Lösung 1.2:4c

Aus Online Mathematik Brückenkurs 1

(Unterschied zwischen Versionen)
Wechseln zu: Navigation, Suche
K
Zeile 3: Zeile 3:
If we calculate the numerator in the main fraction first, we get
If we calculate the numerator in the main fraction first, we get
 +
{{Displayed math||<math>\frac{\,\dfrac{1}{4}-\dfrac{1}{5}\vphantom{\Biggl(}\,}{\dfrac{3}{10}\vphantom{\Biggl(}} =\frac{\,\dfrac{1\cdot 5}{4\cdot 5}-\dfrac{1\cdot 4}{5\cdot 4}\vphantom{\Biggl(}\,}{\dfrac{3}{10}\vphantom{\Biggl(}} =\frac{\,\dfrac{5}{20}-\dfrac{4}{20}\vphantom{\Biggl(}\,}{\dfrac{3}{10}\vphantom{\Biggl(}} = \frac{\,\dfrac{1}{20}\vphantom{\Biggl(}\,}{\,\dfrac{3}{10}\vphantom{\Biggl(}\,}\,</math>.}}
-
<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>,
-
The double fraction on the right-hand side becomes, after multiplying top and bottom by
+
{{Displayed math||<math>\frac{\,\dfrac{1}{20}\vphantom{\Biggl(}\,}{\,\dfrac{3}{10}\vphantom{\Biggl(}\,} = \frac{\,\dfrac{1}{20}\cdot \dfrac{10}{3}\vphantom{\Biggl(}\,}{\,\dfrac{\rlap{/}3}{\rlap{\,/}10}\cdot \dfrac{\rlap{\,/}10}{\rlap{/}3}\vphantom{\Biggl(}\,} = \dfrac{1}{20}\cdot \dfrac{10}{3}\,</math>.}}
-
<math>{10}/{3}\;</math>
+
-
,
+
 +
Then, we remove the common factor 10,
-
<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>
+
{{Displayed math||<math>\dfrac{1}{20}\cdot \dfrac{10}{3}=\dfrac{1}{2\cdot{}\rlap{\,/}10}\cdot \dfrac{\rlap{\,/}10}{3}=\dfrac{1}{2\cdot 3}=\dfrac{1}{6}\,</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
Method 2
-
Another way to calculate the expression is to divide it up into two separate terms:
+
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>
+
{{Displayed math||<math>\frac{\,\dfrac{1}{4}-\dfrac{1}{5}\vphantom{\Biggl(}\,}{\dfrac{3}{10}\vphantom{\Biggl(}} = \frac{\,\dfrac{1}{4}\vphantom{\Biggl(}\,}{\,\dfrac{3}{10}\vphantom{\Biggl(}\,}-\frac{\,\dfrac{1}{5}\vphantom{\Biggl(}\,}{\,\dfrac{3}{10}\vphantom{\Biggl(}\,}\,</math>.}}
-
Instead of multiplying, respectively, by
+
We simplify both double fractions on the right-hand side by multiplying top and bottom by 10/3
-
<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:
+
 +
{{Displayed math||<math>\frac{\,\dfrac{1}{4}\vphantom{\Biggl(}\,}{\,\dfrac{3}{10}\vphantom{\Biggl(}\,}-\frac{\,\dfrac{1}{5}\vphantom{\Biggl(}\,}{\,\dfrac{3}{10}\vphantom{\Biggl(}\,} = \frac{\,\dfrac{1}{4}\cdot \dfrac{10}{3}\vphantom{\Biggl(}\,}{\,\dfrac{\rlap{/}3}{\rlap{\,/}10}\cdot \dfrac{\rlap{\,/}10}{\rlap{/}3}\vphantom{\Biggl(}\,}-\frac{\,\dfrac{1}{5}\cdot \dfrac{10}{3}\vphantom{\Biggl(}\,}{\,\dfrac{\rlap{/}3}{\rlap{\,/}10}\cdot \dfrac{\rlap{\,/}10}{\rlap{/}3}\vphantom{\Biggl(}\,} = \frac{1}{4}\cdot \frac{10}{3}-\frac{1}{5}\cdot \frac{10}{3}\,</math>.}}
-
<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>
+
Instead of multiplying, respectively, by <math>4\cdot 3</math> and <math>5\cdot 3</math>, we keep the numerators factorized and observe that if we multiply the top and bottom of the first fraction by 5 and the second by 4, we obtain the common denominator
-
Because
+
{{Displayed math||<math>\frac{10}{4\cdot 3}-\frac{10}{5\cdot 3}=\frac{10\cdot 5}{4\cdot 3\cdot 5}-\frac{10\cdot 4}{5\cdot 3\cdot 4}=\frac{50-40}{3\cdot 4\cdot 5}=\frac{10}{3\cdot 4\cdot 5}\,</math>.}}
-
<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:
+
 +
Because <math>10=2\cdot 5</math> and <math>4=2\cdot 2</math>, we can cancel out the common factors 2 and 5 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>
+
{{Displayed math||<math>\frac{10}{3\cdot 4\cdot 5}=\frac{\rlap{/}2{}\cdot{}\rlap{/}5}{3\cdot 2\cdot{}\rlap{/}2\cdot{}\rlap{/}5}=\frac{1}{6}\,</math>.}}

Version vom 11:07, 19. Sep. 2008

Method 1

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

Vorlage:Displayed math

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

Vorlage:Displayed math

Then, we remove the common factor 10,

Vorlage:Displayed math


Method 2

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

Vorlage:Displayed math

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

Vorlage:Displayed math

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

Vorlage:Displayed math

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

Vorlage:Displayed math

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_1.2:4c“