Ushtrime - kthimi i thyesave ne numra dhjetor dhe anasjelltas

Shembull 1

Duke zbatuar ato që mësuam mbi kthimin e thyesave ne numra dhjetor dhe anasjelltas, të gjendet vlera e shprehjes:

$\displaystyle \frac{13}{2}+3,6\cdot \frac{9}{5}-1,2:\frac{3}{5}+0,05$

Shohim që emëruesit e thyesave janë 2 dhe 5, pra thyesat kthehen në numra dhjetor.

$\displaystyle =6,5+3,6\cdot 1,6-1,2:0,6+0,05$

$\displaystyle =6,5+5,76-2+0,05$

$\displaystyle =12,26-2+0,05$

$\displaystyle =10,26+0,05$

$\displaystyle =10,31$

Shembull 2

Duke zbatuar ato që mësuam mbi kthimin e thyesave ne numra dhjetor dhe anasjelltas, të gjendet vlera e shprehjes:

$\displaystyle 2\left[ \left( 7\frac{1}{12}-3\frac{17}{36} \right)\cdot 3,6 \right]-4\frac{1}{3}+0,65$

Shohim që thyesat e kësaj shprehje nuk mund të kthehen në numra dhjetor të fundëm, ndaj i bëjmë veprimet me thyesa.

$\displaystyle =2\left[ \left( 7\frac{1}{12}-3\frac{17}{36} \right)\cdot \frac{36}{10} \right]-\frac{13}{3}:\frac{65}{100}$

$\displaystyle =2\left[ \left( \frac{85}{12}-\frac{125}{36} \right)\cdot \frac{36}{10} \right]-\frac{13}{3}\cdot \frac{11}{65}$

$\displaystyle =2\left[ \left( \frac{255}{36}-\frac{125}{36} \right)\cdot \frac{36}{10} \right]-\frac{20}{3}$

$\displaystyle =2\left[ \frac{130}{36}\cdot \frac{36}{10} \right]-\frac{20}{3}$

$\displaystyle =2\cdot \frac{13}{1}-\frac{20}{3}$

$\displaystyle =26-\frac{20}{3}$

$\displaystyle =\frac{26\cdot 3-20}{3}$

$\displaystyle =\frac{58}{3}$

$\displaystyle =19\frac{1}{3}$

Ushtrimi 1

Duke zbatuar ato që mësuam mbi kthimin e thyesave ne numra dhjetor dhe anasjelltas, gjeni vlerën e shprehjeve:

a) $\displaystyle \frac{1}{9}:2+0:\frac{3}{7}-2,5\cdot \frac{3}{4}$

b) $\displaystyle 18:\frac{5}{8}-2,25\cdot 8+1,25\cdot 1\frac{1}{5}$

Zgjidhje

a) $\displaystyle \frac{1}{9}:2+0:\frac{3}{7}-2,5\cdot \frac{3}{4}$

$\displaystyle =\frac{1}{9}\cdot \frac{1}{2}+0:\frac{3}{7}-\frac{25}{10}\cdot \frac{3}{4}$

$\displaystyle =\frac{1}{18}+0-\frac{75}{40}$

$\displaystyle =\frac{1}{18}-\frac{75}{40}$

b) $\displaystyle 18:\frac{5}{8}-2,25\cdot 8+1,25\cdot 1\frac{1}{5}$

$\displaystyle =18\cdot \frac{8}{5}-20+1,25\cdot \frac{6}{5}$

$\displaystyle =\frac{144}{5}-20+\frac{125}{100}\cdot \frac{6}{5}$

$\displaystyle =\frac{144-100}{5}+\frac{75}{50}$

$\displaystyle =\frac{44}{5}+\frac{15}{10}$

$\displaystyle =\frac{88}{10}+\frac{15}{10}$

$\displaystyle =\frac{103}{10}=10,3$

Ushtrimi 2

Duke zbatuar ato që mësuam mbi kthimin e thyesave ne numra dhjetor dhe anasjelltas, të gjenden vlerat e shprehjeve:

a) $\displaystyle \left( 15:\frac{10}{7}-3 \right):\frac{4}{9}-\left( 9:1,5-3 \right):\frac{1}{12}$

b) $\displaystyle \left( 2+\frac{4}{5} \right):\frac{5}{7}+\left( 1+0,6 \right):\frac{16}{15}-\left( \frac{2}{5}+1 \right)$

Zgjidhje

a) $\displaystyle \left( 15:\frac{10}{7}-3 \right):\frac{4}{9}-\left( 9:1,5-3 \right):\frac{1}{12}$

Shohim që thyesat nuk mund të kthehen në numra dhjetor të fundëm:

$\displaystyle =\left( 15\cdot \frac{7}{10}-3 \right):\frac{4}{9}-\left( 9:\frac{15}{10}-3 \right):\frac{1}{12}$

$\displaystyle =\left( \frac{105}{10}-\frac{30}{10} \right):\frac{4}{9}-\left( 9\cdot \frac{10}{15}-3 \right):\frac{1}{12}$

$\displaystyle =\frac{75}{10}:\frac{4}{9}-\left( \frac{90}{15}-3 \right):\frac{1}{12}$

$\displaystyle =\frac{75}{10}\cdot \frac{9}{4}-\left( 6-3 \right):\frac{1}{12}$

$\displaystyle =\frac{675}{40}-3:\frac{1}{12}$

$\displaystyle =\frac{135}{8}-3\cdot \frac{12}{1}$

$\displaystyle =\frac{135}{8}-36$

$\displaystyle =\frac{135-284}{8}$

$\displaystyle =-\frac{149}{8}$

b) $\displaystyle \left( 2+\frac{4}{5} \right):\frac{5}{7}+\left( 1+0,6 \right):\frac{16}{15}-\left( \frac{2}{5}+1 \right)$

$\displaystyle =\left( \frac{10}{5}+\frac{4}{5} \right):\frac{5}{7}+\left( \frac{10}{10}+\frac{6}{10} \right):\frac{16}{15}-\left( \frac{2}{5}+\frac{5}{5} \right)$

$\displaystyle =\frac{14}{5}:\frac{5}{7}+\frac{16}{10}:\frac{16}{15}-\frac{7}{5}$

$\displaystyle =\frac{14}{5}\cdot \frac{7}{5}+\frac{16}{10}\cdot \frac{15}{16}-\frac{7}{5}$

$\displaystyle =\frac{98}{25}+\frac{15}{10}-\frac{7}{5}$

$\displaystyle =\frac{196}{50}+\frac{75}{50}-\frac{70}{50}$

$\displaystyle =\frac{201}{50}$

Ushtrimi 3

Duke zbatuar ato që mësuam mbi kthimin e thyesave ne numra dhjetor dhe anasjelltas, të gjenden vlerat e shprehjeve:

a) $\displaystyle 45+21:7-{{5}^{2}}:5+3-2\cdot 4$

b) $\displaystyle 79\cdot 68+\left[ 1400-\left( 777-687 \right)\cdot 5 \right]\cdot 96$

c) $\displaystyle \left( 0,008+0,992 \right)\cdot \left( 5\cdot \frac{6}{10}-1,4 \right)$

d) $\displaystyle 13,5:1\frac{1}{3}+16\frac{1}{2}\cdot 1\frac{5}{1}+19\frac{1}{4}:0,16$

Zgjidhje

a) $\displaystyle 45+21:7-{{5}^{2}}:5+3-2\cdot 4$

$\displaystyle =45+3-25:5+3-8$

$\displaystyle =45+3-5+3-8$

$\displaystyle =48-5+3-8$

$\displaystyle =46-8$

$\displaystyle =42$

b) $\displaystyle 79\cdot 68+\left[ 1400-\left( 777-687 \right)\cdot 5 \right]\cdot 96$

$\displaystyle =79\cdot 68+\left[ 1400-90\cdot 5 \right]\cdot 96$

$\displaystyle =79\cdot 68+\left[ 1400-450 \right]\cdot 96$

$\displaystyle =79\cdot 68+1350\cdot 96$

$\displaystyle =5372+129600$

$\displaystyle =134972$

c) $\displaystyle \left( 0,008+0,992 \right)\cdot \left( 5\cdot \frac{6}{10}-1,4 \right)$

$\displaystyle =1\cdot \left( 1\cdot \frac{6}{2}-1,4 \right)$

$\displaystyle =3-1,4$

$\displaystyle =1,6$

d) $\displaystyle 13,5:1\frac{1}{3}+16\frac{1}{2}\cdot 1\frac{5}{1}+19\frac{1}{4}:0,16$

$\displaystyle =13,5:\frac{4}{3}+\frac{33}{2}\cdot 6+\frac{77}{4}:0,16$

$\displaystyle =\frac{135}{10}:\frac{4}{3}+33\cdot 3+\frac{77}{4}:\frac{16}{10}$

$\displaystyle =\frac{135}{10}\cdot \frac{3}{4}+99+\frac{77}{4}\cdot \frac{10}{16}$

$\displaystyle =\frac{405}{40}+99+\frac{770}{64}$

Ushtrimi 4

Të gjenden vlerat e shprehjeve:

a) $\displaystyle 18+\left\{ 25+\left[ 31-\left( 26-18+5 \right) \right]-17 \right\}+7-\left( 78+119 \right)$

b) $\displaystyle 117-96-\left\{ 10-\left[ 12-\left( 9-3 \right) \right] \right\}-\left[ 17-\left( 13-7+5\cdot 2 \right) \right]$

c) $\displaystyle \left( \frac{1}{2}\cdot 4:\frac{6}{5}+\frac{2}{3}:\frac{4}{3}\cdot 0,3 \right):\left( 1+0,35:\frac{3}{7} \right)$

d) $\displaystyle \left( \frac{1}{3}+0,25 \right)\cdot \frac{7}{6}+\left( 1,4-\frac{1}{8} \right):\frac{51}{10}$

Zgjidhje

a) $\displaystyle 18+\left\{ 25+\left[ 31-\left( 26-18+5 \right) \right]-17 \right\}+7-\left( 78+119 \right)$

$\displaystyle =18+\left\{ 25+\left[ 31-13 \right]-17 \right\}+7-197$

$\displaystyle =18+\left\{ 25+18-17 \right\}+7-197$

$\displaystyle =18+26+7-197$

$\displaystyle =51-197$

$\displaystyle =-146$

b) $\displaystyle 117-96-\left\{ 10-\left[ 12-\left( 9-3 \right) \right] \right\}-\left[ 17-\left( 13-7+5\cdot 2 \right) \right]$

$\displaystyle =117-96-\left\{ 10-\left[ 12-6 \right] \right\}-\left[ 17-\left( 13-7+10 \right) \right]$

$\displaystyle =117-96-\left\{ 10-6 \right\}-\left[ 17-16 \right]$

$\displaystyle =117-96-4-1$

$\displaystyle =24$

c) $\displaystyle \left( \frac{1}{2}\cdot 4:\frac{6}{5}+\frac{2}{3}:\frac{4}{3}\cdot 0,3 \right):\left( 1+0,35:\frac{3}{7} \right)$

$\displaystyle =\left( 2:\frac{6}{5}+\frac{2}{3}\cdot \frac{3}{4}\cdot \frac{3}{10} \right):\left( 1+\frac{35}{100}\cdot \frac{7}{3} \right)$

$\displaystyle =\left( 2\cdot \frac{5}{6}+\frac{1}{2}\cdot \frac{3}{10} \right):\left( 1+\frac{35}{100}\cdot \frac{7}{3} \right)$

$\displaystyle =\left( \frac{5}{3}+\frac{3}{20} \right):\left( 1+\frac{245}{300} \right)$

$\displaystyle =\left( \frac{5}{3}+\frac{3}{20} \right):\left( 1+\frac{49}{60} \right)$

$\displaystyle =\left( \frac{100}{60}+\frac{9}{60} \right):\left( \frac{60}{60}+\frac{49}{60} \right)$

$\displaystyle =\frac{109}{60}:\frac{109}{60}$

$\displaystyle =1$

d) $\displaystyle \left( \frac{1}{3}+0,25 \right)\cdot \frac{7}{6}+\left( 1,4-\frac{1}{8} \right):\frac{51}{10}$

$\displaystyle =\left( \frac{1}{3}+\frac{25}{100} \right)\cdot \frac{7}{6}+\left( \frac{14}{10}-\frac{1}{8} \right):\frac{51}{10}$

$\displaystyle =\left( \frac{100+25}{300} \right)\cdot \frac{7}{6}+\left( \frac{56}{40}-\frac{5}{40} \right):\frac{51}{10}$

$\displaystyle =\frac{125}{300}\cdot \frac{7}{6}+\frac{51}{40}:\frac{51}{10}$

$\displaystyle =\frac{5}{12}\cdot \frac{7}{6}+\frac{51}{40}:\frac{51}{10}$

$\displaystyle =\frac{35}{72}+\frac{51}{40}\cdot \frac{10}{51}$

$\displaystyle =\frac{35}{72}+\frac{1}{4}$

Ushtrimi 5

Gjeni vlerën e shprehjeve:

a) $\displaystyle \left( 0,8-0,47 \right)\cdot \left( 0,8+0,47 \right):41,91$ b) $\displaystyle \left( 4,5-3\frac{2}{7} \right):\left( 7,5:8\frac{1}{3}+2\frac{1}{7}:3 \right)$

c) $\displaystyle \left[ \left( 6\frac{1}{6}-1\frac{1}{24} \right):\left( 8\frac{3}{8}+7,2 \right)+\frac{418}{623} \right]$

Zgjidhje

a) $\displaystyle \left( 0,8-0,47 \right)\cdot \left( 0,8+0,47 \right):41,91$ $\displaystyle =0,33\cdot 1,27:41,91$

$\displaystyle =0,4191:41,91$

$\displaystyle =41,91:4191$

$\displaystyle =0,01$

b) $\displaystyle \left( 4,5-3\frac{2}{7} \right):\left( 7,5:8\frac{1}{3}+2\frac{1}{7}:3 \right)$ $\displaystyle =\left( \frac{45}{10}-\frac{23}{7} \right):\left( \frac{75}{10}:\frac{25}{3}+\frac{15}{7}:3 \right)$

$\displaystyle =\left( \frac{315}{70}-\frac{230}{70} \right):\left( \frac{75}{10}\cdot \frac{3}{25}+\frac{15}{7}\cdot \frac{1}{3} \right)$

$\displaystyle =\frac{85}{70}:\left( \frac{9}{10}+\frac{5}{7} \right)$

$\displaystyle =\frac{85}{70}:\left( \frac{63}{70}+\frac{50}{70} \right)$

$\displaystyle =\frac{85}{70}:\frac{113}{70}$

$\displaystyle =\frac{85}{70}\cdot \frac{70}{113}$

$\displaystyle =\frac{85}{113}$

c) $\displaystyle \left[ \left( 6\frac{1}{6}-1\frac{1}{24} \right):\left( 8\frac{3}{8}+7,2 \right)+\frac{418}{623} \right]$ $\displaystyle =\left[ \left( \frac{37}{6}-\frac{25}{24} \right):\left( \frac{67}{8}+\frac{72}{10} \right)+\frac{418}{623} \right]$

$\displaystyle =\left[ \left( \frac{148}{24}-\frac{25}{24} \right):\left( \frac{335}{40}+\frac{288}{40} \right)+\frac{418}{623} \right]$

$\displaystyle =\left[ \frac{123}{24}:\frac{623}{40}+\frac{418}{623} \right]$

$\displaystyle =\left[ \frac{123}{24}\cdot \frac{40}{623}+\frac{418}{623} \right]$

$\displaystyle =\frac{123}{3}\cdot \frac{5}{623}+\frac{418}{623}$

$\displaystyle =\frac{615}{1869}+\frac{418}{623}$

$\displaystyle =\frac{615}{1869}+\frac{1254}{1869}$

$\displaystyle =\frac{1869}{1869}$

$\displaystyle =1$

Ushtrimi 6

Gjeni vlerën e shprehjes:

$\displaystyle \frac{0,5+\frac{1}{4}+\frac{1}{6}+0,12:02}{\frac{1}{3}+0,2+\frac{1}{9}:\frac{1}{3}}$

$\displaystyle=\frac{\frac{5}{10}+\frac{1}{4}+\frac{1}{6}+\frac{12}{100}:\frac{2}{10}}{\frac{1}{3}+\frac{2}{10}+\frac{1}{9}:\frac{1}{3}}$

$\displaystyle =\frac{\frac{5}{10}+\frac{1}{4}+\frac{1}{6}+\frac{3}{25}:\frac{1}{5}}{\frac{1}{3}+\frac{2}{10}+\frac{1}{9}\cdot \frac{3}{1}}$

$\displaystyle =\frac{\frac{5}{10}+\frac{1}{4}+\frac{1}{6}+\frac{3}{25}\cdot \frac{5}{1}}{\frac{1}{3}+\frac{2}{10}+\frac{1}{3}}$

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

$\displaystyle =\frac{\frac{30}{60}+\frac{15}{60}+\frac{10}{60}+\frac{36}{60}}{\frac{10}{30}+\frac{6}{30}+\frac{10}{30}}$

$\displaystyle =\frac{\frac{91}{60}}{\frac{26}{30}}$

$\displaystyle =\frac{91}{60}:\frac{26}{30}$

$\displaystyle =\frac{91}{60}\cdot \frac{30}{26}$

$\displaystyle =\frac{91}{2}\cdot \frac{1}{26}$

$\displaystyle =\frac{91}{52}$

Ushtrimi 7

(i pazgjidhur)

Gjeni vlerën e shprehjeve:

a) $\displaystyle \left( 0,\overline{3}+0,4-0,75\cdot 0,\overline{8} \right)+\left( 0,8+5+0,\overline{6}:0,\overline{4}-1 \right)$

b) $\displaystyle \left\{ 1,\overline{16}+0,\overline{6}-0,25\cdot \left[ \left( 0,25+0,\overline{3}-0,5 \right)\cdot 10,8 \right] \right\}$

Ushtrimi 8

Të gjendet vlera e x-it.

a) $\displaystyle x:\left( \frac{3}{20}+\frac{7}{4} \right)=\left( 1-\frac{2}{3}+\frac{1}{4}-\frac{1}{6} \right)$

b) $\displaystyle \left( 0,75+\frac{1}{2} \right):x=2,5:\frac{5}{16}$

Zgjidhje

a) $\displaystyle x:\left( \frac{3}{20}+\frac{7}{4} \right)=\left( 1-\frac{2}{3}+\frac{1}{4}-\frac{1}{6} \right)$

$\displaystyle x:\left( \frac{3}{20}+\frac{35}{20} \right)=\left( \frac{12}{12}-\frac{2}{12}+\frac{1}{12}-\frac{1}{12} \right)$

$\displaystyle x:\frac{38}{20}=\frac{10}{12}$

$\displaystyle x:\frac{19}{10}=\frac{10}{12}$

$\displaystyle x\cdot \frac{19}{10}=\frac{10}{12}$

$\displaystyle \frac{19x}{10}=\frac{10}{12}$

$\displaystyle 19\cdot 12\cdot x=10\cdot 10$

$\displaystyle x=\frac{100}{19\cdot 12}$

$\displaystyle x=\frac{25}{19\cdot 3}$

$\displaystyle x=\frac{25}{57}$

b) $\displaystyle \left( 0,75+\frac{1}{2} \right):x=2,5:\frac{5}{16}$ $\displaystyle =\left( \frac{75}{100}+\frac{1}{2} \right):x=\frac{25}{10}:\frac{5}{16}$