Question: User: $-0.2\overline{2}$
$1.60$
$15.$ $-5 \frac{1}{12} - 3 \frac{2}{3}$
$\frac{61}{12} - \frac{11}{3}$
graph shapes: a table/grid with horizontal and vertical lines; the top-right cell shows "$-0.2\overline{2}$", the middle-right cell shows "$1.60$", and below the grid the worked fraction expression "$15.$ $-5 \frac{1}{12} - 3 \frac{2}{3}$" leads to "$\frac{61}{12} - \frac{11}{3}$". position_hint: center
1. **State the problem:** Simplify the expression $15. -5 \frac{1}{12} - 3 \frac{2}{3}$ and verify the fraction subtraction $\frac{61}{12} - \frac{11}{3}$.
2. **Convert mixed numbers to improper fractions:**
- $-5 \frac{1}{12} = -\left(5 + \frac{1}{12}\right) = -\frac{60}{12} - \frac{1}{12} = -\frac{61}{12}$
- $3 \frac{2}{3} = 3 + \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3}$
3. **Rewrite the expression:**
$$15 - \frac{61}{12} - \frac{11}{3}$$
4. **Convert $15$ to a fraction with denominator $12$ to combine:**
$$15 = \frac{15 \times 12}{12} = \frac{180}{12}$$
5. **Rewrite the expression with common denominator:**
$$\frac{180}{12} - \frac{61}{12} - \frac{11}{3}$$
6. **Convert $\frac{11}{3}$ to denominator $12$:**
$$\frac{11}{3} = \frac{11 \times 4}{3 \times 4} = \frac{44}{12}$$
7. **Now subtract all fractions:**
$$\frac{180}{12} - \frac{61}{12} - \frac{44}{12} = \frac{180 - 61 - 44}{12} = \frac{75}{12}$$
8. **Simplify the fraction $\frac{75}{12}$:**
Divide numerator and denominator by their greatest common divisor $3$:
$$\frac{\cancel{75}^{25}}{\cancel{12}^{4}}$$
So,
$$\frac{75}{12} = \frac{25}{4}$$
9. **Convert $\frac{25}{4}$ to a mixed number:**
$$\frac{25}{4} = 6 \frac{1}{4}$$
**Final answer:**
$$15 - 5 \frac{1}{12} - 3 \frac{2}{3} = 6 \frac{1}{4}$$