Ken and Sam shared some marbles. The ratio of the number of marbles Ken has to the number of marbles Sam has is
5:3. If Ken's marbles increases by 15%, what percentage of Sam's marbles must be decreased so that the total number of marbles they have remained unchanged?
k = (5/3) s
1.15 (5/3)s = new number of kens marbles
increase in kens marbles = .15 (5/3) s
= decrease in sam's marbles = .25 s
(.25 s / s )100 = 25% decrease
To solve this problem, let's break it down into steps:
Step 1: Determine the initial ratio of Ken's marbles to Sam's marbles.
Given that the ratio of the number of marbles Ken has to the number of marbles Sam has is 5:3, we can assume that Ken has 5x marbles and Sam has 3x marbles, where x is a common factor.
Step 2: Calculate the percentage increase in Ken's marbles.
If Ken's marbles increase by 15%, we can calculate the increase as follows:
Increase = (15/100) * (5x)
Increase = (3/20)x
Step 3: Find the new number of marbles Ken has.
The new number of marbles Ken has after the increase can be calculated by adding the increase to the initial number of marbles:
New marbles for Ken = 5x + (3/20)x
New marbles for Ken = (107/20)x
Step 4: Determine the percentage decrease in Sam's marbles.
To keep the total number of marbles unchanged, we need to calculate the decrease in Sam's marbles.
Let the percentage decrease be d. We can express Sam's new marbles as a fraction of his initial marbles:
New marbles for Sam = (1 - d/100) * (3x)
Step 5: Set up the equation and solve for d.
Since the total number of marbles remains unchanged, we can equate Ken's and Sam's new marbles:
(107/20)x = (1 - d/100) * (3x)
Simplifying the equation:
107x/20 = (1 - d/100) * 3x
107x = (20/3) * (1 - d/100) * 3x
107 = (20/3) * (1 - d/100)
Cross-multiplying:
107 * (3/20) = 1 - d/100
321/20 = 1 - d/100
Rearranging the equation:
d/100 = 1 - 321/20
d/100 = 20/20 - 321/20
d/100 = (20 - 321)/20
d/100 = -301/20
Simplifying further:
d/100 = -301/20
d = (-301/20) * 100
d = -1505
Step 6: Determine the positive percentage decrease in Sam's marbles.
Since a negative percentage decrease does not make sense here, we take the positive value:
Positive percentage decrease = |d| = |-1505| = 1505%
Therefore, in order for the total number of marbles they have to remain unchanged, Sam's marbles must be decreased by approximately 1505%.