During her tennis career, a player won a total of 60 grand slam titles in three categories: women's singles, women's doubles, and mixed doubles. Out of this total, her number of wins in women's singles is eight more than her number of in wins in mixed doubles. Her number of wins in women's doubles is 2 more than 3 times her number of wins in mixed doubles. How many Grand Slam titles did the player win in each category?
Let a = no. of singles titles
Let b = no. of women's doubles titles
Let c = no. of mixed doubles titles
a + b + c = 60
a = c+8
b = 2 + 3c
c+8 + 2+3c + c = 60
5c + 10 = 60
5c=50
c=10
so,
a = 18
b = 32
so,
18 = no. of singles titles
32 = no. of women's doubles titles
10 = no. of mixed doubles titles
To solve this problem, let's denote the number of wins in mixed doubles as "x."
According to the problem, the player's number of wins in women's singles is eight more than her number of wins in mixed doubles. So, the number of wins in women's singles can be expressed as "x + 8."
Similarly, the player's number of wins in women's doubles is 2 more than 3 times her number of wins in mixed doubles. Therefore, the number of wins in women's doubles can be expressed as "3x + 2."
We know that the player won a total of 60 grand slam titles in all categories. So, we can write the equation:
(x) + (x + 8) + (3x + 2) = 60
Simplifying the equation, we get:
5x + 10 = 60
Subtracting 10 from both sides:
5x = 50
Dividing both sides by 5:
x = 10
Now that we have found the value of x, we can substitute this back into our expressions to find the number of wins in each category:
Number of wins in women's singles = x + 8 = 10 + 8 = 18
Number of wins in women's doubles = 3x + 2 = 3(10) + 2 = 30 + 2 = 32
Number of wins in mixed doubles = x = 10
Therefore, the player won 18 Grand Slam titles in women's singles, 32 Grand Slam titles in women's doubles, and 10 Grand Slam titles in mixed doubles.