To solve the problem, we need to determine how many pieces of length [tex]\(1 \frac{1}{2} \, \text{m}\)[/tex] can be cut from a rope that is [tex]\(22 \frac{1}{2} \, \text{m}\)[/tex] long and find out if there is any remaining piece of the rope that couldn't be used to make a full piece of [tex]\( 1 \frac{1}{2} \, \text{m} \)[/tex].
1. Convert the mixed numbers to improper fractions or decimals:
- The rope length [tex]\(22 \frac{1}{2} \, \text{m}\)[/tex] can be converted to a decimal:
[tex]\[
22 \frac{1}{2} = 22 + 0.5 = 22.5 \, \text{m}
\][/tex]
- The piece length [tex]\(1 \frac{1}{2} \, \text{m}\)[/tex] can also be converted to a decimal:
[tex]\[
1 \frac{1}{2} = 1 + 0.5 = 1.5 \, \text{m}
\][/tex]
2. Determine how many full pieces can be cut from the rope:
Here, we divide the total rope length by the length of each piece:
[tex]\[
\text{Number of pieces} = \frac{22.5}{1.5}
\][/tex]
On calculating this:
[tex]\[
\frac{22.5}{1.5} = 15
\][/tex]
Therefore, we can cut [tex]\(15\)[/tex] pieces of [tex]\(1.5 \, \text{m}\)[/tex] from the [tex]\(22.5 \, \text{m}\)[/tex] long rope.
3. Calculate the length of the remaining piece:
We then find the remaining length of the rope by using the modulus operation, which gives the remainder of the division:
[tex]\[
\text{Remaining length} = 22.5 \, \text{m} \mod 1.5 \, \text{m}
\][/tex]
Since [tex]\(22.5\)[/tex] is exactly divisible by [tex]\(1.5\)[/tex]:
[tex]\[
22.5 \mod 1.5 = 0
\][/tex]
Hence, there is no remaining rope left after cutting [tex]\(15\)[/tex] pieces of [tex]\(1.5 \, \text{m}\)[/tex].
Conclusion:
We can cut [tex]\(15\)[/tex] pieces, each of [tex]\(1 \frac{1}{2} \, \text{m}\)[/tex], from a rope that is [tex]\(22 \frac{1}{2} \, \text{m}\)[/tex] long. There will be [tex]\(0 \, \text{m}\)[/tex] left over.
