Math Problem Statement

To complete the table, we need to calculate the interest and future value for each year of the investment. The formula for the future value ( A ) of an investment with principal ( P ), annual interest rate ( r ), and time ( t ) in years is given by:

[ A = P(1 + r)^t ]

The interest earned ( I ) is the difference between the future value and the principal:

[ I = A - P ]

Let's break it down step by step.

Year 1:

  1. Principal amount (P): P800,000
  2. Interest rate (r): 7.5% = 0.075
  3. Time (t): 1 year

Using the future value formula:

[ A_1 = 800,000(1 + 0.075)^1 = 800,000 \times 1.075 = 860,000 ]

The interest earned in the first year is:

[ I_1 = A_1 - P = 860,000 - 800,000 = 60,000 ]

So, the completed row for Year 1 is:

[ \begin{array}{|c|c|l|l|c|l|} \hline \text{Time} & \text{Principal amount} & \text{Interest rate} & \text{Solution} & \text{Interest} & \text{Future value} \ \hline 1 & P800,000 & 7.5% & (P800,000)(1 + 0.075) & P60,000 & P860,000 \ \hline \end{array} ]

Year 2:

  1. Principal amount (P): P860,000 (the future value from Year 1)
  2. Interest rate (r): 7.5% = 0.075
  3. Time (t): 1 year

Using the future value formula:

[ A_2 = 860,000(1 + 0.075)^1 = 860,000 \times 1.075 = 924,500 ]

The interest earned in the second year is:

[ I_2 = A_2 - P = 924,500 - 860,000 = 64,500 ]

So, the completed row for Year 2 is:

[ \begin{array}{|c|c|l|l|c|l|} \hline \text{Time} & \text{Principal amount} & \text{Interest rate} & \text{Solution} & \text{Interest} & \text{Future value} \ \hline 2 & P860,000 & 7.5% & (P860,000)(1 + 0.075) & P64,500 & P924,500 \ \hline \end{array} ]

Combining both years, the completed table is:

[ \begin{array}{|c|c|l|l|c|l|} \hline \text{Time} & \text{Principal amount} & \text{Interest rate} & \text{Solution} & \text{Interest} & \text{Future value} \ \hline 1 & P800,000 & 7.5% & (P800,000)(1 + 0.075) & P60,000 & P860,000 \ \hline 2 & P860,000 & 7.5% & (P860,000)(1 + 0.075) & P64,500 & P924,500 \ \hline \end{array} ]

The final answer is:

[ \boxed{P924,500} ]

Solution

Great! Let's go over the calculations step by step and complete the table as needed.

Year 1:

  • Principal amount (P): 800,000
  • Interest rate (r): 7.5% or 0.075
  • Time (t): 1 year

Using the formula for future value:

A1=800,000(1+0.075)1=800,000×1.075=860,000A_1 = 800,000(1 + 0.075)^1 = 800,000 \times 1.075 = 860,000

The interest earned in Year 1:

I1=A1P=860,000800,000=60,000I_1 = A_1 - P = 860,000 - 800,000 = 60,000

Year 2:

  • Principal amount (P): 860,000 (the future value from Year 1)
  • Interest rate (r): 7.5% or 0.075
  • Time (t): 1 year

Using the same formula:

A2=860,000(1+0.075)1=860,000×1.075=924,500A_2 = 860,000(1 + 0.075)^1 = 860,000 \times 1.075 = 924,500

The interest earned in Year 2:

I2=A2P=924,500860,000=64,500I_2 = A_2 - P = 924,500 - 860,000 = 64,500

Completed Table:

\hline \text{Time} & \text{Principal amount} & \text{Interest rate} & \text{Solution} & \text{Interest} & \text{Future value} \\ \hline 1 & P800,000 & 7.5\% & (P800,000)(1 + 0.075) & P60,000 & P860,000 \\ \hline 2 & P860,000 & 7.5\% & (P860,000)(1 + 0.075) & P64,500 & P924,500 \\ \hline \end{array}$$ ### Final Answer: $$\boxed{P924,500}$$ Would you like any further clarifications or more details on this process? --- Here are 5 related questions that expand upon this: 1. How does changing the interest rate affect the future value over multiple years? 2. What would happen if the interest were compounded more frequently (e.g., quarterly)? 3. How would the results change if the investment had a different starting principal? 4. Can you calculate the total interest earned over multiple years? 5. How would you compute the future value for a different number of years? **Tip**: The key to compound interest is the exponential growth—it accumulates faster with higher rates and longer periods.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Compound Interest
Exponential Growth

Formulas

Future Value Formula: A = P(1 + r)^t
Interest Calculation: I = A - P

Theorems

-

Suitable Grade Level

Grades 9-12