From Förberedande kurs i matematik 1

Revision as of 08:17, 9 September 2008 (edit) Tekbot (Talk | contribs) m (Lösning 1.2:2c moved to Solution 1.2:2c: Robot: moved page) ← Previous diff		Current revision (12:15, 22 September 2008) (edit) (undo) Tek (Talk | contribs) m
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-	{{NAVCONTENT_START}}	+	We divide up the two numerators into the smallest possible integer factors,
-	<center> [[Image:1_2_2c.gif]] </center>	+
-	{{NAVCONTENT_STOP}}	+	{{Displayed math||<math>\begin{align}
		+	12 &= 2\cdot 6 = 2\cdot 2\cdot 3\,,\\
		+	14 &= 2\cdot 7\,\textrm{.} \\
		+	\end{align}</math>}}
		+
		+	The expression can thus be written as
		+
		+	{{Displayed math||
		+	<math>\frac{1}{2\cdot 2\cdot 3}-\frac{1}{2\cdot 7}\,</math>.}}
		+
		+	Here, we see that the denominators have a factor 2 in common. We multiply the top and bottom of the first fraction by 7 and the second by <math>2\cdot 3</math>
		+	i.e. we leave out the common factor 2, so that the fractions have the lowest common denominator <math>2\cdot 2\cdot 3\cdot 7</math>,
		+
		+	{{Displayed math||<math>\begin{align}
		+	\frac{1}{12}-\frac{1}{14} &= \frac{1}{2\cdot 2\cdot 3}-\frac{1}{2\cdot 7}\\[5pt]
		+	&= \frac{1}{2\cdot 2\cdot 3}\cdot \frac{7}{7}-\frac{1}{2\cdot 7}\cdot \frac{2\cdot 3}{2\cdot 3}\\[5pt]
		+	&= \frac{7}{2\cdot 2\cdot 3\cdot 7} - \frac{2\cdot 3}{2\cdot 2\cdot 3\cdot 7}\\[5pt]
		+	&= \frac{7}{84} - \frac{6}{84}\,\textrm{.}
		+	\end{align}</math>}}
		+
		+	The lowest common denominator is 84.

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Current revision

We divide up the two numerators into the smallest possible integer factors,

\displaystyle \begin{align} 12 &= 2\cdot 6 = 2\cdot 2\cdot 3\,,\\ 14 &= 2\cdot 7\,\textrm{.} \\ \end{align}

The expression can thus be written as

\displaystyle \frac{1}{2\cdot 2\cdot 3}-\frac{1}{2\cdot 7}\,.

Here, we see that the denominators have a factor 2 in common. We multiply the top and bottom of the first fraction by 7 and the second by \displaystyle 2\cdot 3 i.e. we leave out the common factor 2, so that the fractions have the lowest common denominator \displaystyle 2\cdot 2\cdot 3\cdot 7,

\displaystyle \begin{align} \frac{1}{12}-\frac{1}{14} &= \frac{1}{2\cdot 2\cdot 3}-\frac{1}{2\cdot 7}\\[5pt] &= \frac{1}{2\cdot 2\cdot 3}\cdot \frac{7}{7}-\frac{1}{2\cdot 7}\cdot \frac{2\cdot 3}{2\cdot 3}\\[5pt] &= \frac{7}{2\cdot 2\cdot 3\cdot 7} - \frac{2\cdot 3}{2\cdot 2\cdot 3\cdot 7}\\[5pt] &= \frac{7}{84} - \frac{6}{84}\,\textrm{.} \end{align}

The lowest common denominator is 84.