Question 217635
The directions you describe are a little confusing to me, too. Let me explain what I think they may be:<ul><li><i>Every</i> fraction can be converted to decimal form by long division.</li><li>Some fractions can be converted in a different way which may often be easier than long division. These fractions have denominators which can be changed to powers of ten. And once the denominator is a power of ten, the decimal form can be found by inspection.</li></ul>
From the directions you describe it sounds like your teacher prefers that you use the second procedure when possible.<br>
Here's a procedure to use:<ol><li>Reduce the fraction, if possible.</li><li>If the denominator is already a power of ten (10, 100, 1000, etc.), skip to step #7</li><li>Factor the denominator into prime numbers. (Do not include 1.)</li><li>If the prime factors include any numbers other than 2 or 5, then use long division.</li><li>If the prime factors are just 2's and/or 5's, then figure out the "appropriate factors". The "appropriate factors" are the factors we need so that every factor of 2 can be paired with a factor of 5 and vice versa. Here are some examples:<ul><li>Denominator: 2 "Appropriate factors": 5</li><li>Denominator: 5 "Appropriate factors": 2</li><li>Denominator: 2*2 "Appropriate factors": 5*5</li><li>Denominator: 2*2*2*5 "Appropriate factors": 5*5</li><li>Denominator: 5*5*5 "Appropriate factors": 2*2*2</li><li>Denominator: 2*2*2*5*5*5*5 "Appropriate factors": 2</li></ul></li><li>Multiply the numerator and denominator by the "appropriate factors". (Your denominator will be a 1 followed by as many zeros as you have pairs of 2*5 in the factored denominator.)</li><li>To convert the "power-of-ten" denominator fraction into decimal:<ol><li>The number of zeros in the denominator is the number of decimal places.</li><li>Write the numerator</li><li>Position the decimal point so that you have the correct number of decimal places. This may require placing one or more zeros in front of the numerator</li></ol></li></ol>
Here are some examples which I hope will make this clear:
{{{1/20 = 1/(2*2*5) = (5/5)(1/(2*2*5)) = 5/100 = 0.05}}}
{{{3/8 = 3/(2*2*2) = ((5*5*5)/(5*5*5))(3/(2*2*2)) = 375/1000 = 0.375}}}
{{{7/50 = 7/(2*5*5) = (2/2)(7/(2*5*5)) = 14/100 = 0.14}}}
{{{5/6 = 5/(2*3)}}} Use long division (because of the 3).
{{{3/80 = 3/(2*2*2*2*5) = ((5*5*5)/(5*5*5))(3/(2*2*2*2*5)) = 375/10000 = 0.0375}}}
{{{1/5000 = 1/(2*2*2*5*5*5*5) = (2/2)(1/(2*2*2*5*5*5*5)) = 2/10000 = 0.0002}}}
{{{5/4 = 5/(2*2) = ((5*5)/(5*5))(5/(2*2)) = 125/100 = 1.25}}}