The width of a rectangle is three fourths of the length. The perimeter of the rectangle becomes 50 cm when the length and the width are each increased by 2 cm. Find the length and the width.
To find the length and width of the rectangle, we can set up a system of equations based on the given information.
Let's denote the length of the rectangle as 'L' and the width as 'W'.
According to the problem, the width is three fourths (3/4) of the length. This can be expressed as W = (3/4)L.
We are also told that when the length and width are each increased by 2 cm, the perimeter of the rectangle becomes 50 cm.
The formula for the perimeter of a rectangle is P = 2(L + W).
So, the original perimeter of the rectangle is P = 2(L + W) = 2(L + (3/4)L) = 2(7/4)L = (7/2)L.
When the length and width are increased by 2 cm, the new perimeter becomes 50 cm. So, the new perimeter is 50 cm, which can be expressed as P + 4 = 50 (since both the length and width are increased by 2 cm).
Substituting the original perimeter with its value, we have (7/2)L + 4 = 50.
To solve this equation, we can start by subtracting 4 from both sides: (7/2)L = 50 - 4 = 46.
Then, multiply both sides of the equation by 2/7 to isolate L: L = (46 * 2) / 7 = 92 / 7.
Therefore, the length of the rectangle is L = 92 / 7 cm.
To find the width, we can substitute this value into the equation W = (3/4)L.
W = (3/4)(92 / 7) = (276 / 4) / 7 = 69 / 7.
Therefore, the width of the rectangle is W = 69 / 7 cm.
To summarize, the length of the rectangle is approximately 13.14 cm (92 / 7) and the width is approximately 9.86 cm (69 / 7).
original length --- x
original width ---- 3x/4
new length = x+2
new width = 3x/4 + 2
2(x+2) + 2(3x/4 + 2) = 50
x + 2 + 3x/4 + 2 = 25
times 4
4x + 8 + 3x + 8 = 100
7x = 84
x = 12
length was 12
width was 9