The ratio of the length, the breadth, and the height of a solid cuboid is given as 3 : 2 : 1, and the breadth of this cuboid is given as 12 cm. Find the volume (in cm 3 ) of the cuboid.

Correct — D. The ratio supplies one multiplier and the breadth names it.

Let l : b : h = 3x : 2x : x.
Breadth is the middle term, so 2x = 12 and x = 6 cm
Then l = 3 × 6 = 18 cm and h = 1 × 6 = 6 cm

Volume = l × b × h = 18 × 12 × 6
= 216 × 6 = 1296 cm³ → option (d).

Why the others are wrong

• (a)
  1500 — 1500 ⁄ 12 = 125, so length × height would have to be 125. The ratio makes that product 3x², which at x = 6 is 108.
• (b)
  1250 — 1250 is not divisible by 12 — the division gives 104.17 — so it cannot be the volume of a cuboid one of whose edges is exactly 12 cm.
• (c)
  1200 — 1200 ⁄ 12 = 100, needing length × height = 100. With l = 3x and h = x that product is 3x² = 108 once the breadth has fixed x at 6.

A ratio of dimensions is not a set of dimensions. Write l : b : h = 3x : 2x : x and one measured edge fixes x for all three.

The measured edge here is the breadth, which is the middle term, so x = 12 ⁄ 2 = 6 and not 12. Substituting the given number straight in as the multiplier is the whole trap.

In ratio form the volume is 3x · 2x · x = 6x³, so at x = 6 it is 6 × 216 = 1296 cm³. The same shortcut answers any 3 : 2 : 1 cuboid once you know one edge.

Key facts

• For l : b : h = 3 : 2 : 1 the volume is 6x³, where x is the common multiplier.
• A breadth of 12 cm against ratio term 2 gives x = 6, so the cuboid is 18 × 12 × 6 cm.
• Volume of a cuboid = length × breadth × height, reported in cubic units.

Common traps

• Setting x = 12 because 12 is the number printed, which multiplies the volume by 8 to 10368.
• Attaching the 12 cm to the length instead, giving x = 4 and a volume of 384.
• Reporting a volume in cm² after doing the multiplication correctly.

No directly related past PYQ was found.