From Förberedande kurs i matematik 1

Revision as of 08:20, 9 September 2008 (edit) Tekbot (Talk | contribs) m (Lösning 1.2:5c moved to Solution 1.2:5c: Robot: moved page) ← Previous diff		Revision as of 15:00, 11 September 2008 (edit) (undo) Ian (Talk | contribs) Next diff →
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-	{{NAVCONTENT_START}}	+	Method 1
-	<center> [[Image:1_2_5c-1(2).gif]] </center>	+
-	<center> [[Image:1_2_5c-2(2).gif]] </center>	+	We calculate the numerator and denominator first.
-	{{NAVCONTENT_STOP}}	+
		+
		+	<math>\begin{align}
		+	& \frac{3}{10}-\frac{1}{5}=\frac{3}{10}-\frac{1\centerdot 2}{5\centerdot 2}=\frac{3-2}{10}=\frac{1}{10} \\
		+	& \frac{7}{8}-\frac{3}{16}=\frac{7\centerdot 2}{8\centerdot 2}-\frac{3}{16}=\frac{14-3}{16}=\frac{11}{16} \\
		+	\end{align}</math>
		+
		+	Thus, the expression becomes
		+
		+
		+	<math>\frac{\frac{3}{10}-\frac{1}{5}}{\frac{7}{8}-\frac{3}{16}}=\frac{\frac{1}{10}}{\frac{11}{16}}=\frac{\frac{1}{10}\centerdot \frac{16}{11}}{\frac{11}{16}\centerdot \frac{16}{11}}=\frac{16}{10\centerdot 11}</math>
		+
		+
		+	and because
		+	<math>16=2\centerdot 2\centerdot 2\centerdot 2</math>
		+	and 10=2∙5
		+	<math>10=2\centerdot 5</math>
		+	, the simplified answer is
		+
		+
		+	<math>\frac{16}{10\centerdot 11}=\frac{2\centerdot 2\centerdot 2\centerdot 2}{2\centerdot 5\centerdot 11}=\frac{8}{55}</math>
		+
		+	Method 2
		+
		+	If we look at the individual fractions
		+	<math>{3}/{10}\;,\ \ {1}/{5,\ \ {7}/{8}\;}\;</math>
		+	and
		+	<math>{3}/{16}\;</math>
		+	, we see that the denominators can be factorized as
		+
		+
		+	<math>10=2\centerdot 5,\ 8=2\centerdot 2\centerdot 2</math>
		+
		+	<math>10=2\centerdot 5,\quad 8=2\centerdot 2\centerdot 2</math>
		+	and
		+	<math>16=2\centerdot 2\centerdot 2\centerdot 2</math>
		+
		+
		+	and therefore 2∙2∙2∙2∙5 is the fractions' lowest common denominator.
		+
		+	If we multiply the top and bottom of the main fraction by
		+	<math>80</math>
		+	, then it will be possible to eliminate all denominators at once,
		+
		+
		+	<math>\begin{align}
		+	& \frac{\frac{3}{10}-\frac{1}{5}}{\frac{7}{8}-\frac{3}{16}}=\frac{\left( \frac{3}{10}-\frac{1}{5} \right)\centerdot 80}{\left( \frac{7}{8}-\frac{3}{16} \right)\centerdot 80}=\frac{\frac{3\centerdot 80}{10}-\frac{1\centerdot 80}{5}}{\frac{7\centerdot 80}{8}-\frac{3\centerdot 80}{16}} \\
		+	& \\
		+	& =\frac{\frac{3\centerdot 8\centerdot 10}{10}-\frac{8\centerdot 2\centerdot 5}{5}}{\frac{7\centerdot 8\centerdot 10}{8}-\frac{3\centerdot 16\centerdot 5}{16}}=\frac{3\centerdot 8-8\centerdot 2}{7\centerdot 10-3\centerdot }=\frac{8}{55} \\
		+	& \\
		+	\end{align}</math>

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Revision as of 15:00, 11 September 2008

Method 1

We calculate the numerator and denominator first.

\displaystyle \begin{align} & \frac{3}{10}-\frac{1}{5}=\frac{3}{10}-\frac{1\centerdot 2}{5\centerdot 2}=\frac{3-2}{10}=\frac{1}{10} \\ & \frac{7}{8}-\frac{3}{16}=\frac{7\centerdot 2}{8\centerdot 2}-\frac{3}{16}=\frac{14-3}{16}=\frac{11}{16} \\ \end{align}

Thus, the expression becomes

\displaystyle \frac{\frac{3}{10}-\frac{1}{5}}{\frac{7}{8}-\frac{3}{16}}=\frac{\frac{1}{10}}{\frac{11}{16}}=\frac{\frac{1}{10}\centerdot \frac{16}{11}}{\frac{11}{16}\centerdot \frac{16}{11}}=\frac{16}{10\centerdot 11}

and because \displaystyle 16=2\centerdot 2\centerdot 2\centerdot 2 and 10=2∙5 \displaystyle 10=2\centerdot 5 , the simplified answer is

\displaystyle \frac{16}{10\centerdot 11}=\frac{2\centerdot 2\centerdot 2\centerdot 2}{2\centerdot 5\centerdot 11}=\frac{8}{55}

Method 2

If we look at the individual fractions \displaystyle {3}/{10}\;,\ \ {1}/{5,\ \ {7}/{8}\;}\; and \displaystyle {3}/{16}\; , we see that the denominators can be factorized as

\displaystyle 10=2\centerdot 5,\ 8=2\centerdot 2\centerdot 2

\displaystyle 10=2\centerdot 5,\quad 8=2\centerdot 2\centerdot 2 and \displaystyle 16=2\centerdot 2\centerdot 2\centerdot 2

and therefore 2∙2∙2∙2∙5 is the fractions' lowest common denominator.

If we multiply the top and bottom of the main fraction by \displaystyle 80 , then it will be possible to eliminate all denominators at once,

\displaystyle \begin{align} & \frac{\frac{3}{10}-\frac{1}{5}}{\frac{7}{8}-\frac{3}{16}}=\frac{\left( \frac{3}{10}-\frac{1}{5} \right)\centerdot 80}{\left( \frac{7}{8}-\frac{3}{16} \right)\centerdot 80}=\frac{\frac{3\centerdot 80}{10}-\frac{1\centerdot 80}{5}}{\frac{7\centerdot 80}{8}-\frac{3\centerdot 80}{16}} \\ & \\ & =\frac{\frac{3\centerdot 8\centerdot 10}{10}-\frac{8\centerdot 2\centerdot 5}{5}}{\frac{7\centerdot 8\centerdot 10}{8}-\frac{3\centerdot 16\centerdot 5}{16}}=\frac{3\centerdot 8-8\centerdot 2}{7\centerdot 10-3\centerdot }=\frac{8}{55} \\ & \\ \end{align}