Convert 0.bar(8) into a rational number
To convert the recurring decimal 0.bar(8) into a rational number, we can use algebraic manipulation.
Let x = 0.bar(8)
Multiply both sides of the equation by 10 to move the decimal point:
10x = 8.bar(8)
Subtracting the original equation from this new equation eliminates the recurring part:
10x - x = 8.bar(8) - 0.bar(8)
9x = 8
Divide both sides of the equation by 9:
x = 8/9
Therefore, 0.bar(8) is equivalent to the rational number 8/9.
To convert 0.bar(82) to a rational number, you would set x equal to 0.bar(82) and then multiply both sides of the equation by what number?
To convert 0.bar(82) into a rational number, let's set x equal to 0.bar(82).
Since there are two digits in the recurring part (82), we need to multiply both sides of the equation by a power of 10 that will eliminate the recurring part.
In this case, we multiply by 100 because we want to move the decimal point two places to the right.
So, multiply both sides of the equation by 100:
100x = 82.bar(82)
Now, subtract the original equation from this new equation, which will eliminate the recurring part:
100x - x = 82.bar(82) - 0.bar(82)
99x = 82
Finally, divide both sides of the equation by 99:
x = 82/99
Therefore, 0.bar(82) is equivalent to the rational number 82/99.
To convert the repeating decimal 0.bar(8) into a rational number, we need to consider it as an infinite geometric series.
Let's represent the number 0.bar(8) as x. By multiplying x by 10, we can shift the decimal point to the right, resulting in 10x. Now, if we subtract x from 10x, the repeating part of the decimal will cancel out.
10x - x = 9x
So, we have 9x = 0.bar(8).
Next, by multiplying 9x by 10 to shift the decimal point, we get 90x. Subtracting 9x from 90x, the repeating part cancels out again.
90x - 9x = 81x
Therefore, 81x = 0.bar(8).
To find x, we divide both sides of the equation by 81:
81x / 81 = 0.bar(8) / 81
x = 0.bar(8) / 81
Now, let's calculate the value of x:
To represent the repeating decimal 0.bar(8) as a fraction, we can write it as follows:
0.bar(8) = 8/9
Substituting this value into the equation for x:
x = 8/9 / 81
To simplify further, we need to write 81 as a fraction:
81 = 81/1
Now, we can divide the fractions:
x = 8/9 ÷ 81/1
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
x = 8/9 × 1/81
Simplifying the multiplication:
x = (8 × 1) / (9 × 81)
x = 8 / 729
Therefore, the rational number equivalent to 0.bar(8) is 8/729.