Aus Online Mathematik Brückenkurs 1

Version vom 08:19, 9. Sep. 2008 (bearbeiten) Tekbot (Diskussion | Beiträge) K (Lösning 1.2:4c moved to Solution 1.2:4c: Robot: moved page) ← Zum vorherigen Versionsunterschied		Version vom 13:31, 11. Sep. 2008 (bearbeiten) (rückgängig) Ian (Diskussion | Beiträge) Zum nächsten Versionsunterschied →
Zeile 1:		Zeile 1:
-	{{NAVCONTENT_START}}	+	Method 1
-	<center> [[Image:1_2_4c-1(2).gif]] </center>	+
-	<center> [[Image:1_2_4c-2(2).gif]] </center>	+	If we calculate the numerator in the main fraction first, we get
-	{{NAVCONTENT_STOP}}	+
		+
		+	<math>\frac{\frac{1}{4}-\frac{1}{5}}{\frac{3}{16}}=\frac{\frac{1\centerdot 5}{4\centerdot 5}-\frac{1\centerdot 4}{5\centerdot 4}}{\frac{3}{16}}=\frac{\frac{5}{20}-\frac{4}{20}}{\frac{3}{10}}=\frac{\frac{1}{20}}{\frac{3}{10}}</math>
		+
		+	The double fraction on the right-hand side becomes, after multiplying top and bottom by
		+	<math>{10}/{3}\;</math>
		+	,
		+
		+
		+	<math>\frac{\frac{1}{20}}{\frac{3}{10}}=\frac{\frac{1}{20}\centerdot \frac{10}{3}}{\frac{3}{10}\centerdot \frac{10}{3}}=\frac{1}{20}\centerdot \frac{10}{3}</math>
		+
		+
		+	Then, we remove the common factor 10:
		+
		+
		+	<math>\frac{1}{20}\centerdot \frac{10}{3}=\frac{1}{2\centerdot 10}\centerdot \frac{10}{3}=\frac{1}{2\centerdot 3}=\frac{1}{6}</math>
		+
		+
		+
		+	Method 2
		+
		+	Another way to calculate the expression is to divide it up into two separate terms:
		+
		+
		+	<math>\frac{\frac{1}{4}-\frac{1}{5}}{\frac{3}{16}}=\frac{\frac{1}{4}}{\frac{3}{10}}-\frac{\frac{1}{5}}{\frac{3}{10}}</math>
		+
		+	We simplify both double fractions on the right-hand side by multiplying top and bottom by 10/3:
		+
		+
		+	<math>\frac{\frac{1}{4}}{\frac{3}{10}}-\frac{\frac{1}{5}}{\frac{3}{10}}=\frac{\frac{1}{4}\centerdot \frac{10}{3}}{\frac{3}{10}\centerdot \frac{10}{3}}-\frac{\frac{1}{5}\centerdot \frac{10}{3}}{\frac{3}{10}\centerdot \frac{10}{3}}=\frac{1}{4}\centerdot \frac{10}{3}-\frac{1}{5}\centerdot \frac{10}{3}</math>
		+
		+	Instead of multiplying, respectively, by
		+	<math>4\centerdot 3</math>
		+	and
		+	<math>5\centerdot 3</math>
		+	, we keep the numerators factorized and observe that if we multiply the top and bottom of the first fraction by
		+	<math>5</math>
		+	and the second by
		+	<math>4</math>
		+	, we obtain the common denominator:
		+
		+
		+	<math>\frac{10}{4\centerdot 3}-\frac{10}{5\centerdot 3}=\frac{10\centerdot 5}{4\centerdot 3\centerdot 5}-\frac{10\centerdot 4}{5\centerdot 3\centerdot 4}=\frac{50-40}{3\centerdot 4\centerdot 5}=\frac{10}{3\centerdot 4\centerdot 5}</math>
		+
		+	Because
		+	<math>10=2\centerdot 5</math>
		+	and
		+	<math>4=2\centerdot 2</math>
		+	, we can cancel out the common factors
		+	<math>2</math>
		+	and
		+	<math>5</math>
		+	and obtain the answer:
		+
		+
		+	<math>\frac{10}{3\centerdot 4\centerdot 5}=\frac{2\centerdot 5}{3\centerdot 2\centerdot 2\centerdot 5}=\frac{1}{6}</math>

---

Version vom 13:31, 11. Sep. 2008

Method 1

If we calculate the numerator in the main fraction first, we get

\displaystyle \frac{\frac{1}{4}-\frac{1}{5}}{\frac{3}{16}}=\frac{\frac{1\centerdot 5}{4\centerdot 5}-\frac{1\centerdot 4}{5\centerdot 4}}{\frac{3}{16}}=\frac{\frac{5}{20}-\frac{4}{20}}{\frac{3}{10}}=\frac{\frac{1}{20}}{\frac{3}{10}}

The double fraction on the right-hand side becomes, after multiplying top and bottom by \displaystyle {10}/{3}\; ,

\displaystyle \frac{\frac{1}{20}}{\frac{3}{10}}=\frac{\frac{1}{20}\centerdot \frac{10}{3}}{\frac{3}{10}\centerdot \frac{10}{3}}=\frac{1}{20}\centerdot \frac{10}{3}

Then, we remove the common factor 10:

\displaystyle \frac{1}{20}\centerdot \frac{10}{3}=\frac{1}{2\centerdot 10}\centerdot \frac{10}{3}=\frac{1}{2\centerdot 3}=\frac{1}{6}

Method 2

Another way to calculate the expression is to divide it up into two separate terms:

\displaystyle \frac{\frac{1}{4}-\frac{1}{5}}{\frac{3}{16}}=\frac{\frac{1}{4}}{\frac{3}{10}}-\frac{\frac{1}{5}}{\frac{3}{10}}

We simplify both double fractions on the right-hand side by multiplying top and bottom by 10/3:

\displaystyle \frac{\frac{1}{4}}{\frac{3}{10}}-\frac{\frac{1}{5}}{\frac{3}{10}}=\frac{\frac{1}{4}\centerdot \frac{10}{3}}{\frac{3}{10}\centerdot \frac{10}{3}}-\frac{\frac{1}{5}\centerdot \frac{10}{3}}{\frac{3}{10}\centerdot \frac{10}{3}}=\frac{1}{4}\centerdot \frac{10}{3}-\frac{1}{5}\centerdot \frac{10}{3}

Instead of multiplying, respectively, by \displaystyle 4\centerdot 3 and \displaystyle 5\centerdot 3 , we keep the numerators factorized and observe that if we multiply the top and bottom of the first fraction by \displaystyle 5 and the second by \displaystyle 4 , we obtain the common denominator:

\displaystyle \frac{10}{4\centerdot 3}-\frac{10}{5\centerdot 3}=\frac{10\centerdot 5}{4\centerdot 3\centerdot 5}-\frac{10\centerdot 4}{5\centerdot 3\centerdot 4}=\frac{50-40}{3\centerdot 4\centerdot 5}=\frac{10}{3\centerdot 4\centerdot 5}

Because \displaystyle 10=2\centerdot 5 and \displaystyle 4=2\centerdot 2 , we can cancel out the common factors \displaystyle 2 and \displaystyle 5 and obtain the answer:

\displaystyle \frac{10}{3\centerdot 4\centerdot 5}=\frac{2\centerdot 5}{3\centerdot 2\centerdot 2\centerdot 5}=\frac{1}{6}