# 初中數學/算則觀念/分數

## 分數加減法

${\displaystyle {\frac {1}{3}}+{\frac {1}{4}}}$

${\displaystyle ={\frac {1{\color {red}\times 4}}{3{\color {red}\times 4}}}+{\frac {1{\color {red}\times 3}}{4{\color {red}\times 3}}}}$

${\displaystyle ={\frac {4}{12}}+{\frac {3}{12}}}$

${\displaystyle ={\frac {4+3}{12}}}$

${\displaystyle ={\frac {7}{12}}}$

${\displaystyle {\frac {3}{4}}-{\frac {5}{48}}}$

${\displaystyle ={\frac {3{\color {red}\times 12}}{4{\color {red}\times 12}}}-{\frac {5}{48}}}$

${\displaystyle ={\frac {36}{48}}-{\frac {5}{48}}}$

${\displaystyle ={\frac {36-5}{48}}}$

${\displaystyle ={\frac {31}{48}}}$

## 分子乘除

${\displaystyle {\frac {1}{3}}\times {\frac {1}{4}}}$

${\displaystyle ={\frac {1\times 1}{3\times 4}}}$

${\displaystyle ={\frac {1}{12}}}$

${\displaystyle {\frac {1}{6}}\div {\frac {1}{8}}}$

${\displaystyle ={\frac {1}{6}}\times {\frac {8}{1}}}$

${\displaystyle ={\frac {1\times 8}{6\times 1}}}$

${\displaystyle ={\frac {8/2}{6/2}}}$

${\displaystyle ={\frac {4}{3}}}$

${\displaystyle =1{\frac {1}{3}}}$

## 分數四則運算

• ${\displaystyle (25-{\frac {11}{36}}\times 25)+(11-{\frac {11}{36}}\times 11)}$

${\displaystyle =(25+11)-({\frac {11*25}{36}}+{\frac {11*11}{36}})}$  , (${\displaystyle 25+11=36}$ )

${\displaystyle =36-({\frac {11}{36}}(25+11))}$

${\displaystyle =36-11=25}$

• ${\displaystyle ({\frac {1}{2-{\frac {11}{18}}}}-{\frac {11}{50}})({\frac {36}{11}}-1)}$

${\displaystyle =({\frac {1}{\frac {36-11}{18}}}-{\frac {11}{50}})\times {\frac {36-11}{11}}}$

${\displaystyle =({\frac {18}{36-11}}-{\frac {11}{50}})\times {\frac {36-11}{11}}}$

${\displaystyle ={\frac {36-11}{50}}\times {\frac {36-11}{11}}}$ ,(${\displaystyle 36-11=25}$ )

${\displaystyle ={\frac {25}{50}}\times {\frac {25}{11}}={\frac {25}{22}}=1{\frac {3}{22}}}$

## 圖解

### 乘除的意思

• 乘是「多倍」，5×3或3×5
• 除是「均分」，15÷3為15均分為三份，每份為5
÷3＝
÷3＝

### 分數的意思

「除以分母，乘以分子」。就是均分為分母等份，取其分字等份。

### 整數乘以分數

${\displaystyle 3\times {\frac {1}{4}}={\frac {3}{4}}}$

### 分數除以整數

${\displaystyle {\frac {3}{4}}\div 3={\frac {1}{4}}}$

### 分數乘以分數

${\displaystyle {\frac {1}{3}}\times {\frac {3}{4}}={\frac {4}{12}}\times {\frac {3}{4}}={\frac {4}{12}}}$ 分成四份取其三份${\displaystyle ={\frac {3}{12}}={\frac {1}{4}}}$

### 倒數1

${\displaystyle a\times {\frac {1}{a}}=1}$
${\displaystyle 2\times {\frac {1}{2}}=1}$  =
${\displaystyle 3\times {\frac {1}{3}}=1}$  =
${\displaystyle 4\times {\frac {1}{4}}=1}$  =

### 倒數2

${\displaystyle {\frac {b}{a}}\times {\frac {a}{b}}=1}$

### 除以分數等於乘以其倒數

${\displaystyle {\frac {b}{a}}\div {\frac {d}{c}}=({\frac {b}{a}}\times 1\div {\frac {d}{c}})={\frac {b}{a}}\times ({\frac {d}{c}}\times {\frac {c}{d}})\div {\frac {d}{c}}={\frac {b}{a}}\times {\frac {c}{d}}\times ({\frac {d}{c}}\div {\frac {d}{c}})={\frac {b}{a}}\times {\frac {c}{d}}\times 1={\frac {b}{a}}\times {\frac {c}{d}}}$