How many digits are in the repeating cycle of 17/27?
To find the repeating cycle of the fraction 17/27, we need to perform long division.
____
27|17.00
16
---
100
81
--
190
189
---
100
We see that the remainder is 100, which is the same as the dividend when multiplied by 10.
___
27|100.00
81
---
190
189
---
100
Again, we see that the remainder is 100. So, the process will repeat.
Since the remainder is the same after every division, the repeating cycle in 17/27 is the number 100.
Therefore, the repeating cycle of 17/27 has 3 digits.
To convert 0.82¯¯¯¯¯ to a rational number, we can set x equal to 0.82¯¯¯¯¯ and multiply both sides of the equation by an appropriate power of 10. Since there are two repeating digits after the decimal point, we would multiply both sides of the equation by 100:
100x = 82.82¯¯¯¯¯
This multiplication allows us to shift the decimal point to eliminate the repeating pattern.
Therefore, to convert 0.82¯¯¯¯¯ to a rational number, we would multiply both sides of the equation by 100.
Express 7/13 as a repeating decimal. Mark the repeating digits with a bar.
To express 7/13 as a repeating decimal, we perform long division:
____
13| 7.000
6
---
10
9
--
10
9
--
10
9
The remainder is 10, which is the same as the dividend when multiplied by 10.
____
13| 7.000
6
---
10
9
--
10
9
--
10
9
Again, the remainder is 10, which indicates that the digits will start repeating.
Hence, 7/13 as a repeating decimal is 0.538461 with the digits 538461 repeating. To indicate the repeating digits, we put a bar over them, so the repeating decimal form of 7/13 is 0.538461
Convert 0.8
to a rational number.
To convert 0.8 to a rational number, we can rewrite it as a fraction by placing it over a power of 10.
0.8 can be written as 8/10 because 0.8 is equivalent to 8 tenths.
Next, we can simplify 8/10 by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
8 ÷ 2 = 4
10 ÷ 2 = 5
Therefore, 0.8 as a rational number is 4/5.
Convert 0.8¯¯¯
to a rational number.
To convert 0.8¯¯¯ to a rational number, we need to recognize that the bar indicates a repeating decimal pattern. Let's call the number 0.8x, where x represents the repeating pattern.
To eliminate the decimal, we can multiply both sides of the equation by 10, since there is only one repeating digit after the decimal point.
10 * 0.8x = 8.x
Next, we subtract the original equation from the new equation:
10 * 0.8x - 0.8x = 8.x - 0.8x
Expanding these expressions, we have:
8x = 8.x - 0.8x
8x = 8.8x - 0.8x
8x = 8x
We have obtained an equation where the variable x cancels out. This indicates that x does not depend on any specific value and that the repeating pattern 0.8¯¯¯ can be converted to a rational number without the need for any specific value for x.
Therefore, 0.8¯¯¯ as a rational number is equal to 8/9.
To convert 0.82¯¯¯¯¯
to a rational number, you would set x
equal to 0.82¯¯¯¯¯
and then multiply both sides of the equation by what number?