In the last section, you used a problem-solving process to find the composite area. Another helpful strategy for solving problems involving composite area is estimation.

Using Rounding to Estimate Solutions

Franklin is a contractor who has been asked to replace the floor in a room that is shaped like the one shown below.

Franklin knows that the flooring he will use costs $5.95, which includes all materials, per square foot to install. What is the estimated cost of replacing the floor?

• What information is given in the problem?

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  The dimensions of certain edges of the floor and the price per square foot of the flooring are given.

• What question does the problem ask?

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  The problem is asking you to determine the estimated cost of replacing the floor.

• What is a plan for solving this problem?

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  Possible plan:
  Break the heptagon-shaped floor into polygons that have area formulas you know. Use those polygons to calculate the total area of the floor. Multiply the total area of the floor by the estimated cost per square foot.

• To what whole number could you round the cost per square foot?

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  $5.95 rounds to $6.

• Solve the problem, using your rounded cost per square foot.

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  Possible solution:
  
  
  Since the total area is 233 square feet, multiply 233 square feet by $6 per square foot.
  
  Estimated cost ≈ Total area × Cost per square foot
  Estimated cost ≈ 233 ft² × $6/ft² ≈ $1,398

Rounding can be an effective way to check if your answer to a problem with messy numbers is reasonable. Be careful not to round too much. You want to round to the nearest number so that you can accurately check your work.

Using Compatible Numbers to Estimate Solutions

A pasture is in the shape shown in the figure below.

Each ostrich that is being kept in the pasture requires about 18 square yards of pasture. About how many ostriches can be kept in the pasture?

• What information is given in the problem?

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  The dimensions of certain edges of the pasture and the area of pasture required for each ostrich are given.

• What question does the problem ask?

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  The problem is asking you to determine about how many ostriches can be kept in this pasture.

• What is a plan for solving this problem?

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  Possible plan:
  Break the figure into shapes that have area formulas you know. Use those shapes to estimate the total area of the pasture, rounding the dimensions first. Use compatible numbers to solve a simpler division problem for dividing the approximate area by the amount of area required for each ostrich.

• Solve the problem, using compatible numbers.

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  Since the total area is about 957 square yards, divide 957 square yards by 18 square feet per ostrich. 18 × 5 = 90, so use the compatible numbers of 900 and 18.
  
  Number of ostriches ≈ Total area ÷ Area per ostrich
  Number of ostriches ≈ 900 yd² ÷ 18 yd²/ostrich ≈ 50 ostriches

Pause and Reflect

How is using the strategy of compatible numbers similar to using the strategy of rounding?

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Both methods give an approximate solution to a problem using numbers that make the arithmetic easier to do mentally.

How can you use estimation (rounding or compatible numbers) to determine if your answer is reasonable?

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Estimation provides you with a quick way to check what kind of answer to expect. Estimation can also help you detect a computational error.

Practice

1. The front of a garage needs to be painted. If the cost of paint is $4.75 per square meter, how much will it cost to paint the garage?
   
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   What polygons can you break the front of the garage into in order to calculate the area? How can you use rounding to check your answer for reasonableness?
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2. A track field is in the shape shown below. The field needs to be re-seeded with grass, and the cost of the seed is $1.85 per square yard. What is the cost for re-seeding the field?
   
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   What figures can you break the field into in order to calculate the area? How can you use rounding to check your answer for reasonableness?
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   A = 1 over 2 1 2 πr² ≈ 1 over 2 1 2 (3.14 yd)(22.5 yd)² ≈ 794.8125 yd²
   
   Area of rectangle:
   A = bh = (100 yd)(45 yd) = 4500 yd²
   
   Area of right semicircle:
   A = 1 over 2 1 2 πr² ≈ 1 over 2 1 2 (3.14 yd)(22.5 yd) ≈ 794.8125 yd²
   
   Total area:
   A = 794.8125 yd² + 4500 yd² + 794.8125 yd² = 6089.625 yd²
   
   Cost = Area × Cost per square yard
   Cost = 6089.625 yd² × $1.85/yd² ≈ $11,265.81