How many digits are in the repeating cycle of 1727?
How many digits are in the repeating cycle of 17/27?
Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to check their denominator for any prime factors other than 2 and 5. If a number has only 2 and/or 5 as its prime factors, it can be expressed as a terminating decimal. Otherwise, it will have a repeating decimal representation.
Let's check each number:
- 78: The prime factorization of 78 is 2 * 3 * 13. Since 78 has a prime factor other than 2 and 5, it cannot be expressed as a terminating decimal.
- 512: The prime factorization of 512 is 2^9. Since 512 only has 2 as a prime factor, it can be expressed as a terminating decimal.
- 1725: The prime factorization of 1725 is 3^2 * 5^2 * 7. Since 1725 has prime factors other than 2 and 5, it cannot be expressed as a terminating decimal.
- 3140: The prime factorization of 3140 is 2^2 * 5 * 157. Since 3140 has prime factors other than 2 and 5, it cannot be expressed as a terminating decimal.
Therefore, out of the given numbers, only 512 can be expressed as a terminating decimal. The rest of the numbers (78, 1725, and 3140) will have a repeating decimal representation.
Which of the following rational numbers can be expressed as a repeating decimal: 7/8, 5/12, 17/25, or 31/40?
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to check their denominators for any prime factors other than 2 and 5. If a number has only 2 and/or 5 as its prime factors, it can be expressed as a terminating decimal. Otherwise, it will have a repeating decimal representation.
Let's check each number:
- 7/8: The denominator, 8, has the prime factor 2 but no other prime factors. Therefore, 7/8 can be expressed as a terminating decimal.
- 5/12: The denominator, 12, has the prime factors 2 and 3. Since 12 has a prime factor other than 2 and 5, 5/12 cannot be expressed as a terminating decimal.
- 17/25: The denominator, 25, has the prime factor 5 but no other prime factors. Therefore, 17/25 can be expressed as a terminating decimal.
- 31/40: The denominator, 40, has the prime factors 2 and 5. Since 40 only has 2 and 5 as its prime factors, 31/40 can be expressed as a terminating decimal.
Therefore, out of the given numbers, only 5/12 will have a repeating decimal representation. The rest of the numbers (7/8, 17/25, and 31/40) can be expressed as terminating decimals.
Express 7/13 as a repeating decimal. Mark the repeating digits with a bar.
Note: You will find the option to add a bar above text in the Geometry (∠
) keyboard.
To express 7/13 as a repeating decimal, we can use long division.
Let's perform the long division:
0.538461538...
We can see that the repeating pattern starts to emerge after the decimal point, specifically after the 538. The pattern repeats indefinitely. To indicate this repeating pattern, we write a bar above the repeating digits, which in this case are 538.
Therefore, 7/13 can be expressed as 0.538¯¯¯¯¯¯¯¯¯.
To determine the number of digits in the repeating cycle of a given fraction, we can use long division.
When we divide 1 by 7, the decimal representation is 0.142857 repeating. Similarly, dividing 10 by 7 gives us 1.428571 repeating, and dividing 100 by 7 gives us 14.285714 repeating.
Now, let's divide 1,000 by 7:
142857
-----------
7 | 1,000.000
- 7
30
- 28
20
- 14
60
- 56
40
- 35
50
- 49
10
We can see that after subtracting 7 multiple times, a remainder of 1 appears. This indicates that the repeating cycle of 1727 has 6 digits.
Therefore, the repeating cycle of 1727 has 6 digits.
To determine the number of digits in the repeating cycle of a given fraction, we can use long division.
When we divide 17 by 27, the decimal representation is 0.629629 repeating.
Now, let's divide 629 by 27:
23.2962
----------
27 | 629.0000
- 27
31
- 27
42
- 27
52
- 27
52
- 27
14
--------
51
We can see that after subtracting 27 multiple times, a remainder of 51 appears. This indicates that the repeating cycle of 17/27 has 51 digits.
Therefore, the repeating cycle of 17/27 has 51 digits.