Solution 1.2:3b

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Current revision (08:42, 19 September 2008) (edit) (undo)
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If we divide the denominators in succession by
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If we divide the denominators in succession by 2, we see that
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<math>2</math>
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, we see that
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{{Displayed math||<math>\begin{align}
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24&=2\cdot 2\cdot 2\cdot 3\,,\\
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40&=2\cdot 2\cdot 2\cdot 5\,,\\
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16&=2\cdot 2\cdot 2\cdot 2\,,\\
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\end{align}</math>}}
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<math>\begin{align}
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i.e. they all have a factor <math>2\cdot 2\cdot 2=8</math> in common,
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& 24=2\centerdot 2\centerdot 2\centerdot 3 \\
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& 40=2\centerdot 2\centerdot \centerdot 5 \\
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& 16=2\centerdot 2\centerdot 2\centerdot 2 \\
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\end{align}</math>
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{{Displayed math||<math>\frac{1}{3\cdot 8}+\frac{1}{5\cdot 8}-\frac{1}{2\cdot 8}\,</math>.}}
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i.e. they all have a factor
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Hence we do not need to take 8 as a factor when we multiply the top and bottom of each fraction by the product of the other fractions' denominators, but instead we
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<math>2\centerdot 2\centerdot 2=8</math>
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obtain the lowest common denominator by multiplying top and bottom by the other factors, 2, 3 and 5,
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in common,
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{{Displayed math||<math>\frac{1\cdot 2\cdot 5}{3\cdot 8\cdot 2\cdot 5}+\frac{1\cdot 2\cdot 3}{5\cdot 8\cdot 2\cdot 3}-\frac{1\cdot 3\cdot 5}{2\cdot 8\cdot 3\cdot 5}=\frac{10}{240}+\frac{6}{240}-\frac{15}{240}\,</math>.}}
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The LCD is 240 and the answer is
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<math>\frac{1}{3\centerdot 8}+\frac{1}{5\centerdot 8}-\frac{1}{2\centerdot 8}</math>
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{{Displayed math||<math>\frac{10}{240}+\frac{6}{240}-\frac{15}{240}=\frac{10+6-15}{240}=\frac{1}{240}\,</math>.}}
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,
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and hence we do not need to take
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<math>8</math>
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as a factor when we multiply the top and bottom of each fraction by the product of the other fractions' denominators, but instead we
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obtain the lowest common denominator by multiplying top and bottom by the other factors,
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<math>2,\ 3</math>
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and
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<math>5</math>
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:
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<math>\frac{1\centerdot 2\centerdot 5}{3\centerdot 8\centerdot 2\centerdot 5}+\frac{1\centerdot 2\centerdot 3}{5\centerdot 8\centerdot 2\centerdot 3}-\frac{1\centerdot 3\centerdot 5}{2\centerdot 8\centerdot 3\centerdot 5}=\frac{10}{240}+\frac{6}{240}-\frac{15}{240}</math>
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The LCD is
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<math>240</math>
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and the answer is
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<math>\frac{10}{240}+\frac{6}{240}-\frac{15}{240}=\frac{10+6-15}{240}=\frac{1}{240}</math>
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Current revision

If we divide the denominators in succession by 2, we see that

\displaystyle \begin{align} 
 24&=2\cdot 2\cdot 2\cdot 3\,,\\ 
 40&=2\cdot 2\cdot 2\cdot 5\,,\\ 
 16&=2\cdot 2\cdot 2\cdot 2\,,\\

\end{align}

i.e. they all have a factor \displaystyle 2\cdot 2\cdot 2=8 in common,

\displaystyle \frac{1}{3\cdot 8}+\frac{1}{5\cdot 8}-\frac{1}{2\cdot 8}\,.

Hence we do not need to take 8 as a factor when we multiply the top and bottom of each fraction by the product of the other fractions' denominators, but instead we obtain the lowest common denominator by multiplying top and bottom by the other factors, 2, 3 and 5,

\displaystyle \frac{1\cdot 2\cdot 5}{3\cdot 8\cdot 2\cdot 5}+\frac{1\cdot 2\cdot 3}{5\cdot 8\cdot 2\cdot 3}-\frac{1\cdot 3\cdot 5}{2\cdot 8\cdot 3\cdot 5}=\frac{10}{240}+\frac{6}{240}-\frac{15}{240}\,.

The LCD is 240 and the answer is

\displaystyle \frac{10}{240}+\frac{6}{240}-\frac{15}{240}=\frac{10+6-15}{240}=\frac{1}{240}\,.
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