Question

Akhtar plans to cover a rectangular floor of dimensions $9.5$ m and $11.5$ m using tiles. Two types of square tiles are available: side $1$ m costs \rupee{100, and side $0.5$ m costs \rupee{}30. Tiles can be cut if required. What is the {minimum} cost to cover the entire floor?}

• A. 10930
• B. 10900
• C. 11000
• D. 10950
• E. 10430

Hint:

When cutting is allowed, reduce the problem to buying enough {area}. Compare cost per m$^2$; use the cheaper tile for most of the area and use the other only to cover the fractional remainder so that $x+0.25y$ just meets the required area.

Answer 1

The Correct Option is A

Step 1: Compute the floor area.
\[ \text{Area} = 9.5\times 11.5 = 109.25\ \text{m}^2. \] Step 2: Cost per square metre of each tile.
A $1$ m$\times 1$ m tile covers $1$ m$^2$ at \rupee{}100 $\Rightarrow$ \rupee{}100/m$^2$.
A $0.5$ m$\times 0.5$ m tile covers $0.25$ m$^2$ at \rupee{}30 $\Rightarrow$ \rupee{}120/m$^2$.
{Hence the $1$ m tile is cheaper per area.} We should buy as much $1$ m tile area as possible and use the $0.5$ m tile only to make up any fractional remainder (cutting is allowed, so shapes don’t restrict us). Step 3: Optimize the mix (integer tiles).
Let $x$ be the number of $1$ m tiles and $y$ the number of $0.5$ m tiles. Then \[ x + 0.25y \ge 109.25, \qquad \text{Cost} = 100x + 30y. \] Take $x=\lfloor 109.25\rfloor = 109$; remaining area $= 109.25-109=0.25$ m$^2$ $\Rightarrow$ one $0.5$ m tile ($y=1$) suffices. \[ \text{Cost} = 100(109) + 30(1) = \rupee{}10{,}930. \] Check nearby integer choices.
$x=110,y=0 \Rightarrow$ \rupee{}11{,}000;\quad $x=108,y=5 \Rightarrow$ \rupee{}10{,}950.\; Both are costlier. Thus \rupee{}10{,}930 is minimal. \[ \boxed{\text{Minimum cost}=\ \rupee{}10{,}930} \]