3/1/10 - 15/9*(2/1/4 - 1/7/8)

To solve the expression \( \frac{3}{1}{10} - \frac{15}{9} \left( \frac{2}{1}{4} - \frac{1}{7/8} \right) \), we first need to handle the mixed fractions and the operations step by step.

1. Convert mixed fractions to improper fractions:

   • \( \frac{3}{1}{10} = \frac{3 \times 10 + 1}{10} = \frac{31}{10} \)
   • \( \frac{2}{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4} \)

   For \( \frac{1}{7/8} \):

   • This is the same as \( \frac{8}{7} \).

2. Calculate the inner parentheses: \[ \frac{9}{4} - \frac{8}{7} \]

   To subtract these fractions, we need a common denominator. The least common multiple of 4 and 7 is 28.

   • Convert \( \frac{9}{4} \) to a denominator of 28: \[ \frac{9}{4} = \frac{9 \times 7}{4 \times 7} = \frac{63}{28} \]
   • Convert \( \frac{8}{7} \) to a denominator of 28: \[ \frac{8}{7} = \frac{8 \times 4}{7 \times 4} = \frac{32}{28} \]

   Now subtract: \[ \frac{63}{28} - \frac{32}{28} = \frac{31}{28} \]

3. Multiply: Now we need to multiply this result by \( \frac{15}{9} \): \[ \frac{15}{9} \times \frac{31}{28} = \frac{15 \times 31}{9 \times 28} = \frac{465}{252} \]

   To reduce this fraction, we can find the GCD:

   • Both 465 and 252 can be divided by 3. \[ \frac{465 \div 3}{252 \div 3} = \frac{155}{84} \]

4. Solve the overall expression: Now we need to subtract this from \( \frac{31}{10} \): \[ \frac{31}{10} - \frac{155}{84} \]

   Next, find the common denominator. The least common multiple of 10 and 84 is 420.

   • Convert \( \frac{31}{10} \): \[ \frac{31}{10} = \frac{31 \times 42}{10 \times 42} = \frac{1302}{420} \]
   • Convert \( \frac{155}{84} \): \[ \frac{155}{84} = \frac{155 \times 5}{84 \times 5} = \frac{775}{420} \]

   Now subtract: \[ \frac{1302}{420} - \frac{775}{420} = \frac{1302 - 775}{420} = \frac{527}{420} \]

5. Final result: The answer can be left as \( \frac{527}{420} \) or converted to a mixed number.

   Calculating \( 527 \div 420 \) gives:

   • The quotient is 1 with a remainder of 107.

   Thus: \[ \frac{527}{420} = 1 \frac{107}{420} \]

So, the final answer is:

\[ \frac{527}{420} \text{ or } 1 \frac{107}{420} \]