Solution 1.2:2d

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If we divide up the denominators into their smallest possible integer factors,
If we divide up the denominators into their smallest possible integer factors,
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{{Displayed math||<math>\begin{align}
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<math>\begin{align}
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45&=5\cdot 9=5\cdot 3\cdot 3\,, \\
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& 45=5\centerdot 9=5\centerdot 3\centerdot 3 \\
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75&=3\cdot 25=3\cdot 5\cdot 5\,, \\
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& 75=3\centerdot 25=3\centerdot 5\centerdot 5 \\
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\end{align}</math>}}
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\end{align}</math>
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the expression can be written as
the expression can be written as
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{{Displayed math||<math>\frac{2}{3\cdot 3\cdot 5}+\frac{1}{3\cdot 5\cdot 5}</math>}}
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<math>\frac{1}{5\centerdot 3\centerdot 3}+\frac{1}{3\centerdot 5\centerdot 5}</math>
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and then we see that the denominators have <math>3\cdot 5</math> as a common factor. Therefore, if we multiply the top and bottom of the first fraction by 5
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and the second by 3, the result is the lowest possible denominator
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and then we see that the denominators have
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<math>3\centerdot 5</math>
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as a common factor. Therefore, if we multiply the top and bottom of the first fraction by
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<math>5</math>
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and the second by
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<math>3</math>
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, the result is the lowest possible denominator.
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<math>\begin{align}
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& \frac{2}{5\centerdot 3\centerdot 3}\centerdot \frac{5}{5}+\frac{1}{3\centerdot 5\centerdot 5}\centerdot \frac{3}{3} \\
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& =\frac{2}{5\centerdot 3\centerdot 3\centerdot 5}+\frac{3}{3\centerdot 5\centerdot 5\centerdot 3} \\
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& =\frac{10}{225}+\frac{3}{225} \\
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\end{align}</math>
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{{Displayed math||<math>\begin{align}
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\frac{2}{3\cdot 3\cdot 5}\cdot \frac{5}{5}+\frac{1}{3\cdot 5\cdot 5}\cdot
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\frac{3}{3} &=\frac{2}{3\cdot 3\cdot 5\cdot 5}
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+\frac{3}{3\cdot 5\cdot 5\cdot 3}\\[10pt]
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&= \frac{10}{225}+\frac{3}{225}\,\textrm{.}\\
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\end{align}</math>}}
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The lowest common denominator is
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The lowest common denominator is 225.
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<math>225</math>
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.
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Current revision

If we divide up the denominators into their smallest possible integer factors,

\displaystyle \begin{align} 
45&=5\cdot 9=5\cdot 3\cdot 3\,, \\ 
75&=3\cdot 25=3\cdot 5\cdot 5\,, \\

\end{align}

the expression can be written as

\displaystyle \frac{2}{3\cdot 3\cdot 5}+\frac{1}{3\cdot 5\cdot 5}

and then we see that the denominators have \displaystyle 3\cdot 5 as a common factor. Therefore, if we multiply the top and bottom of the first fraction by 5 and the second by 3, the result is the lowest possible denominator

\displaystyle \begin{align} 
\frac{2}{3\cdot 3\cdot 5}\cdot \frac{5}{5}+\frac{1}{3\cdot 5\cdot 5}\cdot
\frac{3}{3} &=\frac{2}{3\cdot 3\cdot 5\cdot 5}
   +\frac{3}{3\cdot 5\cdot 5\cdot 3}\\[10pt] 
 &= \frac{10}{225}+\frac{3}{225}\,\textrm{.}\\

\end{align}

The lowest common denominator is 225.

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