1. **State the problem:** We want to find how to get the numbers 2, 42, 47, 57, and 58 from a total sum of 180 using scaling fractions. 2. **Understanding the problem:** Scaling fractions means we want to find fractions of 180 that correspond proportionally to these numbers. 3. **Formula:** If $x_i$ are the numbers and $S$ is the total sum, the scaling fraction for each number is given by: $$\text{scaled}_i = \frac{x_i}{\sum x_i} \times S$$ 4. **Calculate the sum of the given numbers:** $$\sum x_i = 2 + 42 + 47 + 57 + 58 = 206$$ 5. **Calculate each scaled number:** - For 2: $$\text{scaled}_1 = \frac{2}{206} \times 180 = \frac{2 \times 180}{206} = \frac{360}{206}$$ - For 42: $$\text{scaled}_2 = \frac{42}{206} \times 180 = \frac{42 \times 180}{206} = \frac{7560}{206}$$ - For 47: $$\text{scaled}_3 = \frac{47}{206} \times 180 = \frac{8460}{206}$$ - For 57: $$\text{scaled}_4 = \frac{57}{206} \times 180 = \frac{10260}{206}$$ - For 58: $$\text{scaled}_5 = \frac{58}{206} \times 180 = \frac{10440}{206}$$ 6. **Simplify each fraction by canceling common factors:** - For 2: $$\frac{360}{206} = \frac{\cancel{2} \times 180}{\cancel{2} \times 103} = \frac{180}{103} \approx 1.75$$ - For 42: $$\frac{7560}{206} = \frac{\cancel{2} \times 3780}{\cancel{2} \times 103} = \frac{3780}{103} \approx 36.70$$ - For 47: $$\frac{8460}{206} = \frac{\cancel{2} \times 4230}{\cancel{2} \times 103} = \frac{4230}{103} \approx 41.07$$ - For 57: $$\frac{10260}{206} = \frac{\cancel{2} \times 5130}{\cancel{2} \times 103} = \frac{5130}{103} \approx 49.71$$ - For 58: $$\frac{10440}{206} = \frac{\cancel{2} \times 5220}{\cancel{2} \times 103} = \frac{5220}{103} \approx 50.68$$ 7. **Interpretation:** These scaled values sum to 180 and maintain the same proportions as the original numbers. **Final scaled numbers:** approximately 1.75, 36.70, 41.07, 49.71, and 50.68 respectively.