Questions: Example 4 Abdulla invests some money into an account for 3 years and receives 612 QR in interest. Saeed invests 700 QR more than Abdulla at the same interest rate for 2 years and gets 492 QAR in interest. How much did Abdulla invest in the account? NOTE: I=PRT, rearrange the formula to solve for R. Make the rates equal to each other. P= initial amount invested, I= interest amount, and T= time in years

Solution Steps

To solve this problem, we need to use the formula for simple interest, \( I = P \times R \times T \), where \( I \) is the interest earned, \( P \) is the principal amount invested, \( R \) is the rate of interest, and \( T \) is the time in years. We have two scenarios: one for Abdulla and one for Saeed. We can set up two equations based on the given information and solve for the unknowns. First, express the interest rate \( R \) in terms of the other variables for both Abdulla and Saeed. Then, set the two expressions for \( R \) equal to each other and solve for the principal amount \( P \) that Abdulla invested.

We start with the simple interest formula \( I = P \times R \times T \). For Abdulla, we have: \[ I_{\text{Abdulla}} = 612 \quad \text{and} \quad T_{\text{Abdulla}} = 3 \] Thus, the equation becomes: \[ 612 = P_{\text{Abdulla}} \times R \times 3 \] For Saeed, who invests \( P_{\text{Abdulla}} + 700 \) for 2 years and earns 492 QR in interest, we have: \[ I_{\text{Saeed}} = 492 \quad \text{and} \quad T_{\text{Saeed}} = 2 \] This gives us the equation: \[ 492 = (P_{\text{Abdulla}} + 700) \times R \times 2 \]

From Abdulla's equation, we can express \( R \): \[ R = \frac{612}{3 P_{\text{Abdulla}}} = \frac{204}{P_{\text{Abdulla}}} \] From Saeed's equation, we express \( R \) as well: \[ R = \frac{492}{2(P_{\text{Abdulla}} + 700)} = \frac{246}{P_{\text{Abdulla}} + 700} \]

Since both expressions represent the same interest rate \( R \), we set them equal: \[ \frac{204}{P_{\text{Abdulla}}} = \frac{246}{P_{\text{Abdulla}} + 700} \]

Cross-multiplying gives: \[ 204(P_{\text{Abdulla}} + 700) = 246 P_{\text{Abdulla}} \] Expanding and rearranging leads to: \[ 204 P_{\text{Abdulla}} + 142800 = 246 P_{\text{Abdulla}} \] \[ 142800 = 246 P_{\text{Abdulla}} - 204 P_{\text{Abdulla}} \] \[ 142800 = 42 P_{\text{Abdulla}} \] Thus, we find: \[ P_{\text{Abdulla}} = \frac{142800}{42} = 3400 \]

Final Answer