st andrews golf course in st andrews scotland is one of the oldest courses in the world. it is an 18-hole course that consists of par-3 holes, par-4 holes and par-5 holes. a golfer who shoots par at the old course at st andrews has a total of 72 strokes for the entire course. there are seven times as many par-4 holes as par- 5 holes,and the sum of the numbers of par-3 and par-5 holes is four.find the numbers of par-3, par-4 and par-5 holes in the course.
Let's set up some variables to keep track of the various items of interest.
a = # par-3 holes
b = # par-4 holes
c = # par-5 holes
We are told:
a+b+c=18 holes
b = 7c
a+c = 4
3a+4b+5c = 72 strokes needed to finish course
3(4-c) + 4(7c) + 5c = 72
12-3c + 28c + 5c = 72
30c = 60
c=2
b=14
a=2
And we didn't even need the fact that a+b+c=18, but it works out just fine.
To solve this problem, we can set up a system of equations based on the information given.
Let's assume there are x par-3 holes, y par-4 holes, and z par-5 holes in the course.
1. We know that the sum of the numbers of par-3 and par-5 holes is four:
x + z = 4
2. We also know that there are seven times as many par-4 holes as par-5 holes:
y = 7z
3. Furthermore, we know that the total number of strokes for the entire course is 72:
3x + 4y + 5z = 72
Now, we can solve this system of equations to find the values of x, y, and z.
Let's start by solving equations 1 and 2 simultaneously:
x + z = 4 ---(equation 1)
y = 7z ---(equation 2)
We can substitute the value of y from equation 2 into equation 3 to eliminate y:
3x + 4(7z) + 5z = 72
3x + 28z + 5z = 72
3x + 33z = 72 ---(equation 4)
We can now solve equations 1 and 4 simultaneously:
x + z = 4 ---(equation 1)
3x + 33z = 72 ---(equation 4)
To eliminate x from the equations, let's multiply equation 1 by 3:
3(x + z) = 3(4)
3x + 3z = 12 ---(equation 5)
Now, subtract equation 5 from equation 4:
(3x + 33z) - (3x + 3z) = 72 - 12
33z - 3z = 60
30z = 60
z = 2
Substituting z = 2 into equation 1, we can find x:
x + 2 = 4
x = 4 - 2
x = 2
Finally, using the obtained values of x and z, we can find y using equation 2:
y = 7z
y = 7(2)
y = 14
Therefore, the numbers of par-3, par-4, and par-5 holes in the course are:
Par-3: 2 holes
Par-4: 14 holes
Par-5: 2 holes