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Convert the decimal number 0.33 in the form qpтАЛ, where p and q are integers and qюАа=0.
991тАЛ
31тАЛ
331тАЛ
91тАЛ
31тАЛ
For a single repeating digit after the decimal point, simply place the digit over a 9. For 0.333..., the result is 3/9=1/3.
The repeating decimal number is 0.333... or 0.3bar.
x=0.n=9nтАЛ
For a single repeating digit after the decimal point, simply place the digit over a 9. For 0.333..., the result is 3/9=1/3.
Confusing 0.3 with 0.33 (which is 33/100) or mistaking the denominator for 99 instead of 9.
Set up the equation
Let x be the repeating decimal.
x=0.3333...
Multiply to shift the decimal
Since one digit repeats, multiply by 10 to move the decimal point once.
10x=3.3333...
Subtract the equations
Subtract the first equation (x=0.3333...) from the second (10x=3.3333...).
10xтИТx=3.3333...тИТ0.3333...тЯ╣9x=3
Solve for x
Divide by 9 to isolate x and simplify the fraction.
x=93тАЛ=31тАЛ
B is correct because the repeating decimal 0.3 simplifies to 93тАЛ, which reduces to 31тАЛ.
This method works for any repeating decimal; for example, 0.ab=99abтАЛ and 0.abc=990abcтИТaтАЛ.
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