Yu Yan makes a wall hanging from a piece of fabric. For the first part, she cuts off \( \frac{1}{3} \) of the leng of the original piece of fabric. For the next two parts, she cuts off 10 inches and 5 inches from the length of the remaining fabric. In total, she cuts \( 35 \frac{1}{2} \) inches from the length of the original piece of fabric. How long was the original piece of fabric? Write and solve an equation. Check your solution.

Let's solve the problem step by step. **Let’s define:** - \( L \) = the original length of the fabric in inches. **Step 1: First Cut** - Yu Yan cuts off \( \frac{1}{3} \) of the original length. - Amount cut: \( \frac{1}{3}L \) - Remaining fabric after first cut: \( L - \frac{1}{3}L = \frac{2}{3}L \) **Step 2: Second and Third Cuts** - She then cuts off 10 inches and 5 inches from the remaining fabric. - Total amount cut in these two cuts: \( 10 + 5 = 15 \) inches - Remaining fabric after these cuts: \( \frac{2}{3}L - 15 \) **Total Fabric Cut** - The total fabric cut is the sum of all cuts: \( \frac{1}{3}L + 15 \) - According to the problem, this total is \( 35\frac{1}{2} \) inches, which is \( 35.5 \) inches. **Setting Up the Equation:** \[ \frac{1}{3}L + 15 = 35.5 \] **Solving the Equation:** 1. Subtract 15 from both sides: \[ \frac{1}{3}L = 35.5 - 15 \] \[ \frac{1}{3}L = 20.5 \] 2. Multiply both sides by 3 to solve for \( L \): \[ L = 20.5 \times 3 \] \[ L = 61.5 \] **Solution:** The original length of the fabric was **61.5 inches**. **Checking the Solution:** 1. **First Cut:** \[ \frac{1}{3} \times 61.5 = 20.5 \text{ inches} \] Remaining fabric: \[ 61.5 - 20.5 = 41 \text{ inches} \] 2. **Second and Third Cuts:** \[ 10 + 5 = 15 \text{ inches} \] Remaining fabric after these cuts: \[ 41 - 15 = 26 \text{ inches} \] 3. **Total Fabric Cut:** \[ 20.5 + 15 = 35.5 \text{ inches} \] This matches the total fabric cut as given in the problem. **Conclusion:** The original piece of fabric was **61.5 inches** long.