Jack invests $1000 at a certain annual interest rate, and he invests another $2000 at an annual rate that is one-half percent higher. If he receives a total of $190 interest in 1 year, at what rate is the $1000 invested?
Let's assume the annual interest rate at which Jack invests the $1000 is "x".
According to the question, he invests $2000 at an annual rate that is one-half percent higher. Therefore, the interest rate for this investment is "x + 0.5".
To calculate the total interest Jack receives in one year, we can use the formula:
$1000 * x + $2000 * (x + 0.5) = $190
Simplifying this equation, we get:
1000x + 2000(x + 0.5) = 190
Expanding the brackets:
1000x + 2000x + 1000 = 190
Combining like terms:
3000x + 1000 = 190
Subtracting 1000 from both sides:
3000x = 90
Dividing both sides by 3000:
x = 0.03
Therefore, the $1000 is invested at a rate of 0.03, or 3% annually.
To solve this problem, let's break it down step by step:
Step 1: Determine the interest earned on the first investment.
Let's assume the interest rate on the first investment is x. The amount invested is $1000, and the interest earned can be calculated using the formula: Interest = Principal * Rate.
Therefore, the interest earned on the first investment is 1000 * (x/100).
Step 2: Determine the interest earned on the second investment.
The interest rate on the second investment is one-half percent (0.5%) higher than the rate on the first investment. So the interest rate on the second investment is (x + 0.5)%.
The amount invested is $2000, and the interest earned can be calculated using the same formula: Interest = Principal * Rate.
Therefore, the interest earned on the second investment is 2000 * ((x + 0.5)/100).
Step 3: Calculate the total interest earned.
We are given that the total interest earned is $190.
So, we can add the interest earned on the first and second investments to get the total interest:
Interest on first investment + Interest on second investment = $190
1000 * (x/100) + 2000 * ((x + 0.5)/100) = $190
Now, you can solve this equation to find the value of x, which represents the rate at which the $1000 investment was made.
1000x + 2000(x+.005) = 190
x = 6%