How many digits are in the repeating cycle of 17/27??
To find the repeating cycle of a fraction, we need to divide the numerator by the denominator. When we calculate 17 divided by 27, the result is 0.629629629... The repeating cycle is "629," which has 3 digits. Therefore, the repeating cycle of 17/27 has 3 digits.
Which of the following rational numbers can be expressed as a repeating decimal:
7/8, 5/12, 17/25, or 31/40?
To determine whether a rational number can be expressed as a repeating decimal, we need to check if the denominator has any prime factors other than 2 and 5.
Let's check each of the given options:
1. 7/8: The denominator is 8, which can be expressed as 2^3. Since the only prime factors of the denominator are 2 and 5, 7/8 can be expressed as a repeating decimal.
2. 5/12: The denominator is 12, which can be expressed as 2^2 * 3. Since it has a prime factor other than 2 and 5, namely 3, 5/12 cannot be expressed as a repeating decimal.
3. 17/25: The denominator is 25, which can be expressed as 5^2. Since the only prime factors of the denominator are 2 and 5, 17/25 can be expressed as a repeating decimal.
4. 31/40: The denominator is 40, which can be expressed as 2^3 * 5. Since it has a prime factor other than 2 and 5, namely 3, 31/40 cannot be expressed as a repeating decimal.
Therefore, the rational numbers that can be expressed as repeating decimals are 7/8 and 17/25.
Wrong
Apologies for the mistake. Let's correctly check each option:
1. 7/8: The denominator is 8, which can be expressed as 2^3. Since the denominator only has the prime factor 2, 7/8 can be expressed as a terminating decimal and not as a repeating decimal.
2. 5/12: The denominator is 12, which can be expressed as 2^2 * 3. Since it has a prime factor other than 2 and 5, namely 3, 5/12 cannot be expressed as a repeating decimal.
3. 17/25: The denominator is 25, which can be expressed as 5^2. Since the only prime factors of the denominator are 2 and 5, 17/25 can be expressed as a repeating decimal.
4. 31/40: The denominator is 40, which can be expressed as 2^3 * 5. Since it has a prime factor other than 2 and 5, namely 3, 31/40 cannot be expressed as a repeating decimal.
Therefore, the rational number that can be expressed as a repeating decimal is 17/25.
It was 5/12
Apologies for the incorrect information. You are correct.
The rational number 5/12 can be expressed as a repeating decimal. The denominator of 12 can be expressed as 2^2 * 3. Since it has a prime factor of 3, the decimal representation of 5/12 will repeat. Therefore, 5/12 can be expressed as a repeating decimal.
Express 7/13 as a repeating decimal. Mark the repeating digits with a bar.
To express 7/13 as a repeating decimal, we need to do long division. Let's start by dividing 7 by 13:
```
0.
----------
13 | 7.000000000...
-6
---
10
```
We get a quotient of 0.53846153846... The remainder is 10, so we need to multiply it by 10 and continue dividing:
```
0.5
----------
13 | 70.00000000...
-65
---
500
-39 (bring down the next digit, which is 0)
---
1100
-104
----
600
- 52
----
480
- 39
----
410
- 39
----
110
- 104
------
60
```
This process continues indefinitely, meaning the decimal representation of 7/13 repeats. The repeating portion is 538461, so the decimal representation is:
7/13 = 0.538461538461... (the repeating portion is marked with a bar over it)
To find the number of digits in the repeating cycle of a decimal representation of a fraction, we need to perform long division until the remainder becomes 0 or we notice a repeating pattern.
Let's perform the long division for 17/27:
__________________
27 | 17.00000
- 16
_______________
100
- 81
____________
190
- 189
______
1000
- 972
_______
2800
- 2700
_______
10000
After carrying out the long division, we can see that the remainder of the division repeats after every fourth decimal place. Therefore, the repeating cycle of 17/27 has 4 digits.
So, the answer is that the repeating cycle of 17/27 has 4 digits.