Clarifying Fraction NotationDate: 07/02/97 at 00:04:39 From: Paul Subject: HELP The problem: 83 1/3 .83 1/3 = --------------------- = ? 100 The answer is supposed to be 5 --- 6 Please help me. My nephew is trying to stump me. Is this a trick problem? Date: 08/29/97 at 14:57:06 From: Doctor Sonya Subject: Re: HELP Hi Paul: It is hard to tell what this notation means. We don't usually include a fraction such as 1/3 after a decimal point. It confuses the meaning of a decimal, a fraction whose denominator is a power of 10, not 10 and 3 as here. The number of digits after the decimal place is the same as the power of 10 in the denominator. For example, 3/10 = 0.3, 40/100 = 0.40, 333/1000 = 0.333, etc. You could rewrite 0.83 1/3 as a series of decimals by separating the digits like this: 0.83 1/3 = 0.8 + 0.03 + 0.00 1/3 which can be written 8/10 + 3/100 + (1/3)/100. The sum of these terms will simplify correctly into the answer you want. But there is a more direct way to solve your nephew's problem, and it involves a different interpretation of the notation. The "decimal" part of the number has two decimal places. Therefore, you can think of 0.83 1/3 as the compound fraction (83 1/3)/100, where the mixed fraction 83 1/3 is the numerator, and 100 is the denominator. Rewrite the mixed fraction in the numerator as the compound fraction 250/3. The expression now looks like (250/3)/100. Dividing by 100 is the same as multiplying by 1/100. Therefore, (250/3)/100 is equivalent to (250/3) x 1/100. Multiply the two fractions: The product is 250/300 which reduces to 25/30 = 5/6. There's also an algebraic way to solve the problem, if you're interested. It is the most sophisticated, but also the most elegant. The number 1/3 as a decimal is 0.3333..., where the ellipsis after the 3 means that the 3s never stop coming. Then, write 0.83 1/3 as 0.83333.... with lots of trailing 3s implied. Let x = 0.833333... Multiply both sides by 10: 10x = 8.3333... Subtract the first equation from the second and all the repeating 3s disappear. 10 x = 8.3333... 1 x = 0.8333... _______________ 9 x = 7.5 (Divide both sides by 9.) x = 7.5/9 = 75/90 = 5/6. Whew! This was a fun problem! -Doctors Chita and Sonya, The Math Forum http://mathforum.org/dr.math/ |
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