(11/2×2/5+3 3/7:9/14):5/12

Question 1

Задача Задача №1698460

А

Аноним

02 октября 2024 03:55 · публичный вопрос

А

Аноним 02 октября 2024 03:55

(11/2×2/5+3 3/7:9/14):5/12

Answer

To solve the expression \( \left( \frac{11}{2} \times \frac{2}{5} + 3 \frac{3}{7} \div \frac{9}{14} \right) \div \frac{5}{12} \), we will simplify it step by step.

First, let's rewrite the mixed number \( 3 \frac{3}{7} \) as an improper fraction: \[ 3 \frac{3}{7} = \frac{3 \times 7 + 3}{7} = \frac{21 + 3}{7} = \frac{24}{7} \]

Now we can rewrite the expression: \[ \left( \frac{11}{2} \times \frac{2}{5} + \frac{24}{7} \div \frac{9}{14} \right) \div \frac{5}{12} \]

Next, we calculate \( \frac{11}{2} \times \frac{2}{5} \): \[ \frac{11 \times 2}{2 \times 5} = \frac{22}{10} = \frac{11}{5} \]

Now for the division \( \frac{24}{7} \div \frac{9}{14} \): Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{24}{7} \div \frac{9}{14} = \frac{24}{7} \times \frac{14}{9} = \frac{24 \times 14}{7 \times 9} \] Now simplifying: \[ \frac{24 \times 14}{7 \times 9} = \frac{336}{63} = \frac{112}{21} \quad (\text{dividing both by } 3) \]

Now we can combine the two fractions \(\frac{11}{5}\) and \(\frac{112}{21}\): First, we need a common denominator. The least common multiple of 5 and 21 is 105. Convert both fractions:

\[ \frac{11}{5} = \frac{11 \times 21}{5 \times 21} = \frac{231}{105} \] \[ \frac{112}{21} = \frac{112 \times 5}{21 \times 5} = \frac{560}{105} \]

Now we sum them: \[ \frac{231}{105} + \frac{560}{105} = \frac{231 + 560}{105} = \frac{791}{105} \]

Now we take the entire expression: \[ \left( \frac{791}{105} \right) \div \frac{5}{12} \] Dividing by a fraction is multiplying by its reciprocal: \[ \frac{791}{105} \times \frac{12}{5} = \frac{791 \times 12}{105 \times 5} \]

Calculating \( 105 \times 5 = 525 \). Calculating \( 791 \times 12 \): \[ 791 \times 12 = 9492 \]

So now we have: \[ \frac{9492}{525} \]

Now, let's simplify \( \frac{9492}{525} \) by finding the greatest common divisor (GCD). Both of these can be simplified; they can both be divided by 3: \[ 9492 \div 3 = 3164 \] \[ 525 \div 3 = 175 \]

So: \[ \frac{9492}{525} = \frac{3164}{175} \]

This fraction does not simplify further. Thus, the final answer is: \[ \boxed{\frac{3164}{175}} \]

Question 2

(11/2×2/5+3 3/7:9/14):5/12

Answer

To solve the expression \( \left( \frac{11}{2} \times \frac{2}{5} + 3 \frac{3}{7} \div \frac{9}{14} \right) \div \frac{5}{12} \), we will simplify it step by step. First, let's rewrite the mixed number \( 3 \frac{3}{7} \) as an improper fraction: \[ 3 \frac{3}{7} = \frac{3 \times 7 + 3}{7} = \frac{21 + 3}{7} = \frac{24}{7} \] Now we can rewrite the expression: \[ \left( \frac{11}{2} \times \frac{2}{5} + \frac{24}{7} \div \frac{9}{14} \right) \div \frac{5}{12} \] Next, we calculate \( \frac{11}{2} \times \frac{2}{5} \): \[ \frac{11 \times 2}{2 \times 5} = \frac{22}{10} = \frac{11}{5} \] Now for the division \( \frac{24}{7} \div \frac{9}{14} \): Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{24}{7} \div \frac{9}{14} = \frac{24}{7} \times \frac{14}{9} = \frac{24 \times 14}{7 \times 9} \] Now simplifying: \[ \frac{24 \times 14}{7 \times 9} = \frac{336}{63} = \frac{112}{21} \quad (\text{dividing both by } 3) \] Now we can combine the two fractions \(\frac{11}{5}\) and \(\frac{112}{21}\): First, we need a common denominator. The least common multiple of 5 and 21 is 105. Convert both fractions: \[ \frac{11}{5} = \frac{11 \times 21}{5 \times 21} = \frac{231}{105} \] \[ \frac{112}{21} = \frac{112 \times 5}{21 \times 5} = \frac{560}{105} \] Now we sum them: \[ \frac{231}{105} + \frac{560}{105} = \frac{231 + 560}{105} = \frac{791}{105} \] Now we take the entire expression: \[ \left( \frac{791}{105} \right) \div \frac{5}{12} \] Dividing by a fraction is multiplying by its reciprocal: \[ \frac{791}{105} \times \frac{12}{5} = \frac{791 \times 12}{105 \times 5} \] Calculating \( 105 \times 5 = 525 \). Calculating \( 791 \times 12 \): \[ 791 \times 12 = 9492 \] So now we have: \[ \frac{9492}{525} \] Now, let's simplify \( \frac{9492}{525} \) by finding the greatest common divisor (GCD). Both of these can be simplified; they can both be divided by 3: \[ 9492 \div 3 = 3164 \] \[ 525 \div 3 = 175 \] So: \[ \frac{9492}{525} = \frac{3164}{175} \] This fraction does not simplify further. Thus, the final answer is: \[ \boxed{\frac{3164}{175}} \]