(d) Calculate the following. $$ \begin{array}{llll} ext { (i) } \frac{1}{3}+\frac{2}{3} & ext { (ii) } \frac{2}{5}+\frac{3}{7} & ext { (iii) } \frac{1}{6}+\frac{4}{5}+\frac{2}{3} & ext { (iv) } \frac{3}{4}+\frac{11}{12}+\frac{5}{6}+\frac{7}{8} \ ext { (v) } \frac{6}{7}-\frac{3}{7} & ext { (vi) } \frac{4}{5}-\frac{1}{6}+\frac{2}{3} & ext { (vii) }-\frac{7}{8}+\frac{4}{5}-\frac{11}{2} & ext { (viii) } \frac{3}{4}-\frac{7}{10}+\frac{5}{16}-\frac{3}{20}\end{array} $$

Question

(d) Calculate the following. $$ \begin{array}{llll}	ext { (i) } \frac{1}{3}+\frac{2}{3} & 	ext { (ii) } \frac{2}{5}+\frac{3}{7} & 	ext { (iii) } \frac{1}{6}+\frac{4}{5}+\frac{2}{3} & 	ext { (iv) } \frac{3}{4}+\frac{11}{12}+\frac{5}{6}+\frac{7}{8} \ 	ext { (v) } \frac{6}{7}-\frac{3}{7} & 	ext { (vi) } \frac{4}{5}-\frac{1}{6}+\frac{2}{3} & 	ext { (vii) }-\frac{7}{8}+\frac{4}{5}-\frac{11}{2} & 	ext { (viii) } \frac{3}{4}-\frac{7}{10}+\frac{5}{16}-\frac{3}{20}\end{array} $$

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$\frac{2}{5}+\frac{3}{7}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{2\times 7}{5\times 7}+\frac{3\times 5}{7\times 5}$$
- step2: Multiply the numbers:
  $$\frac{2\times 7}{35}+\frac{3\times 5}{7\times 5}$$
- step3: Multiply the numbers:
  $$\frac{2\times 7}{35}+\frac{3\times 5}{35}$$
- step4: Transform the expression:
  $$\frac{2\times 7+3\times 5}{35}$$
- step5: Multiply the numbers:
  $$\frac{14+3\times 5}{35}$$
- step6: Multiply the numbers:
  $$\frac{14+15}{35}$$
- step7: Add the numbers:
  $$\frac{29}{35}$$
Calculate or simplify the expression $$ 6/7 - 3/7 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{6}{7}-\frac{3}{7}$$
- step1: Transform the expression:
  $$\frac{6-3}{7}$$
- step2: Subtract the numbers:
  $$\frac{3}{7}$$
Calculate or simplify the expression $$ 3/4 - 7/10 + 5/16 - 3/20 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3}{4}-\frac{7}{10}+\frac{5}{16}-\frac{3}{20}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{3\times 5\times 4}{4\times 5\times 4}-\frac{7\times 2\times 4}{10\times 2\times 4}+\frac{5\times 5}{16\times 5}-\frac{3\times 4}{20\times 4}$$
- step2: Multiply the terms:
  $$\frac{3\times 5\times 4}{80}-\frac{7\times 2\times 4}{10\times 2\times 4}+\frac{5\times 5}{16\times 5}-\frac{3\times 4}{20\times 4}$$
- step3: Multiply the terms:
  $$\frac{3\times 5\times 4}{80}-\frac{7\times 2\times 4}{80}+\frac{5\times 5}{16\times 5}-\frac{3\times 4}{20\times 4}$$
- step4: Multiply the numbers:
  $$\frac{3\times 5\times 4}{80}-\frac{7\times 2\times 4}{80}+\frac{5\times 5}{80}-\frac{3\times 4}{20\times 4}$$
- step5: Multiply the numbers:
  $$\frac{3\times 5\times 4}{80}-\frac{7\times 2\times 4}{80}+\frac{5\times 5}{80}-\frac{3\times 4}{80}$$
- step6: Transform the expression:
  $$\frac{3\times 5\times 4-7\times 2\times 4+5\times 5-3\times 4}{80}$$
- step7: Multiply the terms:
  $$\frac{60-7\times 2\times 4+5\times 5-3\times 4}{80}$$
- step8: Multiply the terms:
  $$\frac{60-56+5\times 5-3\times 4}{80}$$
- step9: Multiply the numbers:
  $$\frac{60-56+25-3\times 4}{80}$$
- step10: Multiply the numbers:
  $$\frac{60-56+25-12}{80}$$
- step11: Calculate:
  $$\frac{17}{80}$$
Calculate or simplify the expression $$ 4/5 - 1/6 + 2/3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{4}{5}-\frac{1}{6}+\frac{2}{3}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{4\times 6}{5\times 6}-\frac{5}{6\times 5}+\frac{2\times 10}{3\times 10}$$
- step2: Multiply the numbers:
  $$\frac{4\times 6}{30}-\frac{5}{6\times 5}+\frac{2\times 10}{3\times 10}$$
- step3: Multiply the numbers:
  $$\frac{4\times 6}{30}-\frac{5}{30}+\frac{2\times 10}{3\times 10}$$
- step4: Multiply the numbers:
  $$\frac{4\times 6}{30}-\frac{5}{30}+\frac{2\times 10}{30}$$
- step5: Transform the expression:
  $$\frac{4\times 6-5+2\times 10}{30}$$
- step6: Multiply the numbers:
  $$\frac{24-5+2\times 10}{30}$$
- step7: Multiply the numbers:
  $$\frac{24-5+20}{30}$$
- step8: Calculate:
  $$\frac{39}{30}$$
- step9: Reduce the fraction:
  $$\frac{13}{10}$$
Calculate or simplify the expression $$ 1/6 + 4/5 + 2/3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{6}+\frac{4}{5}+\frac{2}{3}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{5}{6\times 5}+\frac{4\times 6}{5\times 6}+\frac{2\times 10}{3\times 10}$$
- step2: Multiply the numbers:
  $$\frac{5}{30}+\frac{4\times 6}{5\times 6}+\frac{2\times 10}{3\times 10}$$
- step3: Multiply the numbers:
  $$\frac{5}{30}+\frac{4\times 6}{30}+\frac{2\times 10}{3\times 10}$$
- step4: Multiply the numbers:
  $$\frac{5}{30}+\frac{4\times 6}{30}+\frac{2\times 10}{30}$$
- step5: Transform the expression:
  $$\frac{5+4\times 6+2\times 10}{30}$$
- step6: Multiply the numbers:
  $$\frac{5+24+2\times 10}{30}$$
- step7: Multiply the numbers:
  $$\frac{5+24+20}{30}$$
- step8: Add the numbers:
  $$\frac{49}{30}$$
Calculate or simplify the expression $$ 1/3 + 2/3 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{3}+\frac{2}{3}$$
- step1: Transform the expression:
  $$\frac{1+2}{3}$$
- step2: Add the numbers:
  $$\frac{3}{3}$$
- step3: Reduce the numbers:
  $$\frac{1}{1}$$
- step4: Calculate:
  $$1$$
Calculate or simplify the expression $$ -7/8 + 4/5 - 11/2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{-7}{8}+\frac{4}{5}-\frac{11}{2}$$
- step1: Rewrite the fraction:
  $$-\frac{7}{8}+\frac{4}{5}-\frac{11}{2}$$
- step2: Reduce fractions to a common denominator:
  $$-\frac{7\times 5}{8\times 5}+\frac{4\times 8}{5\times 8}-\frac{11\times 20}{2\times 20}$$
- step3: Multiply the numbers:
  $$-\frac{7\times 5}{40}+\frac{4\times 8}{5\times 8}-\frac{11\times 20}{2\times 20}$$
- step4: Multiply the numbers:
  $$-\frac{7\times 5}{40}+\frac{4\times 8}{40}-\frac{11\times 20}{2\times 20}$$
- step5: Multiply the numbers:
  $$-\frac{7\times 5}{40}+\frac{4\times 8}{40}-\frac{11\times 20}{40}$$
- step6: Transform the expression:
  $$\frac{-7\times 5+4\times 8-11\times 20}{40}$$
- step7: Multiply the numbers:
  $$\frac{-35+4\times 8-11\times 20}{40}$$
- step8: Multiply the numbers:
  $$\frac{-35+32-11\times 20}{40}$$
- step9: Multiply the numbers:
  $$\frac{-35+32-220}{40}$$
- step10: Calculate:
  $$\frac{-223}{40}$$
- step11: Rewrite the fraction:
  $$-\frac{223}{40}$$
Calculate or simplify the expression $$ 3/4 + 11/12 + 5/6 + 7/8 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3}{4}+\frac{11}{12}+\frac{5}{6}+\frac{7}{8}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{3\times 3\times 2}{4\times 3\times 2}+\frac{11\times 2}{12\times 2}+\frac{5\times 2\times 2}{6\times 2\times 2}+\frac{7\times 3}{8\times 3}$$
- step2: Multiply the terms:
  $$\frac{3\times 3\times 2}{24}+\frac{11\times 2}{12\times 2}+\frac{5\times 2\times 2}{6\times 2\times 2}+\frac{7\times 3}{8\times 3}$$
- step3: Multiply the numbers:
  $$\frac{3\times 3\times 2}{24}+\frac{11\times 2}{24}+\frac{5\times 2\times 2}{6\times 2\times 2}+\frac{7\times 3}{8\times 3}$$
- step4: Multiply the terms:
  $$\frac{3\times 3\times 2}{24}+\frac{11\times 2}{24}+\frac{5\times 2\times 2}{24}+\frac{7\times 3}{8\times 3}$$
- step5: Multiply the numbers:
  $$\frac{3\times 3\times 2}{24}+\frac{11\times 2}{24}+\frac{5\times 2\times 2}{24}+\frac{7\times 3}{24}$$
- step6: Transform the expression:
  $$\frac{3\times 3\times 2+11\times 2+5\times 2\times 2+7\times 3}{24}$$
- step7: Multiply the terms:
  $$\frac{18+11\times 2+5\times 2\times 2+7\times 3}{24}$$
- step8: Multiply the numbers:
  $$\frac{18+22+5\times 2\times 2+7\times 3}{24}$$
- step9: Multiply the terms:
  $$\frac{18+22+20+7\times 3}{24}$$
- step10: Multiply the numbers:
  $$\frac{18+22+20+21}{24}$$
- step11: Add the numbers:
  $$\frac{81}{24}$$
- step12: Reduce the fraction:
  $$\frac{27}{8}$$
Here are the calculations for each part:

(i) 
$$
\frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1
$$

(ii) 
To add $$\frac{2}{5} + \frac{3}{7}$$, we need a common denominator. The least common multiple of 5 and 7 is 35.
$$
\frac{2}{5} = \frac{2 \times 7}{5 \times 7} = \frac{14}{35}, \quad \frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35}
$$
Now, add them:
$$
\frac{14}{35} + \frac{15}{35} = \frac{29}{35}
$$

(iii) 
For $$\frac{1}{6} + \frac{4}{5} + \frac{2}{3}$$, we find a common denominator. The least common multiple of 6, 5, and 3 is 30.
$$
\frac{1}{6} = \frac{5}{30}, \quad \frac{4}{5} = \frac{24}{30}, \quad \frac{2}{3} = \frac{20}{30}
$$
Now, add them:
$$
\frac{5}{30} + \frac{24}{30} + \frac{20}{30} = \frac{49}{30} = 1 \frac{19}{30}
$$

(iv) 
For $$\frac{3}{4} + \frac{11}{12} + \frac{5}{6} + \frac{7}{8}$$, the least common multiple of 4, 12, 6, and 8 is 24.
$$
\frac{3}{4} = \frac{18}{24}, \quad \frac{11}{12} = \frac{22}{24}, \quad \frac{5}{6} = \frac{20}{24}, \quad \frac{7}{8} = \frac{21}{24}
$$
Now, add them:
$$
\frac{18}{24} + \frac{22}{24} + \frac{20}{24} + \frac{21}{24} = \frac{81}{24} = 3 \frac{9}{24} = 3 \frac{3}{8}
$$

(v) 
For $$\frac{6}{7} - \frac{3}{7}$$:
$$
\frac{6}{7} - \frac{3}{7} = \frac{3}{7}
$$

(vi) 
For $$\frac{4}{5} - \frac{1}{6} + \frac{2}{3}$$, the least common multiple of 5, 6, and 3 is 30.
$$
\frac{4}{5} = \frac{24}{30}, \quad \frac{1}{6} = \frac{5}{30}, \quad \frac{2}{3} = \frac{20}{30}
$$
Now, add them:
$$
\frac{24}{30} - \frac{5}{30} + \frac{20}{30} = \frac{39}{30} = 1 \frac{9}{30} = 1 \frac{3}{10}
$$

(vii) 
For $$-\frac{7}{8} + \frac{4}{5} - \frac{11}{2}$$, we need a common denominator. The least common multiple of 8, 5, and 2 is 40.
$$
-\frac{7}{8} = -\frac{35}{40}, \quad \frac{4}{5} = \frac{32}{40}, \quad -\frac{11}{2} = -\frac{220}{40}
$$
Now, add them:
$$
-\frac{35}{40} + \frac{32}{40} - \frac{220}{40} = -\frac{223}{40} = -5 \frac{23}{40}
$$

(viii) 
For $$\frac{3}{4} - \frac{7}{10} + \frac{5}{16} - \frac{3}{20}$$, the least common multiple of 4, 10, 16, and 20 is 80.
$$
\frac{3}{4} = \frac{60}{80}, \quad -\frac{7}{10} = -\frac{56}{80}, \quad \frac{5}{16} = \frac{25}{80}, \quad -\frac{3}{20} = -\frac{12}{80}
$$
Now, add them:
$$
\frac{60}{80} - \frac{56}{80} + \frac{25}{80} - \frac{12}{80} = \frac{17}{80}
$$

### Summary of Results:
- (i) $$1$$
- (ii) $$\frac{29}{35}$$
- (iii) $$\frac{49}{30} = 1 \frac{19}{30}$$
- (iv) $$\frac{81}{24} = 3 \frac{3}{8}$$
- (v) $$\frac{3}{7}$$
- (vi) $$\frac{39}{30} = 1 \frac{3}{10}$$
- (vii) $$-\frac{223}{40} = -5 \frac{23}{40}$$
- (viii) $$\frac{17}{80}$$
Here are the results for each calculation: (i) $$1$$ (ii) $$\frac{29}{35}$$ (iii) $$\frac{49}{30} = 1 \frac{19}{30}$$ (iv) $$\frac{81}{24} = 3 \frac{3}{8}$$ (v) $$\frac{3}{7}$$ (vi) $$\frac{39}{30} = 1 \frac{3}{10}$$ (vii) $$-\frac{223}{40} = -5 \frac{23}{40}$$ (viii) $$\frac{17}{80}$$

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