Katie was chosen as a student leader. She got 3 votes for every 2 votes that Amanda got. If 8 people changed their minds and chose Amanda instead of Katie, Katie would only have gotten 1 vote for every two that Amanda got. How many people voted?
I want to know a simple way to solve this.
kat votes = k
Aman votes = (2/3) k
later
Aman gets (2/3) k + 8
Kat gets k - 8
and
(k-8)2 = (2/3) k + 8
or
3 (2 k - 16) = 2 k + 24
6 k - 48 = 2 k + 24
4 k = 72
k = 18
take it from there after checking my sloppy arithmetic.
To solve this problem, let's break it down step by step:
Step 1: Let's assume the number of votes Katie initially received as "3x" and the number of votes Amanda initially received as "2x". This is based on the information that Katie got 3 votes for every 2 votes that Amanda got.
Step 2: According to the problem, if 8 people changed their minds and voted for Amanda instead of Katie, Katie would only get 1 vote for every 2 votes that Amanda gets. So, the new number of votes for Katie would be "x" (reduced from 3x) and the new number of votes for Amanda would be "2x + 8".
Step 3: Now, let's equate the new number of votes for Katie and Amanda: "x = 2x + 8".
Step 4: Solving the equation: Move 2x to the left side, and 8 to the right side of the equation: "x - 2x = 8".
Simplifying the equation: "-x = 8".
Multiplying both sides by -1: "x = -8".
Step 5: Since the number of votes cannot be negative, it means we made a mistake somewhere. Let's revisit Step 1.
Step 1 (revisited): We assumed the number of votes for Katie initially as "3x" and the number of votes for Amanda initially as "2x". However, the problem doesn't provide any specific values for x or the total number of votes.
Step 6: With no specific values, we cannot directly determine the number of people who voted. We can only determine the ratio between the votes for Katie and Amanda (3:2).
To solve this problem, it's important to understand the given information and the relationship between Katie's and Amanda's votes.
Let's assign variables to represent the number of votes each person received. Let's say Katie received K votes, and Amanda received A votes.
According to the first statement, Katie received 3 votes for every 2 votes that Amanda received. So we can set up the equation: K = (3/2)A.
Now, let's consider the second statement. If 8 people changed their minds from Katie to Amanda, then Katie would have received 8 fewer votes, while Amanda would have received 8 more votes. This can be expressed as: K - 8 = (A + 8)/2.
We can now solve these two equations simultaneously to find the values of K and A. Let's substitute the value of K from the first equation into the second equation:
(3/2)A - 8 = (A + 8)/2.
Multiply both sides by 2 to eliminate the fraction:
3A - 16 = A + 8.
Subtract A from both sides:
2A - 16 = 8.
Add 16 to both sides:
2A = 24.
Divide both sides by 2:
A = 12.
Now, substitute the value of A back into the first equation to find K:
K = (3/2)(12) = 18.
So, Katie received 18 votes, and Amanda received 12 votes.
To find the total number of people who voted, you can simply add Katie's and Amanda's votes together:
Total number of people who voted = 18 + 12 = 30.
Therefore, a total of 30 people voted.