The price of a product is decreased by x% and then increased by x%. The final price becomes 10% less than the original. What is the value of x (approximately)?

Correct — A. A fall of x% followed by a rise of x% leaves a net fall of x²⁄100 %.

Net change = −x + x − (x × x) ÷ 100 = −x²⁄100
Set the fall equal to 10: x²⁄100 = 10
x² = 1000
x = √1000 ≈ 31.62, which is 31.6 → option (a)

Check: 100 → 68.38 after the fall → 68.38 × 1.3162 ≈ 90, a 10% fall.

Why the others are wrong

• (b)
  33.6 — 33.6 gives a net fall of 33.6² ÷ 100 ≈ 11.3%, more than the stated 10%.
• (c)
  30.6 — 30.6 gives a net fall of 30.6² ÷ 100 ≈ 9.4%, short of the stated 10%.
• (d)
  39.6 — 39.6 gives a net fall of 39.6² ÷ 100 ≈ 15.7%, far past the stated 10%.

Two successive changes of a% and b% combine to a + b + ab⁄100 %. The cross term ab⁄100 is why a fall and an equal rise do not cancel.

The rise is taken on the already reduced price, a smaller base than the fall was taken on, so the price never climbs back.

With a = −x and b = +x the formula collapses to −x²⁄100, whichever change comes first.

Key facts

• Successive changes of a% and b% give a net change of a + b + ab⁄100 %.
• An equal fall and rise of x%, in either order, give a net fall of x²⁄100 %.
• √1000 ≈ 31.62, since 31.62² ≈ 999.8.

Common traps

• Assuming the fall and the rise cancel: the rise works on a smaller base, so the price ends below where it started.
• Dropping the ÷ 100: x² = 10 gives x ≈ 3.16, a tenth of the right value.

No directly related past PYQ was found.