$$ \begin{array}{lll} ext { Calculate: } & ext { (b) } \frac{2}{3}-\frac{1}{3}+\frac{1}{2} & ext { (c) } \frac{4}{3}+\frac{2}{9}+\frac{3}{4} \ ext { (a) } \frac{1}{5}+\frac{1}{3} imes \frac{1}{2} & ext { (c) } 2 \frac{3}{4}-1 \frac{1}{2} imes 1 \frac{2}{3} & ext { (1) } 2 \frac{1}{4}+13 \frac{1}{2}-2\end{array} $$

Question

$$ \begin{array}{lll}	ext { Calculate: } & 	ext { (b) } \frac{2}{3}-\frac{1}{3}+\frac{1}{2} & 	ext { (c) } \frac{4}{3}+\frac{2}{9}+\frac{3}{4} \ 	ext { (a) } \frac{1}{5}+\frac{1}{3} 	imes \frac{1}{2} & 	ext { (c) } 2 \frac{3}{4}-1 \frac{1}{2} 	imes 1 \frac{2}{3} & 	ext { (1) } 2 \frac{1}{4}+13 \frac{1}{2}-2\end{array} $$

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{5}+\frac{1}{3}\times \frac{1}{2}$$
- step1: Multiply the numbers:
  $$\frac{1}{5}+\frac{1}{6}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{6}{5\times 6}+\frac{5}{6\times 5}$$
- step3: Multiply the numbers:
  $$\frac{6}{30}+\frac{5}{6\times 5}$$
- step4: Multiply the numbers:
  $$\frac{6}{30}+\frac{5}{30}$$
- step5: Transform the expression:
  $$\frac{6+5}{30}$$
- step6: Add the numbers:
  $$\frac{11}{30}$$
Calculate or simplify the expression $$ 2 \frac{1}{4}+13 \frac{1}{2}-2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2\frac{1}{4}+13\frac{1}{2}-2$$
- step1: Convert the expressions:
  $$\frac{9}{4}+\frac{27}{2}-2$$
- step2: Reduce fractions to a common denominator:
  $$\frac{9}{4}+\frac{27\times 2}{2\times 2}-\frac{2\times 2\times 2}{2\times 2}$$
- step3: Multiply the numbers:
  $$\frac{9}{4}+\frac{27\times 2}{4}-\frac{2\times 2\times 2}{2\times 2}$$
- step4: Multiply the numbers:
  $$\frac{9}{4}+\frac{27\times 2}{4}-\frac{2\times 2\times 2}{4}$$
- step5: Transform the expression:
  $$\frac{9+27\times 2-2\times 2\times 2}{4}$$
- step6: Multiply the numbers:
  $$\frac{9+54-2\times 2\times 2}{4}$$
- step7: Multiply the terms:
  $$\frac{9+54-8}{4}$$
- step8: Calculate:
  $$\frac{55}{4}$$
Calculate or simplify the expression $$ \frac{4}{3}+\frac{2}{9}+\frac{3}{4} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{4}{3}+\frac{2}{9}+\frac{3}{4}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{4\times 3\times 4}{3\times 3\times 4}+\frac{2\times 4}{9\times 4}+\frac{3\times 3\times 3}{4\times 3\times 3}$$
- step2: Multiply the terms:
  $$\frac{4\times 3\times 4}{36}+\frac{2\times 4}{9\times 4}+\frac{3\times 3\times 3}{4\times 3\times 3}$$
- step3: Multiply the numbers:
  $$\frac{4\times 3\times 4}{36}+\frac{2\times 4}{36}+\frac{3\times 3\times 3}{4\times 3\times 3}$$
- step4: Multiply the terms:
  $$\frac{4\times 3\times 4}{36}+\frac{2\times 4}{36}+\frac{3\times 3\times 3}{36}$$
- step5: Transform the expression:
  $$\frac{4\times 3\times 4+2\times 4+3\times 3\times 3}{36}$$
- step6: Multiply the terms:
  $$\frac{48+2\times 4+3\times 3\times 3}{36}$$
- step7: Multiply the numbers:
  $$\frac{48+8+3\times 3\times 3}{36}$$
- step8: Multiply the terms:
  $$\frac{48+8+27}{36}$$
- step9: Add the numbers:
  $$\frac{83}{36}$$
Calculate or simplify the expression $$ \frac{2}{3}-\frac{1}{3}+\frac{1}{2} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{2}{3}-\frac{1}{3}+\frac{1}{2}$$
- step1: Subtract the numbers:
  $$\frac{1}{3}+\frac{1}{2}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{2}{3\times 2}+\frac{3}{2\times 3}$$
- step3: Multiply the numbers:
  $$\frac{2}{6}+\frac{3}{2\times 3}$$
- step4: Multiply the numbers:
  $$\frac{2}{6}+\frac{3}{6}$$
- step5: Transform the expression:
  $$\frac{2+3}{6}$$
- step6: Add the numbers:
  $$\frac{5}{6}$$
Calculate or simplify the expression $$ 2 \frac{3}{4}-1 \frac{1}{2} \times 1 \frac{2}{3} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2\frac{3}{4}-1\frac{1}{2}\times 1\frac{2}{3}$$
- step1: Convert the expressions:
  $$\frac{11}{4}-\frac{3}{2}\times \frac{5}{3}$$
- step2: Multiply the numbers:
  $$\frac{11}{4}-\frac{5}{2}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{11}{4}-\frac{5\times 2}{2\times 2}$$
- step4: Multiply the numbers:
  $$\frac{11}{4}-\frac{5\times 2}{4}$$
- step5: Transform the expression:
  $$\frac{11-5\times 2}{4}$$
- step6: Multiply the numbers:
  $$\frac{11-10}{4}$$
- step7: Subtract the numbers:
  $$\frac{1}{4}$$
Let's calculate each expression step by step.

### (a) $$ \frac{1}{5} + \frac{1}{3} \times \frac{1}{2} $$

1. First, calculate $$ \frac{1}{3} \times \frac{1}{2} $$:
   $$
   \frac{1}{3} \times \frac{1}{2} = \frac{1 \times 1}{3 \times 2} = \frac{1}{6}
   $$

2. Now, add $$ \frac{1}{5} + \frac{1}{6} $$:
   - Find a common denominator, which is 30:
   $$
   \frac{1}{5} = \frac{6}{30}, \quad \frac{1}{6} = \frac{5}{30}
   $$
   - Now add:
   $$
   \frac{6}{30} + \frac{5}{30} = \frac{11}{30}
   $$

### (b) $$ \frac{2}{3} - \frac{1}{3} + \frac{1}{2} $$

1. First, calculate $$ \frac{2}{3} - \frac{1}{3} $$:
   $$
   \frac{2}{3} - \frac{1}{3} = \frac{1}{3}
   $$

2. Now add $$ \frac{1}{3} + \frac{1}{2} $$:
   - Find a common denominator, which is 6:
   $$
   \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6}
   $$
   - Now add:
   $$
   \frac{2}{6} + \frac{3}{6} = \frac{5}{6}
   $$

### (c) $$ \frac{4}{3} + \frac{2}{9} + \frac{3}{4} $$

1. Find a common denominator, which is 36:
   $$
   \frac{4}{3} = \frac{48}{36}, \quad \frac{2}{9} = \frac{8}{36}, \quad \frac{3}{4} = \frac{27}{36}
   $$

2. Now add:
   $$
   \frac{48}{36} + \frac{8}{36} + \frac{27}{36} = \frac{83}{36}
   $$

### (d) $$ 2 \frac{3}{4} - 1 \frac{1}{2} \times 1 \frac{2}{3} $$

1. Convert mixed numbers to improper fractions:
   $$
   2 \frac{3}{4} = \frac{11}{4}, \quad 1 \frac{1}{2} = \frac{3}{2}, \quad 1 \frac{2}{3} = \frac{5}{3}
   $$

2. Calculate $$ \frac{3}{2} \times \frac{5}{3} $$:
   $$
   \frac{3}{2} \times \frac{5}{3} = \frac{15}{6} = \frac{5}{2}
   $$

3. Now subtract:
   $$
   \frac{11}{4} - \frac{5}{2} = \frac{11}{4} - \frac{10}{4} = \frac{1}{4}
   $$

### (e) $$ 2 \frac{1}{4} + 13 \frac{1}{2} - 2 $$

1. Convert mixed numbers to improper fractions:
   $$
   2 \frac{1}{4} = \frac{9}{4}, \quad 13 \frac{1}{2} = \frac{27}{2}
   $$

2. Find a common denominator, which is 4:
   $$
   \frac{27}{2} = \frac{54}{4}
   $$

3. Now add and subtract:
   $$
   \frac{9}{4} + \frac{54}{4} - 2 = \frac{9 + 54 - 8}{4} = \frac{55}{4}
   $$

### Summary of Results:
- (a) $$ \frac{11}{30} $$
- (b) $$ \frac{5}{6} $$
- (c) $$ \frac{83}{36} $$
- (d) $$ \frac{1}{4} $$
- (e) $$ \frac{55}{4} $$ or $$ 13 \frac{3}{4} $$
The results are: - (a) $$ \frac{11}{30} $$ - (b) $$ \frac{5}{6} $$ - (c) $$ \frac{83}{36} $$ - (d) $$ \frac{1}{4} $$ - (e) $$ \frac{55}{4} $$ or $$ 13 \frac{3}{4} $$

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