On an 18-hole golf course, there are par-3 holes, par-4 holes and par-5 holes. A golfer shoots a par of 70 for the 18 holes. There are twice as many par-4 holes as par-5 holes. How many of each type of hole are there on the golf course?
number of par 5 ---- x
number of par 4 ---- 2x
number of par 3 ---- 18-x-2x = 18-3x
solve
5x + 4(2x) + 3(18-3x) = 70
..
.
x = 4
par 5 --- 4
par 4 --- 8
par 3 --- 6
To solve this problem, we can use the information given to set up a system of equations.
Let's assume there are x par-3 holes on the golf course. Since par-3 holes have a par score of 3, these holes contribute 3x to the total par score.
Similarly, let's assume there are y par-4 holes and z par-5 holes on the golf course. Since par-4 holes have a par score of 4 and par-5 holes have a par score of 5, these holes contribute 4y and 5z to the total par score, respectively.
According to the given information, the total par score for the 18 holes is 70. Therefore, we have the equation:
3x + 4y + 5z = 70
The problem also states that there are twice as many par-4 holes as par-5 holes. So we can write another equation:
y = 2z
Now we have a system of two equations:
3x + 4y + 5z = 70 (Equation 1)
y = 2z (Equation 2)
We can use these equations to solve for the values of x, y, and z, which represent the number of par-3 holes, par-4 holes, and par-5 holes on the golf course, respectively.
To solve the system, we can substitute Equation 2 into Equation 1:
3x + 4(2z) + 5z = 70
Simplifying, we get:
3x + 8z + 5z = 70
3x + 13z = 70
Now we can solve for x and z in terms of y. Since Equation 2 states that y = 2z, we have:
x = (70 - 13z) / 3
y = 2z
Now we can substitute these expressions for x and y into Equation 2 to solve for z:
(70 - 13z) / 3 = 2z
Multiply both sides of the equation by 3:
70 - 13z = 6z
Combine like terms:
19z = 70
Divide both sides by 19:
z = 70 / 19
Since we can't have a fraction of a hole, we need to find multiples of z that give us whole numbers. By attempting different values of z, we find that z = 4 satisfies the equation.
So, z = 4, and using Equation 2, we can substitute z to find y:
y = 2z = 2(4) = 8
Finally, we can use Equation 1 to find x:
3x + 4y + 5z = 70
3x + 4(8) + 5(4) = 70
3x + 32 + 20 = 70
3x = 70 - 32 - 20
3x = 18
x = 18 / 3
x = 6
Therefore, there are 6 par-3 holes, 8 par-4 holes, and 4 par-5 holes on the golf course.