Mr. Perez invested half of his money at 7% per year, and half at 8% per year. His annual income from the two investments was 7,500 PHP. How much did he invest at each rate?
Let's call the amount of money Mr. Perez invested at 7% per year x.
So, the amount he invested at 8% per year would be (1/2)x.
The income from his investment at 7% per year is 7/100 * x = 7x/100.
The income from his investment at 8% per year is 8/100 * (1/2)x = 8x/200 = x/25.
The total income from both investments is 7x/100 + x/25 = 7500.
Multiplying both sides by 100 to get rid of the denominators, we get 7x + 4x = 750000.
Combining like terms, we have 11x = 750000.
Dividing both sides by 11, we get x = 750000/11 ≈ 68,181.82 .
So, Mr. Perez invested 68,181.82 PHP at 7% per year and 1/2 * 68,181.82 = 34090.91 PHP at 8% per year. Answer: \boxed{68,181.82, 34090.91}.
To determine how much Mr. Perez invested at each rate, we can set up a system of equations based on the given information. Let's denote the amount invested at 7% as x PHP and the amount invested at 8% as y PHP.
According to the problem, Mr. Perez invested half of his money at 7% per year and the other half at 8% per year. Therefore, we have the equation:
x + y = total amount invested
Since he invested half of his money at each rate, we can also express this as:
x = 0.5 * total amount invested
y = 0.5 * total amount invested
Now, let's consider the annual income from these investments. The income from an investment is calculated by multiplying the principal (amount invested) by the interest rate. We can set up the following equation for the total income:
0.07x + 0.08y = 7,500 PHP
Substituting the expressions for x and y from earlier, we have:
0.07 * (0.5 * total amount invested) + 0.08 * (0.5 * total amount invested) = 7,500 PHP
Simplifying this equation gives us:
0.035 * total amount invested + 0.04 * total amount invested = 7,500 PHP
Combining like terms:
0.075 * total amount invested = 7,500 PHP
Now we can solve for the total amount invested:
total amount invested = 7,500 PHP / 0.075
total amount invested = 100,000 PHP
Since Mr. Perez divided his money equally between the two investments, we can now calculate how much he invested at each rate:
x = 0.5 * total amount invested
x = 0.5 * 100,000 PHP
x = 50,000 PHP
y = 0.5 * total amount invested
y = 0.5 * 100,000 PHP
y = 50,000 PHP
Therefore, Mr. Perez invested 50,000 PHP at a 7% interest rate and 50,000 PHP at an 8% interest rate.
Apologies, there seems to have been a mistake in my previous response. Let's solve it correctly:
Let's denote the amount invested at 7% as x PHP and the amount invested at 8% as y PHP.
We know that the total income from the two investments is 7,500 PHP. Therefore, we can set up the equation:
0.07x + 0.08y = 7,500
We also know that Mr. Perez invested half of his money at each rate, so we can set up another equation:
x + y = total amount invested
Since he invested half of his money at each rate, we can express this as:
x = 0.5 * total amount invested
y = 0.5 * total amount invested
To solve this system of equations, we can substitute the expressions for x and y into the first equation:
0.07 * (0.5 * total amount invested) + 0.08 * (0.5 * total amount invested) = 7,500
Simplifying this equation gives us:
0.035 * total amount invested + 0.04 * total amount invested = 7,500
Combining like terms:
0.075 * total amount invested = 7,500
Now we can solve for the total amount invested:
total amount invested = 7,500 / 0.075
total amount invested = 100,000
Since Mr. Perez divided his money equally between the two investments, we can now calculate how much he invested at each rate:
x = 0.5 * total amount invested
x = 0.5 * 100,000
x = 50,000
y = 0.5 * total amount invested
y = 0.5 * 100,000
y = 50,000
Therefore, Mr. Perez invested 50,000 PHP at a 7% interest rate and 50,000 PHP at an 8% interest rate.
Let's assume Mr. Perez had an initial amount of money, which we'll call 'x' PHP.
He invested half of this amount, which is (1/2)x PHP, at 7% per year.
Therefore, the income generated from this investment is ((1/2)x)(0.07) PHP per year.
He also invested the other half, which is also (1/2)x PHP, at 8% per year.
Therefore, the income generated from this investment is ((1/2)x)(0.08) PHP per year.
The total income from both investments is 7,500 PHP.
So, we can write the equation to solve for 'x' as follows:
((1/2)x)(0.07) + ((1/2)x)(0.08) = 7,500
To simplify the equation, we can combine like terms:
0.035x + 0.04x = 7,500
Adding the terms on the left side, we get:
0.075x = 7,500
To solve for 'x', we divide both sides of the equation by 0.075:
x = 7,500 / 0.075
Calculating this expression, we find:
x = 100,000
Therefore, Mr. Perez had 100,000 PHP to invest in total.
He invested half of this, which is (1/2)(100,000) = 50,000 PHP, at 7% per year.
And he invested the other half, which is (1/2)(100,000) = 50,000 PHP, at 8% per year.
So, Mr. Perez invested 50,000 PHP at each interest rate.