Å utvide og forkorte brøker

1.1.9 Utvid brøkene slik at de får like nevnere

$\frac{5}{12}, \frac{7}{3}, \frac{11}{18}, \frac{8}{9}, \frac{13}{6}, \frac{15}{2}, \frac{3}{4}$

vis fasit

Fellesnevneren er 36. Vi utvider brøkene slik at alle får nevner 36.

$\begin{array}{rcl} \frac{5}{12} & = & \frac{5 \cdot 3}{12 \cdot 3} = \frac{15}{36} \\ \frac{7}{3} & = & \frac{7 \cdot 12}{3 \cdot 12} = \frac{84}{36} \\ \frac{11}{18} & = & \frac{11 \cdot 2}{18 \cdot 2} = \frac{22}{36} \\ \frac{8}{9} & = & \frac{8 \cdot 4}{9 \cdot 4} = \frac{32}{36} \\ \frac{13}{6} & = & \frac{13 \cdot 6}{6 \cdot 6} = \frac{78}{36} \\ \frac{15}{2} & = & \frac{15 \cdot 18}{2 \cdot 18} = \frac{270}{36} \\ \frac{3}{4} & = & \frac{3 \cdot 9}{4 \cdot 9} = \frac{27}{36} \end{array}$

1.1.10 Forkort brøkene

$\frac{6}{12}, \frac{21}{3}, \frac{30}{18}, \frac{6}{9}, \frac{84}{6}, \frac{72}{420}, \frac{384}{256}$

vis fasit

$\begin{array}{rcl} \frac{6}{12} & = & \frac{6 : 6}{12 : 6} = \frac{1}{2} \\ \frac{21}{3} & = & \frac{21 : 3}{3 : 3} = 7 \\ \frac{30}{18} & = & \frac{30 : 6}{18 : 6} = \frac{5}{3} \\ \frac{6}{9} & = & \frac{6 : 3}{9 : 3} = \frac{2}{3} \\ \frac{84}{6} & = & \frac{84 : 6}{6 : 6} = 14 \\ \frac{72}{420} & = & \frac{72 : 2}{420 : 2} = \frac{36 : 2}{210 : 2} = \frac{18 : 3}{105 : 3} = \frac{6}{35} \\ \frac{384}{256} & = & \frac{384 : 2}{256 : 2} = \frac{192 : 2}{128 : 2} = \frac{96 : 2}{64 : 2} = \frac{48 : 2}{32 : 2} = \frac{24 : 8}{16 : 8} = \frac{3}{2} \end{array}$

1.1.11 Velg > eller < eller = i hver av rutene nedenfor. Begrunn svarene dine.

a) $\frac{3}{4} \frac{4}{5}$

vis fasit

$\begin{array}{rcl} \frac{3}{4} < \frac{4}{5} \\ \frac{1}{4} & = & 0, 25 \\ \frac{3}{4} & = & 3 \cdot 0, 25 = 0, 75 \\ \frac{1}{5} & = & 0, 2 \\ \frac{4}{5} & = & 0, 2 \cdot 4 = 0, 8 \end{array}$

b) $\frac{12}{18} \frac{2}{3}$

vis fasit

$\begin{array}{rcl} \frac{12}{18} & = & \frac{2}{3} \\ \frac{12}{18} & = & \frac{12 : 6}{18 : 6} = \frac{2}{3} \end{array}$

c) $\frac{37}{18} 2$

vis fasit

$\begin{array}{rcl} \frac{37}{18} > 2 \\ 2 \cdot 18 & = & 36 \\ \frac{36}{18} & = & 2 \end{array}$

d) $\frac{4}{20} \frac{1}{6}$

vis fasit

$\begin{array}{rcl} \frac{4}{20} > \frac{1}{6} \\ \frac{4}{20} & = & \frac{4 : 4}{20 : 4} = \frac{1}{5} \\ \frac{1}{5} & > & \frac{1}{6} \end{array}$

Å trekke sammen brøker med forskjellige nevnere

1.1.12 Trekk sammen

a) $\frac{1}{2} + \frac{2}{3} - \frac{1}{4}$

vis fasit

$= \frac{1 \cdot 6}{2 \cdot 6} + \frac{2 \cdot 4}{3 \cdot 4} - \frac{1 \cdot 3}{4 \cdot 3} = \frac{6}{12} + \frac{8}{12} - \frac{3}{12} = \frac{6 + 8 - 3}{12} = \frac{11}{12}$

b) $\frac{4}{3} + 2 - \frac{1}{7}$

vis fasit

$= \frac{4 \cdot 7}{3 \cdot 7} + \frac{2 \cdot 21}{1 \cdot 21} - \frac{1 \cdot 3}{7 \cdot 3} = \frac{28 + 42 - 3}{21} = \frac{67}{21}$

c) $1 - \frac{1}{2} + \frac{4}{9} - \frac{3}{4} + \frac{7}{18}$

vis fasit

$\begin{array}{lcl} \frac{1 \cdot 36}{1 \cdot 36} - \frac{1 \cdot 18}{2 \cdot 18} + \frac{4 \cdot 4}{9 \cdot 4} - \frac{3 \cdot 9}{4 \cdot 9} + \frac{7 \cdot 2}{18 \cdot 2} & = \\ \frac{36 - 18 + 16 - 27 + 14}{36} = \frac{21 : 3}{36 : 3} = \frac{7}{12} \end{array}$

1.1.13 Trekk sammen

a) $\frac{1}{4} + \frac{5}{6}$

vis fasit

$= \frac{1 \cdot 3}{4 \cdot 3} + \frac{5 \cdot 2}{6 \cdot 2} = \frac{3}{12} + \frac{10}{12} = \frac{13}{12}$

b) $\frac{3}{4} - 2$

vis fasit

$= \frac{3}{4} - \frac{2}{1} = \frac{3}{4} - \frac{2 \cdot 4}{1 \cdot 4} = \frac{3 - 8}{4} = \frac{- 5}{4} = - \frac{5}{4}$

c) $3 - \frac{1}{6} + \frac{2}{3}$

vis fasit

$= \frac{3 \cdot 6}{1 \cdot 6} - \frac{1}{6} + \frac{2 \cdot 2}{3 \cdot 2} = \frac{18}{6} - \frac{1}{6} + \frac{4}{6} = \frac{18 - 1 + 4}{6} = \frac{21}{6} = \frac{7}{2}$

d) $2 - \frac{1}{4} + \frac{2}{3}$

vis fasit

$= \frac{2 \cdot 12}{1 \cdot 12} - \frac{1 \cdot 3}{4 \cdot 3} + \frac{2 \cdot 4}{3 \cdot 4} = \frac{24}{12} - \frac{3}{12} + \frac{8}{12} = \frac{24 - 3 + 8}{12} = \frac{29}{12}$

e) $1 + \frac{1}{4} - \frac{7}{9}$

vis fasit

$= \frac{1 \cdot 36}{1 \cdot 36} + \frac{1 \cdot 9}{4 \cdot 9} - \frac{7 \cdot 4}{9 \cdot 4} = \frac{36}{36} + \frac{9}{36} - \frac{28}{36} = \frac{36 + 9 - 28}{36} = \frac{17}{36}$

f) $2 \frac{1}{5} - 3 \frac{2}{3}$

vis fasit

$\begin{array}{rcl} \frac{2}{1} + \frac{1}{5} - \frac{3}{1} - \frac{2}{3} & = & \frac{2 \cdot 15}{1 \cdot 15} + \frac{1 \cdot 3}{5 \cdot 3} - \frac{3 \cdot 15}{1 \cdot 15} - \frac{2 \cdot 5}{3 \cdot 5} = \\ \frac{30 + 3 - 45 - 10}{15} & = & - \frac{22}{15} \end{array}$

g) $\frac{1}{5} + 2 - 1 \frac{1}{4}$

vis fasit

$\begin{array}{lcl} \frac{1 \cdot 4}{5 \cdot 4} + \frac{2 \cdot 20}{1 \cdot 20} - \frac{1 \cdot 20}{1 \cdot 20} - \frac{1 \cdot 5}{4 \cdot 5} & = & \frac{4}{20} + \frac{40}{20} - \frac{20}{20} - \frac{5}{20} = \\ \frac{4 + 40 - 20 - 5}{20} = \frac{19}{20} \end{array}$

Introduksjon