Shortcut Trick
Case 1: Removal of 780 units machine reduces average by 30 ⇒ 780 − k = 30(n − 1)
Case 2: Addition of 0 units machine reduces average by 90 ⇒ k − 0 = 90(n + 1)
Substitute k = 90n + 90 in first equation: 780 − (90n + 90) = 30n − 30
690 − 90n = 30n − 30 ⇒ 120n = 720 ⇒ n = 6
Average k = 90(6 + 1) = 630 units. Total output = 6 × 630 = 3780 units.
∴ The correct answer is 3780 units.
Alternate Method
Given:
Number of machines = n | Average output = k units/day
Value of machine excluded = 780 units | Value of machine included = 0 units
Formula Used:
Total Output = Number of Machines × Average Output
Calculations:
⇒ Total initial output = n × k
⇒ When 780 is excluded: (nk − 780) ÷ (n − 1) = k − 30
⇒ nk − 780 = (n − 1)(k − 30) = nk − 30n − k + 30
⇒ 30n + k = 810 ... (Eq. 1)
⇒ When a dead machine (0 units) is included: (nk + 0) ÷ (n + 1) = k − 90
⇒ nk = (n + 1)(k − 90) = nk − 90n + k − 90
⇒ k − 90n = 90 ⇒ k = 90n + 90 ... (Eq. 2)
⇒ Substituting (Eq. 2) in (Eq. 1): 30n + (90n + 90) = 810
⇒ 120n = 720 ⇒ n = 6
⇒ k = 90(6) + 90 = 630
⇒ Total Output = nk = 6 × 630 = 3780 units
∴ The correct answer is 3780 units.
Additional Information
Net Deviation Principle
In any average problem, the sum of deviations from the average is always zero. If an item is added or removed, the total change in the sum equals the deviation of that item from the original average.
Impact of Adding Zero
When a zero-value item is added to a set, the total sum remains constant, but the average decreases because the denominator (count) increases: New Avg = Old Sum ÷ (n + 1).
Average Property
If every element in a set is increased or decreased by a constant value 'x', the average also increases or decreases by 'x' respectively.