You inherit one million dollars. You invest it all in three accounts for one year. The first account pays \( 4 \% \) compounded annually, the second account pays \( 5 \% \) compounded annually, and the third account pays \( 3 \% \) compounded annually. After one year, you earn \( \$ 44,000 \) in interest. If you invest four times the money into the account that pays \( 4 \% \) compared to \( 3 \% \), how much did you invest in each account? Investment in the \( 4 \% \) account is \( \$ \square \) Investment in the \( 5 \% \) account is \( \$ \square \) Investment in the \( 3 \% \) account is \( \$ \square \)

Let - \( x \) be the amount invested at \( 4\% \), - \( y \) be the amount invested at \( 5\% \), - \( z \) be the amount invested at \( 3\% \). We are given three conditions: 1. The total investment is one million dollars: \[ x + y + z = 1,\!000,\!000. \] 2. The total interest earned after one year is \$44,000: \[ 0.04x + 0.05y + 0.03z = 44,\!000. \] 3. The investment in the \( 4\% \) account is four times that in the \( 3\% \) account: \[ x = 4z. \] **Step 1. Substitute \( x = 4z \) into the total investment equation** \[ 4z + y + z = 1,\!000,\!000 \quad \Longrightarrow \quad 5z + y = 1,\!000,\!000. \] Thus, \[ y = 1,\!000,\!000 - 5z. \] **Step 2. Substitute \( x = 4z \) into the interest equation** \[ 0.04(4z) + 0.05y + 0.03z = 44,\!000. \] Simplify: \[ 0.16z + 0.05y + 0.03z = 44,\!000 \quad \Longrightarrow \quad 0.19z + 0.05y = 44,\!000. \] **Step 3. Substitute \( y = 1,\!000,\!000 - 5z \) into the interest equation** \[ 0.19z + 0.05(1,\!000,\!000 - 5z) = 44,\!000. \] Distribute \( 0.05 \): \[ 0.19z + 50,\!000 - 0.25z = 44,\!000. \] Combine like terms: \[ -0.06z + 50,\!000 = 44,\!000. \] **Step 4. Solve for \( z \)** Subtract \( 50,\!000 \) from both sides: \[ -0.06z = 44,\!000 - 50,\!000 = -6,\!000. \] Divide by \(-0.06\): \[ z = \frac{-6,\!000}{-0.06} = 100,\!000. \] **Step 5. Calculate \( x \) and \( y \) using \( z \)** Since \( x = 4z \): \[ x = 4(100,\!000) = 400,\!000. \] And since \( y = 1,\!000,\!000 - 5z \): \[ y = 1,\!000,\!000 - 5(100,\!000) = 1,\!000,\!000 - 500,\!000 = 500,\!000. \] **Final Answer** - Investment in the \( 4\% \) account is \( \$400,\!000 \). - Investment in the \( 5\% \) account is \( \$500,\!000 \). - Investment in the \( 3\% \) account is \( \$100,\!000 \).