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. Apply both changes as multipliers and set the result to 0.9 (10% less).

(1 − x⁄100)(1 + x⁄100) = 0.9
Expand: 1 − x²⁄10000 = 0.9
Rearrange: x²⁄10000 = 0.1, so x² = 1000

Square root: x = √1000 ≈ 31.62
Nearest option: 31.6 → option (a)

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 more than the stated 10%.

A fall of x% followed by a rise of the same x% never returns to the start, because the rise is taken on the smaller, reduced price.

The net change is −x²⁄100 %, from (1 − x⁄100)(1 + x⁄100) = 1 − x²⁄10000.

Set x²⁄100 = 10 and x = √1000, just over 31.6.

Key facts

• Successive changes of −x% and +x% give a net change of −x²⁄100 %.
• The order does not matter: +x% then −x% gives the same net fall.
• √1000 ≈ 31.62, since 31.6² = 998.56 and 31.7² = 1004.89.

Common traps

• Expecting equal changes to cancel. A 10% cut and then a 10% rise ends 1% below the start.
• Taking the net fall as x% itself and answering near 10, which confuses x² ⁄ 100 with x.

No directly related past PYQ was found.