Simplify : i. `(1 (5)/(7) xx (7)/(10) ) - ((3)/(5) div (9)/(10))`

Answer

To simplify the expression \(1 \frac{5}{7} \times \frac{7}{10} - \frac{3}{5} \div \frac{9}{10}\), we will follow these steps:

### Step 1: Convert the mixed number to an improper fraction
The mixed number \(1 \frac{5}{7}\) can be converted to an improper fraction.

\[ 1 \frac{5}{7} = \frac{(1 \times 7) + 5}{7} = \frac{12}{7} \]

**Hint:** To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator, then place this value over the original denominator.

### Step 2: Multiply the improper fraction by \(\frac{7}{10}\)
Now we multiply \(\frac{12}{7}\) by \(\frac{7}{10}\):

\[ \frac{12}{7} \times \frac{7}{10} = \frac{12 \times 7}{7 \times 10} \]

The \(7\) in the numerator and denominator cancels out:

\[ = \frac{12}{10} \]

**Hint:** When multiplying fractions, you can simplify by canceling out common factors in the numerator and denominator.

### Step 3: Simplify \(\frac{12}{10}\)
Now, we simplify \(\frac{12}{10}\):

\[ \frac{12}{10} = \frac{6}{5} \quad (\text{dividing both numerator and denominator by 2}) \]

**Hint:** To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

### Step 4: Handle the division \(\frac{3}{5} \div \frac{9}{10}\)
Next, we need to simplify \(\frac{3}{5} \div \frac{9}{10}\). To do this, we will multiply by the reciprocal of \(\frac{9}{10}\):

\[ \frac{3}{5} \div \frac{9}{10} = \frac{3}{5} \times \frac{10}{9} \]

**Hint:** To divide by a fraction, multiply by its reciprocal (flip the fraction).

### Step 5: Multiply the fractions
Now we multiply:

\[ \frac{3 \times 10}{5 \times 9} = \frac{30}{45} \]

### Step 6: Simplify \(\frac{30}{45}\)
Now we simplify \(\frac{30}{45}\):

\[ \frac{30}{45} = \frac{2}{3} \quad (\text{dividing both numerator and denominator by 15}) \]

**Hint:** Again, simplify by dividing both the numerator and denominator by their GCD.

### Step 7: Subtract the two results
Now we have \( \frac{6}{5} - \frac{2}{3} \). To subtract these fractions, we need a common denominator. The least common multiple (LCM) of \(5\) and \(3\) is \(15\).

### Step 8: Convert to a common denominator
Convert both fractions to have a denominator of \(15\):

\[ \frac{6}{5} = \frac{6 \times 3}{5 \times 3} = \frac{18}{15} \]

\[ \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \]

### Step 9: Perform the subtraction
Now we can subtract:

\[ \frac{18}{15} - \frac{10}{15} = \frac{18 - 10}{15} = \frac{8}{15} \]

### Final Answer
The simplified result of the expression \(1 \frac{5}{7} \times \frac{7}{10} - \frac{3}{5} \div \frac{9}{10}\) is:

\[ \frac{8}{15} \]

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