Question

Mrs. Kam plans to buy a house with $4,000,000 as an investment. She expects that the price of this house will have increased by r% one year later and increased by (r + 5)% after another year such that the price of the house will be $5,520,000 in two years' time. Show that (100 + r)(105 + r) = 13,800 and hence find the value of r.

Answer 1

To find the rate of increase (r) in the value of Mrs. Kam's investment, you need to express the value of her property after two years using the compound interest formula and show that (100 + r)(105 + r) = 13,800. Then, solve the quadratic equation to find the value of r.

Step-by-step explanation:

Mrs. Kam's investment problem can be solved by setting up an equation based on the compound interest formula. The question asks us to show that (100 + r)(105 + r) = 13,800 which represents the percentage increase in value of the house over two years. Here's how you can solve it:

• At the end of the first year, the value increases by r%, so the house value becomes $4,000,000 × (1 + r/100).
• After the second year, the value increases by (r + 5)%, so the new value becomes $4,000,000 × (1 + r/100) × (1 + (r + 5)/100).
• According to the information given, this value is $5,520,000. So we can write the equation $4,000,000 × (1 + r/100) × (1 + (r + 5)/100) = $5,520,000.
• Dividing both sides of the equation by $4,000,000 gives (1 + r/100) × (1 + (r + 5)/100) = 5,520,000/4,000,000 = 1.38.
• Converting into percentage form, we have (100 + r)(105 + r) = 138 × 100 = 13,800, as required.
• To find the value of r, you would solve the quadratic equation (100 + r)(105 + r) = 13,800 through factoring, completing the square, or using the quadratic formula.

By solving the quadratic equation, you can determine the value of r, which is the rate of increase in the value of Mrs. Kam's property investment.