Question

Rectangle \(A\) has a length of 12 inches and a width of 5 inches. Rectangle \(B\) has a length of 9 inches and a width of 10 inches. By what number must the area of rectangle \(A\) be multiplied in order to get the area of rectangle \(B ?\)

Short answer

1.5

Step-by-step solution

Find the area of Rectangle A

The area of a rectangle can be found by multiplying its length by its width. For Rectangle A, the length is 12 inches and the width is 5 inches. Therefore, the area of Rectangle A is given by: \[ \text{Area of Rectangle A} = 12 \times 5 = 60 \text{ square inches} \]

Find the area of Rectangle B

For Rectangle B, the length is 9 inches and the width is 10 inches. Multiply the length by the width to find the area: \[ \text{Area of Rectangle B} = 9 \times 10 = 90 \text{ square inches} \]

Determine the multiplication factor

To find the number by which the area of Rectangle A must be multiplied to equal the area of Rectangle B, divide the area of Rectangle B by the area of Rectangle A: \[ \text{Multiplication factor} = \frac{\text{Area of Rectangle B}}{\text{Area of Rectangle A}} = \frac{90}{60} = 1.5 \]

Key concepts

Area calculation

In geometry, calculating the area of shapes is a basic yet essential skill. For rectangles, the formula is straightforward: Multiply the length by the width. This gives you the total space inside the shape.
For example: If you have a rectangle with a length of 12 inches and a width of 5 inches, you simply multiply these two numbers:
\[ \text{Area} = 12 \times 5 = 60 \text{ square inches} \]
Remember, areas are always measured in square units because they cover two dimensions: length and width. This same formula applied to another rectangle with a length of 9 inches and a width of 10 inches would yield:
\[ \text{Area} = 9 \times 10 = 90 \text{ square inches} \]
Knowing how to calculate the area helps in a variety of real-world problems, from determining how much paint you need for a wall to figuring out the amount of grass seed required for a lawn.

Rectangle properties

Rectangles are four-sided shapes known as quadrilaterals. What sets them apart is that they have four right angles.
This makes all their angles equal to 90 degrees. Additionally, opposite sides of a rectangle are equal in length.
For instance, if one side of the rectangle is 12 inches long, the opposite side is also 12 inches long.
Here are key properties to remember:

• Opposite sides are equal
• All angles are 90 degrees
• The length and the width are always perpendicular

These properties make it easier to measure and compare rectangles. For instance, when you know the length and width, you can easily calculate other aspects like perimeter (sum of all sides) and the diagonal using the Pythagorean theorem.

Mathematical ratios

A ratio is a way to compare two quantities. In the context of rectangles and their areas, a ratio can help describe how much larger or smaller one area is compared to another.
To find the ratio of two areas, you divide one area by the other.
Take our example: If Rectangle A has an area of 60 square inches and Rectangle B has an area of 90 square inches:
\[ \text{Ratio} = \frac{90 \text{ square inches}}{60 \text{ square inches}} = 1.5 \]
This ratio tells you that Rectangle B's area is 1.5 times that of Rectangle A.
Ratios are particularly useful for scale models and drawings, helping to maintain proportionality. For example, in architectural plans, knowing the ratio ensures that rooms and spaces are scaled correctly.