### 5. Calculating the Final Score of Gandhi House
To find the final score of Gandhi House in the five rounds, we simply sum up all the given scores:
Given scores:
[tex]\[ -10, +5, -4, -3, +15 \][/tex]
Let's add them step by step:
1. Start with -10
2. Add 5: [tex]\( -10 + 5 = -5 \)[/tex]
3. Add -4: [tex]\( -5 + (-4) = -9 \)[/tex]
4. Add -3: [tex]\( -9 + (-3) = -12 \)[/tex]
5. Add 15: [tex]\( -12 + 15 = 3 \)[/tex]
Hence, the final score of Gandhi House is:
[tex]\[ \boxed{3} \][/tex]
### 6. Finding the Sum for Each Sub-question
#### (i) Calculate the sum of:
[tex]\[ 100 + (-18) + (-25) + (-63) \][/tex]
Step-by-step:
1. Start with 100
2. Add -18: [tex]\( 100 + (-18) = 82 \)[/tex]
3. Add -25: [tex]\( 82 + (-25) = 57 \)[/tex]
4. Add -63: [tex]\( 57 + (-63) = -6 \)[/tex]
So, the sum is:
[tex]\[ \boxed{-6} \][/tex]
#### (ii) Calculate the sum of:
[tex]\[ 1295 + (-410) + (-878) + 312 \][/tex]
Step-by-step:
1. Start with 1295
2. Add -410: [tex]\( 1295 + (-410) = 885 \)[/tex]
3. Add -878: [tex]\( 885 + (-878) = 7 \)[/tex]
4. Add 312: [tex]\( 7 + 312 = 319 \)[/tex]
So, the sum is:
[tex]\[ \boxed{319} \][/tex]
#### (iii) Calculate the sum of:
[tex]\[ 1000 + 873 + (-874) + (-999) \][/tex]
Step-by-step:
1. Start with 1000
2. Add 873: [tex]\( 1000 + 873 = 1873 \)[/tex]
3. Add -874: [tex]\( 1873 + (-874) = 999 \)[/tex]
4. Add -999: [tex]\( 999 + (-999) = 0 \)[/tex]
So, the sum is:
[tex]\[ \boxed{0} \][/tex]
#### (iv) Calculate the sum of:
[tex]\[ 2004 + 516 + (-520) - 1999 \][/tex]
Step-by-step:
1. Start with 2004
2. Add 516: [tex]\( 2004 + 516 = 2520 \)[/tex]
3. Add -520: [tex]\( 2520 + (-520) = 2000 \)[/tex]
4. Subtract 1999: [tex]\( 2000 - 1999 = 1 \)[/tex]
So, the sum is:
[tex]\[ \boxed{1} \][/tex]
#### (v) Calculate the sum of:
[tex]\[ (-7) + (+10) + (-6) + (-11) + (+9) + (+2) \][/tex]
Step-by-step:
1. Start with -7
2. Add 10: [tex]\( -7 + 10 = 3 \)[/tex]
3. Add -6: [tex]\( 3 + (-6) = -3 \)[/tex]
4. Add -11: [tex]\( -3 + (-11) = -14 \)[/tex]
5. Add 9: [tex]\( -14 + 9 = -5 \)[/tex]
6. Add 2: [tex]\( -5 + 2 = -3 \)[/tex]
So, the sum is:
[tex]\[ \boxed{-3} \][/tex]
These are the detailed solutions for each part of the question.
