Simplify : j. `((21)/(10) xx (2)/(14) ) - ((1)/(7) div (36)/(28))`

Answer

To simplify the expression \(\left(\frac{21}{10} \times \frac{2}{14}\right) - \left(\frac{1}{7} \div \frac{36}{28}\right)\), we will follow these steps:

### Step 1: Simplify the multiplication
We start with the multiplication part of the expression, \(\frac{21}{10} \times \frac{2}{14}\).

\[ \frac{21}{10} \times \frac{2}{14} = \frac{21 \times 2}{10 \times 14} \]

### Step 2: Factor and simplify
Now, we can simplify the fraction before multiplying:

\[ \frac{21}{14} = \frac{21 \div 7}{14 \div 7} = \frac{3}{2} \]
\[ \frac{2}{10} = \frac{2 \div 2}{10 \div 2} = \frac{1}{5} \]

Now we can multiply:

\[ \frac{3}{2} \times \frac{1}{5} = \frac{3 \times 1}{2 \times 5} = \frac{3}{10} \]

### Step 3: Simplify the division
Next, we simplify the division part of the expression, \(\frac{1}{7} \div \frac{36}{28}\).

To divide by a fraction, we multiply by its reciprocal:

\[ \frac{1}{7} \div \frac{36}{28} = \frac{1}{7} \times \frac{28}{36} \]

### Step 4: Simplify the fraction
Now we simplify \(\frac{28}{36}\):

\[ \frac{28}{36} = \frac{28 \div 4}{36 \div 4} = \frac{7}{9} \]

Now we multiply:

\[ \frac{1}{7} \times \frac{7}{9} = \frac{1 \times 7}{7 \times 9} = \frac{1}{9} \]

### Step 5: Combine the results
Now we combine the results from Step 2 and Step 4:

\[ \frac{3}{10} - \frac{1}{9} \]

### Step 6: Find a common denominator
The least common multiple (LCM) of 10 and 9 is 90. We will convert both fractions to have a denominator of 90:

\[ \frac{3}{10} = \frac{3 \times 9}{10 \times 9} = \frac{27}{90} \]
\[ \frac{1}{9} = \frac{1 \times 10}{9 \times 10} = \frac{10}{90} \]

### Step 7: Subtract the fractions
Now we can subtract:

\[ \frac{27}{90} - \frac{10}{90} = \frac{27 - 10}{90} = \frac{17}{90} \]

### Final Answer
The simplified expression is:

\[ \frac{17}{90} \]

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