The length of a rectangle is twice the width , if its perimeter is 80, what is the dimension of the rectangle?

Question

10. Two-thirds of a number increased by three-fourth is five-third, what is the number?

Answer 1

Hi Jihen,

 

We'll take each question in turn.

 

 

Question #1 (dimensions of rectangle?):

 

Let's begin by assigning letters to represent our two unknowns:

    l = length

    w = width

 

Next, let's write two equations which we can use to solve for our two unknowns and solve them for our two unknowns:

    Equation #1 (length is twice the width):

        l = 2w

    Equation #2 (perimeter of rectangle):

        P = 2l + 2w

             P = 80 (given)

             l = 2w (from Equation #1)

        80 = 2(2w) + 2w

        80 = 4w + 2w

        6w = 80

        w = 80/6

        w = 40/3 (width of rectangle)

        w = 13 and 1/3 (width of rectangle)

     Equation #1 (length is twice the width):

         l = 2w

              w = 40/3

         l = 2(40/3)

         l = 80/3 (length of rectangle)

         l = 26 and 2/3 (length of rectangle)

 

Finally, we can verify our answers by plugging them into Equation #2 (perimeter of rectangle) and see if both sides balance:

    P = 2l + 2w (Equation #2)

    80 = 2(80/3) + 2(40/3)

    80 = 160/3 + 80/3

    80 = 240/3

    80 = 80 (both sides balance)

 

Since both sides balance, we are confident that our answers are correct.

 

 

Question #2 (what is the number?):

 

Let's begin by assigning a letter to represent our unknown:

    x = the number

 

Next, let's write an equation which can use to solve for our unknown and then solve it for our unknown:

    Equation (2/3 of a number increased by 3/4 is 5/3):

        (2/3)x + 3/4 = 5/3

        2x/3 + 3/4 = 5/3

        12(2x/3) + 12(3/4) = 12(5/3)

        24x/3 + 36/4 = 60/3

        8x + 9 = 20

        8x = 20 - 9

        8x = 11

          x = 11/8 (the number)

          x = 1 and 3/8 (the number)

 

Finally, we can verify our answer by plugging it back into our equation and see if both sides balance:

    (2/3)x + 3/4 = 5/3 (Equation)

    (2/3)(11/8) + 3/4 = 5/3

    22/24 + 3/4 = 5/3

    12(11/12) + 12(3/4) = 12(5/3)

    132/12 + 36/4 = 60/3

    11 + 9 = 20

    20 = 20 (both sides balance)

 

Since both sides balance, we are confident that our answer is correct.

 

Thanks for submitting these questions & glad to help.

 

God bless, Jordan.

 

Answer 2

In this problem, you have two variables which makes it seem daunting, but it's not that bad! Let's look at what is given using L for length and W for width.

You know that the perimeter of the rectangle is 80 yd, which is the sum of all sides when added together. For that shape, there are 2 lengths and 2 widths, so 2L+2W= 80.

You also know that the length is twice as large as the width, so you can represent the length as such: L= 2W.

Now because there are two variables, substitute L for "2W" into the other equation, so it looks like this: 2(2W) +2W=80. Using the distributive property, "partner up" the 2 with each of the terms inside of the ( ) by multiplication, so the equation becomes 2•2+2•W +2W=80 which is 2W+2W+2W=80.

If you look there are like terms that can be combined (terms that have the same "tag") which in this case is the terms with W, so now the equation is 6W=80. You want W to be isolated, so we must remove the coefficient (the number in front of the variable) by doing the opposite operation, which is division: 6W /6= 80/6.

80/6 will not give a whole number, so when you divide 80 by 6 you will have a decimal. A fraction is division and division is a fraction so 80/6 is the same as 80÷6. As a result, W=13 1/3 or 13.3333...! So now we can go back and replace W with the value. Either equation would work, but since L is isolated in L=2W, the math is easier. Doing this, L= 2(13.333...) which is 26.66667 so L=26.66667.

Check to see if it works by using the other equation, L as 26.66666 and W as 13.33333: 2(26.666)+ 2(13.3333)= 80. 52.332+26.666=80 which is 80=80. It works AND you know that 26.666 is twice more than 13.3333.. so you now know that L=26.6667 and W=13.3333