To determine the total length of the wire Rohini initially had, we can follow a step-by-step process, keeping in mind the wire usage and the remaining wire.
1. Let the total length of the wire be denoted as [tex]\( x \)[/tex].
2. Rohini used [tex]\(\frac{1}{2}\)[/tex] of the wire in wiring the room.
- This means, she has used [tex]\(\frac{1}{2} x\)[/tex] of the total wire.
- Thus, the remaining wire after wiring the room is:
[tex]\[
x - \frac{1}{2} x = \frac{1}{2} x
\][/tex]
3. She gave [tex]\(\frac{1}{3}\)[/tex] of the remaining wire to her friend.
- The remaining wire after wiring the room is [tex]\(\frac{1}{2} x\)[/tex].
- The wire given to her friend is:
[tex]\[
\frac{1}{3} \times \frac{1}{2} x = \frac{1}{6} x
\][/tex]
4. Wire left with Rohini after giving some to her friend:
- The wire left with her after wiring the room is [tex]\(\frac{1}{2} x\)[/tex].
- After giving [tex]\(\frac{1}{6} x\)[/tex] to her friend, the remaining wire with her is:
[tex]\[
\frac{1}{2} x - \frac{1}{6} x
\][/tex]
5. Simplify this to find the remaining wire:
- Finding a common denominator for [tex]\(\frac{1}{2} x\)[/tex] and [tex]\(\frac{1}{6} x\)[/tex]:
[tex]\[
\frac{3}{6} x - \frac{1}{6} x = \frac{2}{6} x = \frac{1}{3} x
\][/tex]
6. According to the problem, 30 meters of wire remained with her:
- So, we set up the equation:
[tex]\[
\frac{1}{3} x = 30
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
- To isolate [tex]\( x \)[/tex], multiply both sides of the equation by 3:
[tex]\[
x = 30 \times 3
\][/tex]
[tex]\[
x = 90
\][/tex]
8. Conclusion
- Therefore, the total length of the wire Rohini initially had is 90 meters.
