Express 7/13 as a repeating decimal. Mark the repeating digits with a bar.
To convert 7/13 to a repeating decimal, we need to perform long division.
Let's start by dividing 7 by 13:
```
0. (The quotient would be 0)
________
13| 7.000000
```
Since 7 is smaller than 13, we bring down the next digit of 0 after the decimal point.
```
0.5
________
13| 7.000000
0 (quotient obtained so far; there will be repeating digits)
```
Next, we add a decimal point and continue dividing. Bring down a zero again.
```
0.53
________
13| 7.000000
0
_______
70 (subtract 13 × 5 = 65 from 70)
```
Subtract 65 from 70 which gives 5. Bring down another zero.
```
0.538
________
13| 7.000000
0
_______
70
65 (subtract 13 × 0.5 = 6.5 from 50)
_______
50 (bring down another zero)
```
Since we obtained the remainder of 50, we bring another zero and continue dividing.
```
0.5384
________
13| 7.000000
0
_______
70
65
_______
50
39 (subtract 13 × 0.53 ≈ 6.89 from 50)
_______
110 (bring down another zero)
```
Again, we obtain a remainder of 110, so we bring another zero and continue dividing.
```
0.53846...
________
13| 7.000000
0
_______
70
65
_______
50
39
_______
110
104 (subtract 13 × 0.5384 ≈ 6.99792 from 110)
_______
6 (bring down another zero; remainder is 6)
```
Since we have a remainder of 6, we bring a zero and continue dividing.
```
0.538461...
________
13| 7.000000
0
_______
70
65
_______
50
39
_______
110
104
_______
6
```
We obtained a remainder of 6, which is the same as the remainder we had previously. Hence, we have a repeating decimal.
Thus, 7/13 can be expressed as a repeating decimal: 0.538461... (with 6 and 538461 repeating)
Convert 0.8¯¯¯ to a rational number
To convert the repeating decimal 0.8¯¯¯ to a rational number, we can express it as a fraction.
Let's denote x = 0.8¯¯¯ and write it as an equation:
x = 0.8¯¯¯
Next, we multiply both sides of the equation by 10 to eliminate the repeating decimal:
10x = 8.8¯¯¯
Subtracting the two equations, we can eliminate the repeating decimal on the right side:
10x - x = 8.8¯¯¯ - 0.8¯¯¯
9x = 8
Dividing both sides of the equation by 9, we can solve for x:
x = 8/9
Therefore, the repeating decimal 0.8¯¯¯ is equivalent to the rational number 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?
To convert the repeating decimal 0.82¯¯¯¯¯ to a rational number, we would set x equal to 0.82¯¯¯¯¯ and then multiply both sides of the equation by a suitable power of 10.
Since the repeating part consists of 2 digits, we need to multiply both sides by a power of 10 that has the same number of digits as the repeating part. In this case, we have two repeating digits, so we will multiply both sides by 100.
Therefore, we would multiply both sides of the equation x = 0.82¯¯¯¯¯ by 100:
100x = 82.82¯¯¯¯¯
By subtracting the two equations, we can eliminate the repeating part on the right side:
100x - x = 82.82¯¯¯¯¯ - 0.82¯¯¯¯¯
99x = 82
Finally, we divide both sides of the equation by 99 to solve for x:
x = 82/99
So, the repeating decimal 0.82¯¯¯¯¯ can be expressed as the rational number 82/99.