Lina Chen invested $24,000, part at 8% and the rest at 7.2%. How much did she invest at each rate if her income from the 8% investment is two thirds that of the 7.2% investment?
To solve this problem, let's break it down step by step.
Let's assume that Lina Chen invested x dollars at 8% and the remaining (24000 - x) dollars at 7.2%.
According to the problem, the income from the 8% investment is two-thirds (2/3) that of the 7.2% investment.
To find the income from the 8% investment, we multiply the investment amount (x dollars) by the interest rate (8%) and divide it by 100:
Income from the 8% investment = (x * 8) / 100
To find the income from the 7.2% investment, we multiply the investment amount (24000 - x dollars) by the interest rate (7.2%) and divide it by 100:
Income from the 7.2% investment = ((24000 - x) * 7.2) / 100
According to the problem, the income from the 8% investment is two-thirds that of the 7.2% investment:
(x * 8) / 100 = (2/3) * ((24000 - x) * 7.2) / 100
Now, let's solve this equation to find the value of x.
Multiply both sides by 100 to remove the denominators:
8x = (2/3) * 7.2 * (24000 - x)
Simplify:
8x = (2/3) * (7.2 * 24000 - 7.2x)
Distribute (2/3) * (7.2 * 24000 - 7.2x):
8x = (2/3) * (172800 - 7.2x)
Multiply both sides by 3 to eliminate the fraction:
24x = 2 * (172800 - 7.2x)
Simplify:
24x = 345600 - 14.4x
Add 14.4x to both sides:
24x + 14.4x = 345600
Combine like terms:
38.4x = 345600
Divide both sides by 38.4 to solve for x:
x = 345600 / 38.4
Calculate:
x = 9000
Therefore, Lina Chen invested $9000 at 8% and the remaining ($24000 - $9000 = $15000) at 7.2%.