decimal expression format (dominator) for terminating divide
Answers
Step-by-step explanation:
Any rational number (that is, a fraction in lowest terms) can be written as either a terminating decimal or a repeating decimal . Just divide the numerator by the denominator . If you end up with a remainder of 0 , then you have a terminating decimal.
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Step-by-step explanation:
Rational Numbers
Integers are positive and negative whole numbers, including zero. Here are the integers:
{ … -3, -2, -1, 0, 1, 2, 3, …}
When we take a ratio of two integers, we get a rational number.
A rational number is any number of the form a/b, where a & b are integers, and b ≠ 0.
Rational numbers are the set of all fractions made with integer ingredients. Notice that all integers are included in the set of rational numbers, because, for example, 3/1 = 3.
Rational Numbers as Decimals
When we make a decimal out of a fraction, one of two things happens. It either terminates (comes to an end) or repeats (goes on forever in a pattern). Terminating rational numbers include:
1/2 = 0.5
1/8 = 0.125
3/20 = 0.15
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9/160 = 0.05625
Repeating rational numbers include:
1/3 = 0.333333333333333333333333333333333333…
1/7 = 0.142857142857142857142857142857142857…
1/11 = 0.090909090909090909090909090909090909…
1/15 = 0.066666666666666666666666666666666666…
When Do Rational Numbers Terminate?
The GMAT won’t give you a complicated fraction like 9/160 and expect you to figure out what its decimal expression is. BUT, the GMAT could give you a fraction like 9/160 and ask whether it terminates or not. How do you know?
Well, first of all, any terminating decimal (like 0.0376) is, essentially, a fraction with a power of ten in the dominator; for example, 0.0376 = 376/10000 = 47/1250. Notice we simplified this fraction, by cancelling a factor of 8 in the numerator. Ten has factors of 2 and 5, so any power of ten will have powers of 2 and powers of 5, and some might be canceled by factors in the numerator , but no other factors will be introduced into the denominator. Thus, if the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, then it can be written as something over a power of ten, which means its decimal expression will terminate.
If the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, the decimal expression terminates. If there is any prime factor in the denominator other than 2 or 5, then the decimal expression repeats.
Here’s some examples of this concept at work:
1/24 repeats (there’s a factor of 3)
1/25 terminates (just powers of 5)
1/28 repeats (there’s a factor of 7)
1/32 terminates (just powers of 2)
1/40 terminates (just powers of 2 and 5)
Notice, as long as the fraction is in lowest terms, the numerator doesn’t matter at all. Since 1/40 terminates, then 7/40, 13/40, or any other integer over 40 also terminates. Since 1/28 repeats, then 5/28 and 15/28 and 25/28 all repeat; notice, though that 7/28 doesn’t repeat, because of the cancellation: 7/28 = 1/4 = 0.25.
Shortcut Decimals to Know
There are certain decimals that are good to know as shortcuts, both for fraction-to-decimal conversions and for fraction-to-percent conversions. These are:
1/2 = 0.5
1/3 = 0.33333333333333333333333333…
2/3 = 0.66666666666666666666666666…
1/4 = 0.25
3/4 = 0.75
1/5 = 0.2 (and times 2, 3, and 4 for other easy decimals)
1/6 = 0.166666666666666666666666666….
5/6 = 0.833333333333333333333333333…
1/8 = 0.125
1/9 = 0.111111111111111111111111111… (and times other digits for other easy decimals)
1/11 = 0.09090909090909090909090909… (and times other digits for other easy decimal