schao wrote:Q: Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 24, 2.82, and 5.096 are three terminating decimals. If r and s are positive integers and the ratio r/s is expressed as a decimal, is r/s a termiating decimal?
1) 90<r<100
2) s=4
My problem is that I still don't get what's a terminating decimal and how can I approach this question??
Thanks.
Sherry
A TERMINATING decimal has a FINITE NUMBER OF DIGITS:
.5
.123
.8730253.
A NON-TERMINATING decimal has an INFINITE NUMBER OF DIGITS:
.33333....
.121212....
.871871871...
To determine whether a fraction will yield a terminating decimal:
1. Put the fraction in its MOST REDUCED FORM.
2. PRIME-FACTORIZE the denominator.
If the prime-factorization of the denominator includes ONLY 2's AND/OR 5's, the fraction will yield a TERMINATING decimal.
If the prime-factorization of the denominator includes ANY OTHER PRIME FACTOR, the fraction will yield a NON-TERMINATING decimal.
Case 1: 3/120
In its most reduced form, 3/120 = 1/40.
40 = 2² * 5.
Since the the prime-factorization of the denominator includes only 2's and 5's, 3/120 will yield a TERMINATING DECIMAL:
3/120 = .025
Case 2: 15/110
In its most reduced form, 15/110 = 3/22.
22 = 2*11.
Since the prime-factorization of the denominator includes 11 -- a prime factor OTHER THAN 2 OR 5 -- 15/110 will yield a NON-TERMINATING DECIMAL:
15/110 = .1363636...
Onto the problem at hand.
Question rephrased: When r/s is in its most reduced form, will the prime-factorization of the denominator include only 2's and/or 5's?
Statement 1: 90<r<100
No information about s.
INSUFFICIENT.
Statement 2: s=4
Since s=2², when r/s is in its most reduced form, the prime-factorization of the denominator will not include a prime factor other than 2.
Thus, r/s will yield a terminating decimal.
SUFFICIENT.
The correct answer is
B.
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