Question

A rectangular poster is 3 times as long as it is wide. A rectangular banner is 5 times as long as it is wide. Both the banner and the poster has a perimeter of 24 inches. What are the lengths and widths of the poster and the banner

Answer 1

"A poster is 3 times as long as it is wide"

This means that the length is three times larger than the width of the poster (which is in the shape of a rectangle).

"A banner is 5 times as long as it is wide"

This is exactly the same as the one for the poster, except the length of the banner is FIVE times larger than the width, instead of three.

"Both the banner and the poster have perimeter of 24 inches"

Both the banner and poster are rectangles. The formula for perimeter for rectangles is P = 2L + 2W (L = length, W = width).Perimeter of the Poster = 2L + 2WPerimeter of the Banner = 2L + 2WFor the poster, it was mentioned that the length is 3x larger than the width, so that must mean L = 3 * W = 3W. So therefore:Perimeter of the Poster = 2(3W) + 2W = 6W + 2W = 8W.For the banner, it was mentioned that the length is 5x larger than the width, so that must mean L = 5 * W = 5W. So therefore:Perimeter of the Banner = 2(5W) + 2W = 10W + 2W = 12W.So we found our perimeters, which are 8W for the poster and 12W for the banner. But the question also states that BOTH perimeters are 24 inches. So we must set 8W equal to 24 and 12W equal to 24.8W = 24W = 3 inches (For the poster)12W = 24W = 2 inches (For the banner)So we found our widths of both the poster AND the banner. Now we just need to find the lengths. But remember how I stated earlier that the length for the poster is 3W (L = 3W)? And for the banner, it's L = 5W? We can use that to find the lengths since we already know values for the widths.For the poster:L = 3W = 3(3) = 9 inchesFor the banner:L = 5W = 5(2) = 10 inchesSo the poster has a length of 9 inches and width of 3 inches, and the banner has a length of 10 inches and a width of 2 inches

Answer 2

A) For the poster: length = 9 inches and width = 3 inches

B) For the banner: length = 10 inches and width = 2 inches

Explanation:

A) The perimeter was given as 24 inches, so that;

P = 2 (l+w)

But, l = 3w, where w is the width. So that;

P = 2 (3w + w)

24 = 2 (4w)

24 = 8w

⇒ w = 3 inches

But, l = 3w = 3 × 3 = 9 inches

Therefore for the poster, length is 9 inches and width is 3 inches.

B) With the same value of perimeter, 24 inches:

p = 2 (l+w)

But, l = 5w, where w is the width. So that:

p = 2( 5w + w)

24 = 2 (6w)

24 = 12w

⇒ w = 2 inches

But, l = 5w = 5 × 2 = 10 inches

Therefore, for the banner, length is 10 inches and width is 2 inches.