What should be subtracted from `((9)/(10) + (13)/(20))` to get `(1)/(12)` ?

Answer

To solve the problem of what should be subtracted from \(\frac{9}{10} + \frac{13}{20}\) to get \(\frac{1}{12}\), we can follow these steps:

### Step 1: Combine the fractions \(\frac{9}{10}\) and \(\frac{13}{20}\)
To add these fractions, we need a common denominator. The denominators are 10 and 20. The least common multiple (LCM) of 10 and 20 is 20.

- Convert \(\frac{9}{10}\) to have a denominator of 20:
\[ \frac{9}{10} = \frac{9 \times 2}{10 \times 2} = \frac{18}{20} \]

Now we can add:
\[ \frac{18}{20} + \frac{13}{20} = \frac{18 + 13}{20} = \frac{31}{20} \]

### Step 2: Set up the equation
Now we have:
\[ \frac{31}{20} - x = \frac{1}{12} \]

### Step 3: Isolate \(x\)
To find \(x\), we can rearrange the equation:
\[ -x = \frac{1}{12} - \frac{31}{20} \]
Multiplying both sides by -1 gives:
\[ x = -\left(\frac{1}{12} - \frac{31}{20}\right) \]

### Step 4: Find a common denominator for \(\frac{1}{12}\) and \(\frac{31}{20}\)
The least common multiple of 12 and 20 is 60. We convert both fractions:
\[ \frac{1}{12} = \frac{1 \times 5}{12 \times 5} = \frac{5}{60} \]
\[ \frac{31}{20} = \frac{31 \times 3}{20 \times 3} = \frac{93}{60} \]

### Step 5: Subtract the fractions
Now we can subtract:
\[ x = -\left(\frac{5}{60} - \frac{93}{60}\right) = -\left(\frac{5 - 93}{60}\right) = -\left(\frac{-88}{60}\right) = \frac{88}{60} \]

### Step 6: Simplify \(\frac{88}{60}\)
We can simplify this fraction:
\[ \frac{88}{60} = \frac{44}{30} = \frac{22}{15} \]

### Final Answer
Thus, the value of \(x\) that should be subtracted is:
\[ x = \frac{22}{15} \]

---