the decimal expansion of120/(3^2)is
Answers
Answer:
How many places will the decimal expansion of 147/120 terminate?
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Here’s a procedure you can use to answer this question for any fraction, not just 147120 .
Step 1: Write out the prime factorizations for the numerator and denominator.
Every composite number can be written as a unique product of prime numbers. For example, the number 4 can be written as 2*2 (two is the only even prime number) and the number 6 can be written as 2*3.
The prime factorization of 147 is 3*7*7. The prime factorization of 120 is 2*2*2*3*5.
147120=3⋅7⋅72⋅2⋅2⋅3⋅5
NOTE: If the numerator or denominator is 1, you can simply put 1. 1 is not a prime number, but it won’t mess up the rest of the process, so just go with it.
Step 2: Eliminate any prime factors that appear in both the numerator and the denominator.
This is just reducing the fraction to lowest terms. If the fraction was already in lowest terms, there won’t be any prime factors that appear in both the numerator and the denominator.
In this case, the prime factor 3 appears in both the numerator and the denominator, which means that this fraction is not in lowest terms. Strike out the 3 above and below.
3⋅7⋅72⋅2⋅2⋅3⋅5=7⋅72⋅2⋅2⋅5
Step 3: Now let’s determine if the decimal expansion terminates or repeats. Look at the remaining prime factors in the denominator. If there are only 2s and 5s, then the decimal expansion will terminate. If there are any prime factors in the denominator other than 2 or 5, the decimal expansion will repeat forever.
This is because the decimal expansion is base 10, and the only prime factors of 10 are 2 and 5. If the denominator contains even one prime factor that doesn’t divide 10, the decimal expansion will be unable to reach a point where the remainder is zero.
In this case, the denominator’s prime factorization (after reducing) has three factors of 2 and one of 5, so we know that its decimal expansion will terminate.
If the decimal expansion repeats, there are ways to figure out how many digits must pass before the sequence starts over…but that’s a topic for another day.
Step 4: Once you’ve established that the decimal expansion terminates, determine whether there are more 2s or 5s in the decimal expansion. Count the number of instances of whichever factor appears more frequently. That’s how many places the decimal expansion will run on before it terminates. If there are equal numbers of 2s and 5s, use either number as your guide.
Our fraction has three instances of 2 and one of 5 in the denominator. There are more 2s than 5s. In fact, there are three 2s, which means that the decimal expansion runs for three digits before terminating. And indeed it does:
147120=1.225
Now, I haven’t tested this method with every fraction to see if it yields the correct number of decimal places, but based on how fractions are turned into decimal expansions, I believe it should work. If anybody knows of a counter-example, I hope they’ll be kind enough to bring it to my attention.
Also, there’s a much simpler way to find out the answer to your question…
Step 1. Punch 147÷120 into a calculator. Count the number of digits after the decimal on the display screen.
But hey, you don’t really learn the process doing it that way.