Mathematics

Question

A rectangular yard has a width-to-length ratio of 5:9. If the distance around the yard is 2800 ft. What are the dimension of the yard?

Answer 1

Hello,

Let x = the width
Let y = the length

We are told in the problem that the "distance around the yard is 2800 feet," right? That means 2800 is the perimeter, correct? 

The formula for the perimeter is  as follows:  P = 2(w +l) or, since we arbitrarily assigned the values of x and y for width and length we can simply substitute those values, right?  

Or, put another way due to the distributive property in math: P = 2x + 2y  We good so far? Great. 


Given that we KNOW the perimeter is 2800 feet we can then come up with the following: 

2l + 2w = 2800 

This implies that x + y = 1400 Do you see why? From an equation stand point we divided two from each term on the left and did the same on the right, correct? 

Therefore, x + y = 1400     If this is true, which we KNOW it is, then y = 1400 - x, correct? 

You also know the ratio as it is given in the problem, right? width to length is 5:9 or 5/9 

Since we know that ratio, we can now set up a proportional equation and then cross-multiply. 

So, 5/9 = x/1400-x 

5(1400-x) = 9x 

Solving THIS equation then, we now have the following: 

7000 -5x = 9x 
7000 = 14x
x = 500 


Remember, we said at the beginning that x is the width correct, and we know that the width to length ratio is 5:9 

This means 

Width = 500 
Length = 900 

Now how do you KNOW this to be true? Simple: Take your original ratio 5/9 = 500/y or after cross multiplying 5y = 4500 or y = 900. Do you see how we proofed our work as we went along? Now check your work: P = 2(900 + 500) Plugging the above values of 500 and 900 equals 2800. 

Please let me know if this makes sense to you. I hope you have a great week in school. 

Answer 2

Let the length of yard is 9x then width is 5x.

 

Since distance around the yard is 2800 ft, we have

 

2* (5x + 9x) = 2800

 

i.e. 5x + 9x = 1400

 

i.e. 14x = 1400

 

i.e. x = 100

 

Hence width = 5x = 5*100 = 500 ft 

 

and length = 9x = 9*100 = 900 ft