Problem 42 - Thinking Mathematically

Question

A rectangular room measures 12 feet by 15 feet. The entire room is to be covered with rectangular tiles that measure 3 inches by 2 inches. If the tiles are sold at ten for \(\$ 1.20\), what will it cost to tile the room?

Step-by-Step Solution

Answer

The cost of tiling the room will be $518.40

Step 1: Convert Dimensions of the Room to Inches

First, convert the dimensions from feet to inches, because the measurements for the tiles are given in inches. As 1 foot is equal to 12 inches, the room dimensions would therefore be \(12 feet \times 12 inches/foot = 144 inches\) and \(15 feet \times 12 inches/foot = 180 inches\).

Step 2: Calculate the Area of the Room

The area of the room is calculated by multiplying the length and the width of the room. The Area of the room is therefore \(144 inches \times 180 inches = 25,920 square inches\).

Step 3: Calculate the Area of One Tile

The area of one tile is calculated by multiplying its given length and width. Thus, the area of one tile is \(3 inches \times 2 inches = 6 square inches\).

Step 4: Determine the Number of Tiles Needed

To find out how many tiles you will need, divide the total area of the room by the area of one tile. Thus, we get \(25,920 square inches / 6 square inches = 4,320 tiles.\)

Step 5: Calculate the Total Cost

To calculate the total cost, first determine how many packs of tiles are needed, then multiply number of packs by the cost of each pack. As each pack costs $1.20 and contains 10 tiles, the number of packs needed is \(4320 tiles/10 tiles = 432 packs.\) So, the total cost would be \(432 packs \times $1.20/pack = $518.40.\)

Key Concepts

Area Calculation

To effectively cover a rectangular room with tiles, understanding how to calculate the area is essential. The area of any rectangle is found by multiplying its length by its width.
This gives us the amount of space within the boundary of the room. In this example, the room's length and width in inches are 144 inches and 180 inches, respectively. Thus, the room's area is calculated as follows:

• Length: 144 inches
• Width: 180 inches
• Area: \(144 \times 180 = 25,920\) square inches

This value represents the total surface area of the floor that needs to be covered by tiles.

Unit Conversion

Unit conversion is crucial in this problem because the room's dimensions are given in feet while the tiles' dimensions are provided in inches. To work with consistent measurements, we must convert units. Here, the room dimensions need to be converted from feet to inches.

• Remember: 1 foot = 12 inches
• Room Length: 12 feet = \(12 \times 12 = 144\) inches
• Room Width: 15 feet = \(15 \times 12 = 180\) inches

This step ensures that all calculations are in the same unit, facilitating precise area computation and further mathematical operations.

Rectangular Tiles

When tiling a space, each tile has its own specified area, which is an important piece of information to determine how many tiles are needed. Here, each tile measures 3 inches by 2 inches. To find the area of one tile, multiply its length by its width.

• Tile Length: 3 inches
• Tile Width: 2 inches
• Tile Area: \(3 \times 2 = 6\) square inches

This value aids in calculating how many such tiles are required to cover the entire room's surface area. Now, to cover the total room area of 25,920 square inches, we divide:

• Total Room Area: 25,920 square inches
• Tile Area: 6 square inches
• Total Tiles Required: \(25,920 \div 6 = 4,320\) tiles

Cost Calculation

After determining how many tiles are needed, calculating the cost becomes the next vital step. Since the tiles are packaged and sold in sets, understanding the pricing format is important. In this case, tiles are sold in packs of 10 for \(\\(1.20\).
To find out how many packs are necessary, divide the total number of tiles by the number of tiles per pack:

• Total Tiles Needed: 4,320
• Tiles Per Pack: 10
• Packs Needed: \(4,320 \div 10 = 432\) packs

Finally, multiply the number of packs needed by the cost per pack:

• Packs Needed: 432
• Cost Per Pack: \(\\)1.20\)
• Total Cost: \(432 \times 1.20 = \$518.40\)

This calculation helps determine the total monetary expense required to purchase all the tiles necessary to cover the room.