Find Paint Cans Needed for Wall Minus Whiteboard

What You Will Learn

• How to calculate area when you need to exclude sections
• When and why to round up in real-world problems
• Converting area calculations into practical purchasing decisions
• The difference between total coverage and usable coverage
• How geometric visualization helps solve word problems

Solution: Method 1 — The Subtraction Approach

The most intuitive way is to find the total wall area, then subtract the whiteboard area that won't be painted.

Step 1 — Calculate the total wall area

The wall is a rectangle, so its area is length times width:

Step 2 — Calculate the whiteboard area

The whiteboard is also rectangular:

Step 3 — Find the area that needs paint

Subtract the whiteboard area from the total wall area:

Step 4 — Calculate how many cans are needed

Divide the paintable area by the coverage per can:

Step 5 — Round up to whole cans

Since Ms. Baker can't buy 0.68 of a can, she needs to round up to the next whole number:

Solution: Method 2 — The Panel-by-Panel Approach

Another way is to think of the wall as separate rectangular panels around the whiteboard.

Step 1 — Identify the four panels

Looking at our diagram, we can divide the paintable area into four rectangles:

• Top panel: above the whiteboard
• Bottom panel: below the whiteboard
• Left panel: to the left of the whiteboard
• Right panel: to the right of the whiteboard

Step 2 — Calculate each panel area

If the whiteboard is centered on the wall (we'll assume this for simplicity):

Step 3 — Add up all panels

Step 4 — Convert to cans and round up

Notice we get the same answer! This confirms our subtraction method was correct.

Verification

Let's verify our answer makes sense:

Check the area calculation:

• Wall area: 24 × 15 = 360 ft² ✓
• Whiteboard area: 12 × 7 = 84 ft² ✓
• Paintable area: 360 - 84 = 276 ft² ✓

Check the paint calculation:

• 3 cans cover: 3 × 75 = 225 ft² (not enough)
• 4 cans cover: 4 × 75 = 300 ft² (enough for 276 ft²) ✓

The verification confirms that 4 cans is correct.

Does This Seem Reasonable?

Let's do a quick sanity check on our answer.

The whiteboard takes up 84 ÷ 360 = 23% of the wall. So we're painting about 77% of the wall.

If the whole wall needed painting, it would require 360 ÷ 75 = 4.8 cans, so 5 cans total. Since we're painting about 77% of it, we'd expect roughly 5 × 0.77 = 3.85 cans.

Our answer of 4 cans is right in that ballpark, so it passes the smell test!

Common Pitfalls

The General Formula

This problem follows the standard "area with exclusion" pattern:

The ceiling symbol ⌈⌉ means "round up to the next integer." This formula works for any paint problem where you need to avoid certain areas.

You'll see this pattern again with:

• Painting rooms with windows and doors
• Carpeting floors with built-in furniture
• Fertilizing lawns with gardens or pools
• Any "coverage minus obstacles" problem

Where This Shows Up in Real Life

Home renovation: Contractors constantly calculate paint, tile, or carpet needed while excluding fixtures, cabinets, and appliances. Getting this wrong means costly return trips to the store.

Commercial flooring: Office managers need to know how much carpet to order for a floor plan with cubicles, pillars, and permanent fixtures already in place.

Landscaping: Calculating mulch or grass seed needed for a yard minus the area occupied by trees, flower beds, patios, and walkways.

What If?