Mickey and Minnie shared a box of matches. If Mickey was given 10 fewer matches, Minnie would have 5 times as many matches as Mickey.
If Mickey was given 10 more matches, he would have 1/2 as many matches as Minnie.
What percentage of matches did Mickey receive?
let Minnie has = x matches
Mickey has = y matches
Now,
x = 5 (y - 10)
=> x = 5y - 50
=> x - 5y = - 50 — (1)
Now,
y + 10 = 1/2x
=> 2y + 20 = x => x - 2y = 20 — (2)
from (1) & (2)
-3y = -70 => y = 70/3
=> x = 2 * 70/3 + 20 = 210/3
Mickey = 70/3/210/3 * 100 = 25.92%
Let's assume Mickey initially had 'x' matches.
According to the first statement, if Mickey was given 10 fewer matches, Minnie would have 5 times as many matches as Mickey. Therefore, Minnie would have (x - 10) * 5 matches.
According to the second statement, if Mickey was given 10 more matches, he would have 1/2 as many matches as Minnie. Therefore, Mickey would have (x + 10) = 1/2 * (x - 10) * 5.
Let's solve this equation step-by-step to find the value of 'x'.
Expanding the equation:
2(x + 10) = 5(x - 10)
Simplifying the equation:
2x + 20 = 5x - 50
Bringing like terms to one side:
3x = 70
Dividing both sides by 3:
x = 70/3
Now that we know the value of 'x', we can calculate the percentage of matches Mickey received. So, the percentage of matches Mickey received is given by:
[(x + 10) / (x + (x - 10) * 5)] * 100
Substituting the value of 'x' we just calculated:
[(70/3 + 10) / (70/3 + (70/3 - 10) * 5)] * 100
Simplifying the equation:
[(70/3 + 10) / (70/3 + 20/3 * 5)] * 100
Calculating:
[(70/3 + 10) / (70/3 + 100/3)] * 100
Simplifying further:
[(70/3 + 10) / (70/3 + 100/3)] * 100 = (100/3) * 100/ (170/3)
Calculating:
= (100/3) * (3/170) * 100
Simplifying and calculating:
≈ 58.8235294117647
Therefore, Mickey received approximately 58.82% of the matches.
To solve this problem, let's assign variables to represent the numbers of matches that Mickey and Minnie initially received.
Let's say Mickey initially received M matches, and Minnie initially received N matches.
From the given information, we can extract two equations:
1) If Mickey was given 10 fewer matches, Minnie would have 5 times as many matches as Mickey:
N + 10 = 5(M - 10)
2) If Mickey was given 10 more matches, he would have 1/2 as many matches as Minnie:
M + 10 = 1/2(N + 10)
Let's solve this system of equations to find the values of M and N.
1) N + 10 = 5(M - 10)
N + 10 = 5M - 50 (distribute 5)
N = 5M - 60 (move 10 to the other side of the equation)
2) M + 10 = 1/2(N + 10)
2(M + 10) = N + 10 (multiply by 2 to eliminate the fraction)
2M + 20 = N + 10 (distribute 2)
N = 2M + 10 (move 20 to the other side of the equation)
Now we have two expressions for N:
N = 5M - 60 ...(1)
N = 2M + 10 ...(2)
We can equate these two expressions:
5M - 60 = 2M + 10
3M = 70 (move 2M to the right and 10 to the left)
M = 70/3
So, the number of matches that Mickey initially received, M, is approximately 23.333 (rounded to three decimal places).
Now, to find the percentage of matches that Mickey received, we need to calculate M as a percentage of the total matches initially shared.
Let's say the total matches initially shared is T.
M / T * 100 = percentage of matches Mickey received
To find T, we can add the numbers of matches that Mickey and Minnie initially received:
T = M + N
= M + (2M + 10)
= 3M + 10
Substituting the value of M we found earlier:
T = 3 * (70/3) + 10
= 70 + 10
= 80
So, the total number of matches initially shared, T, is 80.
Now, we can calculate the percentage of matches that Mickey received:
M / T * 100 = 23.333 / 80 * 100
≈ 29.1667
Therefore, Mickey received approximately 29.1667% of the matches initially shared.