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Express 0.512 in the form qpтАЛ, where p and q are integers and qюАа=0.
999512тАЛ
99952тАЛ
99512тАЛ
999502тАЛ
999512тАЛ
For a pure recurring decimal with 'n' repeating digits, write the digits in the numerator and 'n' nines in the denominator.
Recurring decimal number x = 0.512 (bar over 512)
x=qpтАЛ=10nтИТ1repeating┬аpartтАЛ
For a pure recurring decimal with 'n' repeating digits, write the digits in the numerator and 'n' nines in the denominator.
Students sometimes mistake the bar notation or include 0 as a repeating digit incorrectly, or fail to simplify if the fraction is reducible.
Define the variable
Let x be the repeating decimal 0.512. This means x=0.512512512...
x=0.512
Multiply by power of 10
Since there are 3 repeating digits under the bar, multiply both sides of the equation by 103=1000.
1000x=512.512512...
Subtract the original equation
Subtract the equation x=0.512512... from 1000x=512.512512... to eliminate the decimal part.
1000xтИТx=512.512512...тИТ0.512512...
Solve for x
Simplify the resulting equation 999x=512 to find the fraction form.
x=999512тАЛ
A is correct because the repeating decimal 0.512 converts directly to the fraction 999512тАЛ by placing the repeating sequence over as many nines as there are repeating digits.
This concept is a precursor to understanding geometric series where тИСn=1тИЮтАЛarnтИТ1=1тИТraтАЛ, as recurring decimals are essentially infinite geometric series.
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