Neil needs to cut 3 long boards into 9 smaller boards. The first is 10 ft, the second is 16 ft, and the third is 18 ft. The table lists the smaller boards Neil needs. Use a drawing to show how he can divide the 3 boards so there is no waste.

Length of board Number needed
4 ft 3
5 ft 4
6 ft 2

3ft

I think that the picture if there is would help me understand what the problem is about

To divide the long boards into smaller boards without wasting any material, Neil can follow these steps:

Step 1: Start with the first board, which is 10 ft long.

- Cut it into three 4 ft long pieces for the boards that need to be 4 ft in length.
- Cut it into four 5 ft long pieces for the boards that need to be 5 ft in length.
- Cut it into two 6 ft long pieces for the boards that need to be 6 ft in length.

Now, Neil has used up the first board completely and obtained the required number of smaller boards.

Step 2: Move on to the second board, which is 16 ft long.

- Cut it into three 4 ft long pieces for the boards that need to be 4 ft in length.
- Cut it into four 5 ft long pieces for the boards that need to be 5 ft in length.

After completing these cuts, Neil will have some leftover material from the second board. Since there are no more requirements for 6 ft long boards, this leftover material will be wasted.

Step 3: Finally, Neil cuts the third board, which is 18 ft long.

- Cut it into three 4 ft long pieces for the boards that need to be 4 ft in length.
- Cut it into four 5 ft long pieces for the boards that need to be 5 ft in length.
- Cut it into two 6 ft long pieces for the boards that need to be 6 ft in length.

Like with the first board, Neil has used up the third board entirely and obtained the required number of smaller boards.

In summary, Neil can divide the three long boards as follows:
- First board (10 ft): 3 boards of 4 ft, 4 boards of 5 ft, and 2 boards of 6 ft.
- Second board (16 ft): 3 boards of 4 ft and 4 boards of 5 ft. (with some waste)
- Third board (18 ft): 3 boards of 4 ft, 4 boards of 5 ft, and 2 boards of 6 ft.

Note: A drawing of these cuts can be illustrated as a series of horizontal lines representing the long boards, along with vertical lines indicating the different cuts made to create the smaller boards.

To divide the long boards into the smaller boards without any waste, we need to find the common length that can be evenly divided from each long board to obtain the desired smaller boards.

Let's start by looking for the greatest common divisor (GCD) of the three long board lengths: 10 ft, 16 ft, and 18 ft.

The prime factorization of each long board length is as follows:
- 10 ft = 2^1 * 5^1
- 16 ft = 2^4
- 18 ft = 2^1 * 3^2

Now, we can find the GCD. We look for the common factors with the lowest exponent for each prime factor:
GCD = 2^1 = 2 ft.

So, the greatest common divisor of the three long boards is 2 ft. This means we can divide each long board into smaller boards of 2 ft length with no waste.

Now, let's visualize how Neil can divide the long boards:

Long Board 1 (10 ft):
- Divide into 5 smaller boards of 2 ft each (10 ft / 2 ft = 5)
- Distribute them as: 3 x 2 ft, 2 x 2 ft (as per the requirement)

Long Board 2 (16 ft):
- Divide into 8 smaller boards of 2 ft each (16 ft / 2 ft = 8)
- Distribute them as: 4 x 2 ft, 4 x 2 ft (as per the requirement)

Long Board 3 (18 ft):
- Divide into 9 smaller boards of 2 ft each (18 ft / 2 ft = 9)
- Distribute them as: 4 x 2 ft, 5 x 2 ft (as per the requirement)

Here is a visual representation:

Board 1 (10 ft): [2 ft] [2 ft] [2 ft] [2 ft] [2 ft]
Board 2 (16 ft): [2 ft] [2 ft] [2 ft] [2 ft] [2 ft] [2 ft] [2 ft] [2 ft]
Board 3 (18 ft): [2 ft] [2 ft] [2 ft] [2 ft] [2 ft] [2 ft] [2 ft] [2 ft] [2 ft]

By dividing the long boards into smaller boards with a length of 2 ft, Neil can obtain the required number of smaller boards without any waste.