A bag contains 10 white gold balls and 6 striped gold balls. A golfer wants to add 112 golf balls to the bag. He wants the ratio of whit to striped golf balls to remain the same. How many of each should he add?
Let's say the golfer adds $w$ white golf balls and $s$ striped golf balls to the bag.
The current ratio of white golf balls to striped golf balls is $10/6 = 5/3$.
The ratio after adding $w$ white golf balls and $s$ striped golf balls should still be $5/3$.
So, we can set up the following equation based on the problem statement:
\[
\frac{10+w}{6+s} = \frac{5}{3}
\]
Let's cross multiply and simplify:
\[
3(10+w) = 5(6+s)
\]
\[
30 + 3w = 30 + 5s
\]
\[
3w = 5s
\]
Since we want to add a total of 112 golf balls, we also know that $w + s = 112$.
We have two equations with two variables:
\begin{align*}
3w &= 5s \\
w + s &= 112
\end{align*}
Multiplying the second equation by 3, we can substitute the value of $112 - s$ for $w$:
\begin{align*}
3w &= 5s \\
3(112-s) + s &= 336 \\
336 - 3s + s &= 336 \\
-2s &= 0 \\
\end{align*}
This gives us $s = 0$, which is not a valid solution for this problem. It means there are already enough white golf balls in the bag to maintain the ratio.
Therefore, the golfer should add 0 white golf balls and 112 striped golf balls.
To keep the ratio of white to striped golf balls the same, we need to determine the ratio of white to striped golf balls in the initial bag.
The initial ratio of white to striped golf balls is:
White balls: 10
Striped balls: 6
We need to find the common ratio between the white and striped balls in order to add the same ratio to the bag.
To find the common ratio, we divide the number of white balls by the number of striped balls:
Common ratio = 10 / 6 = 1.67
Now, we can determine how many white and striped golf balls the golfer should add to the bag.
Let's assume the golfer adds x white balls and y striped balls.
The ratio of white to striped balls after adding x and y balls is:
(10 + x) / (6 + y)
According to the problem statement, the ratio of white to striped balls should remain the same, so we have the equation:
(10 + x) / (6 + y) = 1.67
To solve for x and y, we need to find the values that satisfy this equation.
Multiplying both sides of the equation by (6 + y) gives:
10 + x = 1.67 * (6 + y)
Expanding the right side of the equation gives:
10 + x = 10.02 + 1.67y
Rearranging the equation gives:
x - 1.67y = 0.02
We also know that the golfer wants to add a total of 112 balls:
x + y = 112
Now we have a system of equations:
x - 1.67y = 0.02
x + y = 112
We can solve this system of equations using substitution or elimination. Let's use the elimination method.
Multiplying both sides of the second equation by 1.67 to make the coefficients of y in both equations equal, we have:
1.67x + 1.67y = 186.96
Adding the two equations gives:
(x - 1.67y) + (1.67x + 1.67y) = 0.02 + 186.96
Simplifying, we get:
3.34x = 187.98
Dividing both sides by 3.34, we have:
x = 56.31
Substituting x into the second equation, we get:
56.31 + y = 112
Solving for y, we have:
y = 55.69
Since we can't have fractions of golf balls, we need to round x and y to the nearest whole number.
Therefore, the golfer should add approximately 56 white golf balls and 56 striped golf balls to the bag in order to maintain the same ratio.