A town has 4900 registered voters, each of whom is either a Republican, a Democrat or an Independent. In the last primary election, 1150 people voted, and the number of Republicans who voted was 720 fewer than 30% of the total number of Democrats plus Independents combined. Also, 20% of the Republicans, 25% of the Democrats, and 25% of the Independents voted in the primary. How many Republicans, Democrats and Independents are there in the town?
A man has $210,000 invested in three properties. One earns 12%, one 10% and one 8%. His annual income from the properties is $20,100 and the amount invested at 8% is twice that invested at 12%. How much is invested in each property? at 12% 10% and 8%
What is the annual income from each property?
12 10 and 8%
To solve this problem, we need to set up a system of equations based on the information provided. Let's use the following variables:
Let R represent the number of Republicans.
Let D represent the number of Democrats.
Let I represent the number of Independents.
From the problem, we know the following information:
1) The total number of registered voters is 4900, so R + D + I = 4900. (Equation 1)
2) In the last primary election, 1150 people voted, so the total number of voters who voted is R + D + I = 1150. (Equation 2)
3) The number of Republicans who voted was 720 fewer than 30% of the total number of Democrats plus Independents combined. So, R = 0.3(D + I) - 720. (Equation 3)
4) The percentage of Republicans, Democrats, and Independents who voted in the primary was 20%, 25%, and 25%, respectively. So we can convert these percentages to actual numbers by multiplying them by the total number of each group. We get the following equations:
0.2R + 0.25D + 0.25I = 1150. (Equation 4)
Now we have a system of four equations with three variables. We can solve it using substitution or elimination. Let's use substitution.
From Equation 1, we have R = 4900 - D - I. (Equation 5)
Substituting Equation 5 into Equation 3, we get:
4900 - D - I = 0.3(D + I) - 720.
Simplifying, we have:
4900 - 720 = 0.3D + 0.3I - D - I.
Subtracting like terms, we get:
4180 = -0.7D - 0.7I.
Dividing both sides by -0.7, we have:
D + I = -4180 / -0.7.
Simplifying further:
D + I = 5971.43. (Equation 6)
Substituting Equation 5 into Equation 4, we get:
0.2(4900 - D - I) + 0.25D + 0.25I = 1150.
Simplifying, we have:
980 - 0.2D - 0.2I + 0.25D + 0.25I = 1150.
Combining like terms, we get:
0.05D + 0.05I = 170.
Dividing both sides by 0.05, we have:
D + I = 170 / 0.05.
Simplifying further:
D + I = 3400. (Equation 7)
Now, we have a system of two equations with two variables: Equation 6 and Equation 7.
Subtracting Equation 6 from Equation 7, we get:
D + I - (D + I) = 3400 - 5971.43.
Simplifying, we have:
0 = -2571.43.
This is not possible, so there must be an error in the information provided or in the calculations. Please double-check the given information and try again.