Let's solve the problem step by step: ### **Given:** - Principal (\(P\)) = \$1578.00 - Interest (\(I\)) = \$51.26 - Annual Simple Interest Rate (\(R\)) = \(12\frac{3}{4}\%\) per annum = \(12.75\%\) per annum - Time (\(T\)) = ? (in days) ### **Step 1: Write the Simple Interest Formula** \[ I = P \times R \times T \] Where: - \(I\) = Interest - \(P\) = Principal - \(R\) = Rate per period (as a decimal) - \(T\) = Time (in years) ### **Step 2: Convert the Rate to a Decimal** \[ R = \frac{12.75}{100} = 0.1275 \] ### **Step 3: Express Time in Years** Let \(d\) be the number of days. Since there are 365 days in a year, \[ T = \frac{d}{365} \] ### **Step 4: Plug Values into the Formula and Solve for \(d\)** \[ 51.26 = 1578 \times 0.1275 \times \frac{d}{365} \] First, calculate \(1578 \times 0.1275\): \[ 1578 \times 0.1275 = 201.195 \] So, \[ 51.26 = 201.195 \times \frac{d}{365} \] \[ 51.26 = \frac{201.195d}{365} \] Multiply both sides by 365: \[ 51.26 \times 365 = 201.195d \] \[ 18709.9 = 201.195d \] \[ d = \frac{18709.9}{201.195} \approx 92.97 \] ### **Step 5: Round Up to the Nearest Day** \[ \boxed{93} \] --- ## **Final Answer** **The number of days required is 93 days.**