uptowngirl92 wrote:hey guys can someone give me a more theoritical approach to this?I remember reading somewhere that dividing any number by 3 or 11(i.e if the denominator is 3 or 11)will
always produce a recurring fraction.knowing that,we can eliminate options (a),(b),(d) straight away!However,my memory falls short there
Can anybody tell me such observations which will help me eliminate (c)?Can it be said that since 99 is a multiple of 11 that too will provide a recurring decimal??
It's actually true that *every time* you divide one integer by another, one of two things must happen:
-you get a 'terminating decimal', i.e. a decimal that stops. 1/2 = 0.5, or 1/8 = 0.125 are two examples;
-you get a 'recurring decimal', i.e. a decimal that eventually 'loops' in a repeating pattern forever. 1/3 = 0.3333.... or 1/11 = 0.090909.... or 5/27 = 0.185185185.... are three examples.
So in the above question, 23/37 will also produce a recurring decimal (if you calculate it, you'll find it equals 0.621621621...). And if you do calculate it using long division, you should see why the decimal must eventually repeat. When we divide 23 by 37, to find the tenths digit of the decimal, we ask "how many times can we divide 230 by 37?". The answer is 6, with a remainder of 8. We then use that remainder to find the next digit. Since at each stage, we are just finding remainders when dividing by 37, eventually we have to get back to a remainder we've already used, and the digits will begin to repeat. I hope that's clear - it would be easier to demonstrate if I could write out a long division properly here - but it should be easier to see why the digits must repeat if you actually perform the steps in the long division.
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While it's not likely to be important to know the following for the GMAT, there are numbers which do not have any repeating pattern of digits in their decimal. Pi is one famous example, as is the square root of 2 (or any other root which isn't an integer). This is what makes memorizing the digits of Pi considerably more challenging than memorizing the digits of, say, the decimal equivalent of 1/3.
Because the decimal equivalents of Pi and root(2) do not repeat, it is impossible to write these numbers as fractions involving integers. That is, you cannot possibly find two integers a and b for which Pi = a/b, or for which root(2) = a/b. Decimals which cannot be written as fractions involving integers are known as 'irrational numbers' in mathematics, while numbers that can be written as fractions using only integers are called 'rational numbers'. The 'real numbers' include all of the rational and irrational numbers, and nothing else.
