Problem 60 - Algebra 1: Concepts and Skills

Question

ESTIMATING AREA Estimate the area of a rectangle whose sides are given. First round each side length to the nearest whole number. Then multiply to find the area. $$ 100.4 \text { by } 7.0 $$

Step-by-Step Solution

Answer

The estimated area of the rectangle is 700 square units.

Step 1: Rounding the Dimensions

Round each given dimension to the nearest whole number. Here the dimensions 100.4 and 7.0 are rounded off to 100 and 7 respectively.

Step 2: Calculation of Area

The area of a rectangle is calculated by multiplying its length by its width which here is 100 * 7.

Step 3: Result

The result from Step 2 gives the estimated area of the rectangle.

Key Concepts

Rounding Numbers

Rounding numbers is a basic skill that helps simplify calculations and make estimations easier. When you round a number, you replace it with another number that is close to the original, but simpler to use. Typically, this involves looking at how close the number is to the nearest whole number or another specified place value.

Here's how you can round numbers:

• Look at the digit to the right of the place value you're rounding to. If that digit is 5 or greater, you add 1 to the number you are rounding. If it's less than 5, you leave the number as it is.
• All digits to the right of the rounding point become zero after rounding.

In our example, 100.4 is rounded to 100. Since the digit after the decimal (4) is less than 5, we do not increase 100. Similarly, 7.0 remains 7 because it is already a whole number.

Area of a Rectangle

Finding the area of a rectangle is one of the most common problems you'll encounter in geometry. The area represents the amount of space contained within the rectangle's borders. To find the area, we use the formula:

• Area = length x width

This formula implies that you take the measurement of one side (the length) and multiply it by the measurement of the adjacent side (the width). Both dimensions must be in the same units to get a correct area measurement.

In practical scenarios, like estimating the area of a physical space, rounding the dimensions can help you quickly find a reasonable area estimate. In our task, the dimensions given were 100.4 by 7.0. After rounding, the rectangle becomes 100 by 7, making the multiplication and hence the estimation straightforward.

Multiplication

Multiplication is one of the fundamental arithmetic operations, crucial in calculating areas. It involves adding a number to itself a certain number of times. In the context of a rectangle, multiplication helps in determining how much total space is covered.

For a rectangle, you multiply the length by the width to obtain the area. This means you are calculating how many square units fit into the space horizontally and vertically. In our case, once the dimensions were rounded, the length became 100 and the width was 7. Therefore, to compute the area, we performed the multiplication:

• 100 (length) x 7 (width) = 700

Thus, the estimated area of the rectangle is 700 square units. Multiplying rounded numbers helps simplify the arithmetic, making it faster and easier to get an approximate result.