / 8 = 35 / 56 5 / 7 – 5 / 8 = ? Substitute 40 / 56 for 5 / 7 and 35 / 56 for 5 / 8 40 / 56 - 35 / 56 = ? When adding or subtracting fractions, the denominator does not change. Only the numerators are added or subtracted. = (40 - 35) / 56 = (5) / 56 = 5 / 56 The final answer is: 5 / 56 5 / 7 - 5 / 8 = 5 / 56 Check the answer with a calculator: 5 / 7 - 5 / 8 = 5 / 56 The ≈ is used below (instead of an equal sign) because 5 / 7 and 5 / 56 are repeating decimals 0.714286 - 0.625 ≈ 0.0892857 0.089286 ≈ 0.0892857 Since 0.089286 is the same answer obtained by converting 5 / 56 to its decimal equivalent, the answer of 5 / 56 is correct. ---------------------------------------------- A final note: Why should you learn how to subtract fractions when you can use a calculator and compute the final answer immediately? With a calculator, you don’t need to know anything about the details of the subtraction process. There are a couple of reasons for learning the process: 1. The most important reason is: YOU WILL NOT ALWAYS BE ADDING OR SUBTRACTING NUMBERS. In general, a calculator can do this: 5 / 7 - 5 / 8 . . . but, it cannot do this: x / y - 5x / y² To complete the second problem (x / y - 5x / y²) you need to know and understand the process. 2. The second most important reason is: Converting fractions to decimals will often yield approximate results. 0.714286 - 0.625 ≈ 0.0892857 is approximate. 5 / 7 - 5 / 8 = 5 / 56 is exact. Thanks for writing. Staff www.solving-math-problems.com

 Sep 30, 2011 Subtracting Fractions with Different Denominators by: Staff Part I The question: 5/7 - 5/8 = ? The answer: 5/7 - 5/8 = ? As you already know, fractions cannot be added or subtracted unless they have the same denominator. 5/7 and 5/8 do not have the same denominator. Therefore, each fraction must be rewritten so that both fractions have the same common denominator. The new common denominator can be the least common multiple (LCM) of the two original denominators: 7 and 8 ------------------------------------------------ To find the LCM, you can use the prime factorization method (this is a method you can use to compute the LCM with any fractions) - Factor the numbers 7 and 8 into prime factors. Use exponents to show how many times each prime factor is listed. 7 = 7¹ 8 = 2³ - List every prime number listed as a factor of either 7 or 8. Prime Number Factors Listed: 7, 2 - List these same prime factors again, but include the highest exponent listed for each Prime Number Factors with highest exponent: 2³, 7¹ Multiply the factors with the highest exponent to compute the LCM: LCM = 2³ * 7¹ = 56 ------------------------------------------------ Convert the denominators of both fractions 56 5/7 - 5/8 “IF” the denominator of 5/7 could by multiplied by 8, the new denominator would be 56: 7*8 = 56 This is exactly what you want. However, there is a problem. 5/7 does not have the same value as 5/56. In order to preserve the value of the original fraction (5/7), both the numerator and denominator must each be multiplied by the same amount: the number 8. To apply this concept, multiply the original fraction by 8/8. The fraction 8/8 is equal to 1, so the original fraction is merely being multiplied by 1. As you can see by the following illustration, its value has not been changed. = [original fraction] = [original fraction] * 8/8 = [original fraction] * 1 = [original fraction] Therefore, = [original fraction] * 8/8 = 5/7 * 8/8 When multiplying two fractions, multiply the both numerators and multiply the both denominators. = 5/7 * 8/8 = (5 * 8) / (7 * 8) = (40) / (56) = 40 / 56 -----------------------------------------------------------

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