Members of the West High School Band were hard at work practicing for the annual Homecoming Parade. First, they tried marching in rows of 12, but himself to bring up the rear left Andrew. Then the director told the band members to march in columns of eight, but Andrew was still left to march alone. Even when the band marched in rows of three, Andrew was left out. Finally, in exasperation, Andrew told the hand director that they should march in rows of five in order to have all the rows filled. He was right. Given that there were at least 45 musicians on the field but fewer than 200 musicians. How many students were there in the West High.
Clearly the number must be a multiple of 5, so it could be
45 50 55 60 65 ...145 ... 200
when divided by 12, the remainder is 1, so it could be
49 61 73 85 97 109 121 133 145 .... 193
when divided by 8, the remainder is 1, so it could be
49 57 65 73 81 89 97105 113 121 129 137 145 ...
ahhh, 145 is it!
Let's use a step-by-step approach to solve this problem.
Step 1: Start by assuming the total number of musicians in the West High School Band as 'x'.
Step 2: According to the information given, when marching in rows of 12, Andrew was left out. This means that the total number of musicians, 'x', is not divisible by 12. So, we can write this as an equation:
x is not divisible by 12 -> x % 12 ≠ 0
Step 3: Similarly, when the band members marched in columns of 8, Andrew was still left to march alone. So, the total number of musicians, 'x', is also not divisible by 8. We can write this as an equation:
x is not divisible by 8 -> x % 8 ≠ 0
Step 4: Again, when they marched in rows of 3, Andrew was left out. This means that the total number of musicians, 'x', is not divisible by 3. We can write this as an equation:
x is not divisible by 3 -> x % 3 ≠ 0
Step 5: Finally, when they marched in rows of 5, Andrew pointed out that all the rows would be filled. So, the total number of musicians, 'x', must be divisible by 5. We can write this as an equation:
x is divisible by 5 -> x % 5 = 0
Step 6: Now, we need to find a number that satisfies all these conditions, i.e., is not divisible by 12, 8, or 3 but is divisible by 5.
Step 7: Start by finding the first number that satisfies these conditions. Begin with a number larger than 200 (the maximum limit given) and check if it satisfies the conditions. Let's start with 201.
Step 8: Check if 201 is not divisible by 12, 8, or 3:
201 % 12 ≠ 0 -> 9 ≠ 0
201 % 8 ≠ 0 -> 1 ≠ 0
201 % 3 ≠ 0 -> 0 ≠ 0
Step 9: Check if 201 is divisible by 5:
201 % 5 = 1 ≠ 0
Step 10: Since 201 does not satisfy all the conditions, let's move on and check the next number, 202.
Step 11: Check if 202 is not divisible by 12, 8, or 3:
202 % 12 ≠ 0 -> 10 ≠ 0
202 % 8 ≠ 0 -> 2 ≠ 0
202 % 3 ≠ 0 -> 1 ≠ 0
Step 12: Check if 202 is divisible by 5:
202 % 5 ≠ 0 -> 2 ≠ 0
Step 13: Continue this process until you find a number that satisfies all the conditions.
Step 14: Let's skip ahead and try 215:
215 % 12 ≠ 0 -> 11 ≠ 0
215 % 8 ≠ 0 -> 7 ≠ 0
215 % 3 ≠ 0 -> 0 ≠ 0
215 % 5 = 0
Step 15: We have found a number, 215, that satisfies all the conditions.
Therefore, there are 215 students in the West High School Band.
To solve this problem, we need to find the number of musicians in the West High School Band. We are given that the band initially tried marching in rows of 12, then columns of 8, and finally rows of 3, but Andrew was left out each time.
Let's analyze the information step by step:
1. Andrew was left out when marching in rows of 12.
This means that the total number of musicians in the band is not a multiple of 12.
2. Andrew was left out when marching in columns of 8.
This indicates that the total number of musicians in the band is not a multiple of 8.
3. Andrew was left out when marching in rows of 3.
This suggests that the total number of musicians in the band is not a multiple of 3.
4. Andrew was included when marching in rows of 5.
This implies that the total number of musicians in the band is a multiple of 5.
Based on the given information, we can conclude that the total number of musicians in the band is both a multiple of 5 and not a multiple of 3, 8, or 12.
Now, let's find the possible values for the total number of musicians:
- Since the total number of musicians is between 45 and 200, we can start by finding the multiples of 5 within this range.
The multiples of 5 between 45 and 200 are: 50, 55, 60, 65, 70, 75, ... , 195.
- Now, we need to check which of these multiples meet the conditions of not being a multiple of 3, 8, or 12.
Starting from 50:
- 50 is divisible by 5 but also divisible by 10 (which means it is a multiple of 2 and 5), so it is not a valid answer.
Continue this process until you find a valid multiple.
Once you find a valid multiple, you will have the total number of musicians in the West High School Band, which is also the answer to the question.