1. **Problem statement:** The sum of three numbers is 174. The ratio of the second number to the third number is 9:16, and the ratio of the first number to the third number is 1:4. We need to find the second number. 2. **Set variables:** Let the third number be $x$. 3. **Express other numbers in terms of $x$:** - First number = $\frac{1}{4}x$ - Second number = $\frac{9}{16}x$ 4. **Write the sum equation:** $$\frac{1}{4}x + \frac{9}{16}x + x = 174$$ 5. **Find common denominator and combine:** $$\frac{4}{16}x + \frac{9}{16}x + \frac{16}{16}x = 174$$ $$\frac{4 + 9 + 16}{16}x = 174$$ $$\frac{29}{16}x = 174$$ 6. **Solve for $x$:** $$x = 174 \times \frac{16}{29} = 174 \times \frac{16}{29}$$ Calculate: $$174 \div 29 = 6$$ $$6 \times 16 = 96$$ So, $x = 96$. 7. **Find the second number:** $$\text{Second number} = \frac{9}{16} \times 96 = 9 \times 6 = 54$$ **Final answer:** The second number is **54**.