An arithmetic check in the reduction of levels by the rise and fall method is stated as Last RL – First RL is equal to (i). Sum of rises – Sum of falls (ii). Sum of backsights – sum of foresights (iii). Sum of backsights – Sum of intermediate sights (iv). Sum of foresights – Sum of intermediate sights
- AOnly (i)
- B(i) and (ii)Correct
- C(i) and (iii)
- D(i) and (iv)
Solution
Concept
In levelling, arithmetic checks ensure that reduced levels are correctly calculated without numerical errors.
Formula
∑ BS - ∑ FS = ∑ Rise - ∑ Fall = Last RL - First RL
Step-by-step solution
1. In the Rise and Fall method of levelling, three arithmetic checks are used to verify the calculations. 2. The mathematical check equation is: Sum of Backsights (∑ BS) - Sum of Foresights (∑ FS) = Sum of Rises - Sum of Falls = Last RL - First RL. 3. Here, BS is Backsight, FS is Foresight, and RL is Reduced Level. 4. Therefore, Last RL - First RL equals both (i) Sum of rises - Sum of falls and (ii) Sum of backsights - sum of foresights. Answer: (B) (i) and (ii)
Explanation
In the Rise and Fall method, the difference between the Last RL and First RL must equal the difference between the total Backsights and total Foresights, as well as the total Rises and total Falls. This full check ensures every intermediate step and reduced level is mathematically correct.
Why the other options are wrong
A: It misses statement (ii) which is also a valid check for Last RL - First RL. C: Statement (iii) involves Intermediate Sights (IS) incorrectly; IS is included inside the rise/fall calculations. D: Statement (iv) incorrectly uses intermediate sights in place of backsights or rises/falls.
Shortcut
Remember the three-part check for Rise & Fall method: ∑ BS - ∑ FS = ∑ Rise - ∑ Fall = Last RL - First RL.
Common mistake
Confusing the Height of Instrument (HI) method check with the Rise and Fall method. HI method only checks ∑ BS - ∑ FS = Last RL - First RL.
Correct answer: B) (i) and (ii)