5/12 of a number is subtracted from 3/4 of the number. Their positive difference is 7 less than 5/6 of the number. Find the number.

Let the number be $x$.
"$\frac{5}{12}$ of a number is subtracted from $\frac{3}{4}$ of the number" translates to:

$$\frac{3}{4}x - \frac{5}{12}x$$

To simplify this, find a common denominator, which is 12:

$$\frac{9}{12}x - \frac{5}{12}x = \frac{4}{12}x = \frac{1}{3}x$$

The positive difference is $\frac{1}{3}x$ (assuming $x$ is positive).

"Their positive difference is 7 less than $\frac{5}{6}$ of the number."
"$\frac{5}{6}$ of the number" is $\frac{5}{6}x$.
"7 less than $\frac{5}{6}$ of the number" is:

$$\frac{5}{6}x - 7$$

Equate the expressions from Step 1 and Step 2:

$$\frac{1}{3}x = \frac{5}{6}x - 7$$

To eliminate the fractions, multiply every term in the equation by the least common multiple (LCM) of the denominators (3 and 6), which is 6:

$$6 \left( \frac{1}{3}x \right) = 6 \left( \frac{5}{6}x \right) - 6(7)$$ $$2x = 5x - 42$$

Subtract $5x$ from both sides:

$$2x - 5x = -42$$ $$-3x = -42$$

Divide by -3:

$$x = \frac{-42}{-3}$$ $$x = 14$$

The number is 14.