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NCERT Solutions Ganita Prakash (Part 2) Chapter 5 In-text Questions — The Balancing Act

Book page 105 Updated on2026-09-05

Q1.
Will including a new value in the data increase or decrease the mean?
Answer

It depends entirely on where the new value sits compared with the present mean.

  • New value greater than the mean → the mean increases.
  • New value less than the mean → the mean decreases.
  • New value equal to the mean → the mean does not change.
Old data 4, 6, 8 → mean 6
Include 12 → (4 + 6 + 8 + 12) ÷ 4 = 30 ÷ 4 = 7.5 (up)
Include 2 → (4 + 6 + 8 + 2) ÷ 4 = 20 ÷ 4 = 5 (down)
Why it happens: with n values and mean a, adding x gives a new mean of (na + x) ÷ (n + 1). Subtracting the old mean,
(na + x)/(n + 1) − a = (na + x − na − a)/(n + 1) = (x − a) ÷ (n + 1).
So the shift has the same sign as x − a. In balance language: a value dropped on the right makes that side heavier, so the balance point has to slide right until the distances even out again.
Tip: the formula also tells you how much the mean moves — the gap x − a shared out over n + 1 values. A big class barely notices one new value; a small group feels it a lot.
Q2.
What happens to the mean when an existing value is removed? When will the mean increase, decrease, or stay the same?
Answer

Removing a value does the opposite of adding it.

  • Remove a value greater than the mean → the mean decreases.
  • Remove a value less than the mean → the mean increases.
  • Remove a value equal to the mean → the mean stays the same.
Data 4, 6, 8, 12 → mean 7.5
Remove 12 → (4 + 6 + 8) ÷ 3 = 6 (down)
Remove 4 → (6 + 8 + 12) ÷ 3 = 26 ÷ 3 ≈ 8.67 (up)
Why it happens: with n values and mean a, removing x leaves (na − x) ÷ (n − 1). The change works out to (a − x) ÷ (n − 1) — the mirror image of the ‘adding’ formula. Taking away weight from the heavy side lets the balance point slide back towards the other side.
Q3.
What happens to the mean if a value equal to the mean is included or removed? Try to explain this using the fair-share interpretation of mean that we studied last year.
Answer

Nothing changes — the mean stays exactly where it was.

Data 4, 6, 8, 12, mean 7.5
Include 7.5 → (30 + 7.5) ÷ 5 = 37.5 ÷ 5 = 7.5
Remove 7.5 from that new list → 30 ÷ 4 = 7.5
Fair-share explanation: the mean is what each person would get if everything were pooled and shared equally. Suppose five friends pool their money and each ends up with ₹7.5. Now a sixth friend joins carrying exactly ₹7.5 — she already has her fair share, so nothing has to be passed to her and nothing has to be taken from her. The equal share stays ₹7.5. In the balance picture, a dot placed exactly at the balance point has zero distance on either side, so it tips nothing.
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