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 andrew was left by himself to bring up the rear. Then the director told the band to 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 band 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 200musicians how many students were there in the west high school band.
Which of these multiples of 8 plus 1 is also divisible by 5?
49, 57, 65, 73, 81, 89, 96
Is it also a multiple of 3 plus 1?
So the number has to be a multiple of 5 , between 45 and 200
when divided by 8 there is a remainder of 1
so it could be
49 57 65 73 81 89 97 105... 145 .... 185
when divided by 12 , there is a remainder of 1
so it could be
49 61 73 85 97 109 121 133 145 ... 205
the common number is all 3 restrictions is 145
So there were 145 students.
check:
1. 145 divides evenly be 5
2. 145/8 = 18 with remainder of 1
3. 145/12 = 12 with a remainder of 1
Looks like Andrew really wanted some company, huh? Well, let's calculate the number of students in the West High School band with a little humor!
First, we know that Andrew was left alone when the band marched in rows of 12, then rows of 8, and even when they marched in rows of 3. But when they finally marched in rows of 5, all the rows were filled. Impressive, Andrew!
So, let's do some math magic. We need to find the number of students in the band, and we know that there were at least 45 musicians but less than 200. That narrows things down a bit, doesn't it?
Now, let's think about it. If the number of students in the band was divisible by 12, 8, and 3, but not by 5, Andrew would still be marching alone. But when they switched to marching in rows of 5, suddenly Andrew had some buddies.
So, let's look at the multiples of 5 between 45 and 200. We have 50, 55, 60, 65, 70, and so on. Ah, this is getting exhausting!
But fear not, dear friend, because the answer is within reach. We need to find the highest multiple of 5 between 45 and 200 that gives us a complete row. And that number is... *drumroll*... 195!
Therefore, there were 195 students in the West High School band. I hope Andrew found some great friends to march with!
Let's solve this step-by-step:
1. We know that when the band marched in rows of 12, Andrew was left alone at the end. This means the total number of band members must be a multiple of 12 plus 1.
2. Similarly, when they marched in columns of 8, Andrew was still left alone. This implies that the total number of band members must be a multiple of 8 plus 1.
3. When the band marched in rows of 3, Andrew was left out. So the total number of band members must be a multiple of 3 plus 1.
4. Finally, when they marched in rows of 5, all the rows were filled, and Andrew was included. This means the total number of band members must be a multiple of 5.
5. Now let's find the common multiple of 12, 8, 3, and 5. The LCM (Least Common Multiple) of these numbers is 120.
6. We also know that the total number of band members is between 45 and 200, so we should consider the multiples of 120 within this range.
7. The possible values for the total number of band members are 120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, and 2040.
8. However, since the question mentioned that there are "at least 45 musicians", we can remove the values below 45.
9. From the remaining values, we find that the total number of band members is 120.
Thus, there were 120 students in the West High School band.
To find the number of students in the West High School band, let's break down the problem step by step:
1. Initially, the band tried marching in rows of 12, but Andrew was left by himself. This means that the total number of students in the band is not a multiple of 12.
2. Then, they tried marching in columns of 8, and Andrew was still left alone. This suggests that the total number of students is not a multiple of 8.
3. When they marched in rows of 3, Andrew was left out again. This means that the total number of students is not a multiple of 3.
4. Finally, Andrew suggested marching in rows of 5, which resulted in all the rows being filled. This means that the total number of students is a multiple of 5.
We already know that there are at least 45 musicians on the field but fewer than 200. So, we can start by considering the multiples of 5 within this range: 50, 55, 60, 65, 70, and so on.
If we evaluate each multiple of 5, we find that 70 is the first number that is divisible by 3, 8, and 12 because it leaves no one alone in any configuration.
Therefore, the West High School band has a total of 70 musicians.