A rectangular school banner has a length of 54 inches and a width of 36 inches. A sign is made that is similar to the school banner and has a length of 17 inches. What is the ratio of the area of the school banner to the area of the sign?
Well, it looks like we have some banner math to do! To find the ratio of the area of the school banner to the area of the sign, we first need to calculate the areas of both.
The area of the school banner can be found by multiplying the length (54 inches) by the width (36 inches): 54 inches x 36 inches = 1944 square inches.
The area of the sign can be found by multiplying the length (17 inches) by itself, since it's a square: 17 inches x 17 inches = 289 square inches.
Now, to find the ratio, we divide the area of the school banner by the area of the sign: 1944 square inches / 289 square inches = 6.72.
So, the ratio of the area of the school banner to the area of the sign is approximately 6.72. It's kind of like comparing the size of a whoopie cushion to the size of a clown's red nose - quite a big difference!
To find the ratio of the area of the school banner to the area of the sign, we need to calculate the areas of both the banner and the sign.
The area of the rectangular banner can be found by multiplying its length by its width:
Area of banner = Length × Width = 54 inches × 36 inches
Similarly, the area of the sign can be found by multiplying its length by its width:
Area of sign = Length × Width = 17 inches × ? inches
However, we don't know the width of the sign. To be similar, the ratio of the lengths and widths of the banner and the sign should be the same.
Since the length of the sign is given as 17 inches, and the length of the banner is 54 inches, we can set up the following proportion to find the width of the sign:
Width of sign / Width of banner = Length of sign / Length of banner
Therefore, we have:
Width of sign / 36 inches = 17 inches / 54 inches
To find the width of the sign, we cross-multiply:
Width of sign × 54 inches = 36 inches × 17 inches
Dividing both sides by 54 inches gives us the width of the sign:
Width of sign = (36 inches × 17 inches) / 54 inches
Now we can substitute the width of the sign into the equation to find its area:
Area of sign = 17 inches × (36 inches × 17 inches) / 54 inches
Finally, we can calculate the ratio of the area of the banner to the area of the sign:
Ratio = Area of banner / Area of sign = (54 inches × 36 inches) / (17 inches × (36 inches × 17 inches) / 54 inches)
Simplifying the expression, we get:
Ratio = (54 inches × 36 inches) / (17 inches × (36 inches × 17 inches) / 54 inches)
Ratio = (54 inches × 36 inches) / ((17 inches)^2)
Therefore, the ratio of the area of the school banner to the area of the sign is:
Ratio = (54 inches × 36 inches) / ((17 inches)^2)
To find the ratio of the area of the school banner to the area of the sign, we need to calculate the area of each and then divide them.
The area of a rectangle is calculated by multiplying its length by its width. So, the area of the school banner is:
Area of school banner = Length * Width = 54 inches * 36 inches
Now, let's calculate the area of the sign. Since the sign is similar to the school banner, it means that the ratio of their lengths is the same as the ratio of their widths. In other words, the lengths and widths are proportional.
We can set up the following proportion:
School banner length / Sign length = School banner width / Sign width
Using the information given, we have:
54 inches / 17 inches = 36 inches / Sign width
Cross multiplying the proportion, we get:
54 inches * Sign width = 17 inches * 36 inches
To solve for Sign width, we divide both sides of the equation by 54 inches:
Sign width = (17 inches * 36 inches) / 54 inches
Now that we have the width of the sign, we can calculate its area:
Area of sign = Sign length * Sign width = 17 inches * (17 inches * 36 inches) / 54 inches
Finally, we can find the ratio of the area of the school banner to the area of the sign:
Ratio = Area of school banner / Area of sign
I will calculate these values for you.
The areas of similar shapes are proportional to the square of their corresponding sides
area of banner : area of sign = 54^2 : 17^2
= 2916 : 289
= appr 10.1 : 1
OR, the long way ....
area of banner = 54x36 = 1944 inches^2
for the sign, we need the width
17/54 = w/36
w = 36(17/54) = 612/54 = 34/3
area of sign = 17(34/3) = 578/3
ratio of areas
= 1944 : 578/3
= 5832 : 578
= 2916 : 289 , same as above