A bag contains a total of 16 golf balls, 6 of which are white. A golfer wants to add 112 golf balls to the bag. He wants the ratio of white to striped golf balls to remain the same. How many of each should he add?(3 points)
Certainly! To maintain the 3:5 ratio of white to striped golf balls, divide the 112 additional balls into 8 equal parts (since 3+5=8). Each part consists of 112/8 = 14 balls. Multiply by the ratio: 3 parts of white balls is 3*14 = 42 and 5 parts of striped balls is 5*14 = 70. Thus, add 42 white balls and 70 striped balls.
Thank you so much for everything!! Btw, I'm sorry, I didn't mean to make you answer the question so many times like that. Thank you though!!
No problem at all! I'm here to help, and it's always good to explore different methods of solving a problem. Don't hesitate to ask if you have any more questions or need further assistance. Have a great day!
Currently, there are 6 white golf balls in the bag. This means there are 16 - 6 = 10 striped golf balls in the bag. We have a white to striped ratio of 6:10, which can be simplified to 3:5. This is the ratio we want to maintain when the golfer adds 112 new balls.
Let's call the number of white balls to be added x and the number of striped balls to be added y. We then have:
1. x/y = 3/5 (to maintain the same ratio)
2. x + y = 112 (because he's adding 112 balls in total)
From equation 1,
x = (3/5)y
We can substitute this into equation 2:
(3/5)y + y = 112
(3/5)y + (5/5)y = 112
(3y + 5y) / 5 = 112
8y / 5 = 112
8y = 112 * 5
8y = 560
y = 560 / 8
y = 70
Now that we know the number of striped balls to add is 70, we can find the number of white balls to add using equation 1:
x = (3/5)y
x = (3/5) * 70
x = 210/5
x = 42
Therefore, the golfer should add 42 white balls and 70 striped balls to maintain the same ratio of white to striped balls.
can you show me another way to solve this
Certainly! We can approach this problem by using proportions. Since we want the ratio of white to striped golf balls to remain the same, we will keep the ratio of 6:10 consistent as we add golf balls.
The current ratio of white to striped is 6 white balls to 10 striped balls. So for every 6 white balls, there are 10 striped balls. When we add more balls, we want to maintain this 6:10 ratio, which is equivalent to a 3:5 ratio.
Let's use the variables \( x \) for the number of white balls and \( y \) for the number of striped balls he needs to add. We want to maintain the ratio of \( \frac{3}{5} = \frac{x}{y} \). We also know that \( x + y = 112 \) because he wants to add a total of 112 balls.
Express the ratio of added white balls to striped balls:
\[ \frac{x}{y} = \frac{3}{5} \]
Multiply both sides to move out of the fraction and solve for one of the variables:
\[ 5x = 3y \]
\[ y = \frac{5}{3}x \]
Now let's use the total number of additional balls to find the actual numbers:
\[ x + y = 112 \]
Substitute \( y \) with \( \frac{5}{3}x \) in the above equation:
\[ x + \frac{5}{3}x = 112 \]
Combine like terms by finding a common denominator:
\[ \frac{3}{3}x + \frac{5}{3}x = \frac{8}{3}x = 112 \]
Now, to solve for \( x \):
\[ \frac{8}{3}x = 112 \]
\[ x = \frac{112 \cdot 3}{8} \]
\[ x = 14 \cdot 3 \]
\[ x = 42 \]
Now that we have \( x \), the number of white balls to add, we can substitute back into the equation \( y = \frac{5}{3}x \) to find \( y \), the number of striped balls to add:
\[ y = \frac{5}{3} \cdot 42 \]
\[ y = 5 \cdot 14 \]
\[ y = 70 \]
So the golfer should add 42 white balls and 70 striped balls to the bag to maintain the same ratio of white to striped golf balls.
what is the simplest way to solve this?
The simplest way to solve this problem is to recognize that the ratio of white to striped golf balls needs to stay the same, and use that ratio directly to divide the total number of golf balls to be added.
Let's break it down step by step:
1. We know there are 6 white and 10 striped balls, which gives us a simplified ratio of 3:5 (since 6/2 = 3 and 10/2 = 5).
2. The golfer wants to add 112 golf balls in total. We need to split these 112 golf balls in the same 3:5 ratio.
3. To determine how many parts the total number of balls is being split into according to the ratio, we add the numbers of the ratio together: 3 parts (white) + 5 parts (striped) = 8 parts in total.
4. Each part, therefore, consists of the total number of new balls divided by the total number of parts: 112 balls / 8 parts = 14 balls per part.
5. Now we just need to multiply the number of balls per part by the number of parts for white and striped:
- White balls: 3 parts * 14 balls per part = 42 white balls
- Striped balls: 5 parts * 14 balls per part = 70 striped balls
So, the golfer should add 42 white balls and 70 striped balls to the bag. This maintains the ratio of 3:5 (white:striped) and adds up to the total number of 112 balls he wanted to add.