1. The problem is to find how to get the numbers 1, 27, 35, 44, and 52 from their sum 172 using a scaling factor. 2. First, calculate the sum of the original numbers: $$1 + 27 + 35 + 44 + 52 = 159$$. 3. The scaling factor is the ratio of the desired sum to the original sum: $$\text{scaling factor} = \frac{172}{159}$$. 4. Calculate the scaling factor: $$\frac{172}{159} \approx 1.0811$$. 5. Multiply each original number by the scaling factor to get the scaled numbers: - $$1 \times 1.0811 \approx 1.0811$$ - $$27 \times 1.0811 \approx 29.1897$$ - $$35 \times 1.0811 \approx 37.8385$$ - $$44 \times 1.0811 \approx 47.5684$$ - $$52 \times 1.0811 \approx 56.2189$$ 6. These scaled numbers sum approximately to 172, confirming the scaling factor works. 7. In summary, the formula used is $$\text{scaled number} = \text{original number} \times \frac{\text{desired sum}}{\text{original sum}}$$. 8. This method preserves the proportions of the original numbers while adjusting their total to the desired sum.