finding the perimeter

**anshulbshah** · October 1st 2009, 09:07 PM

Seven congruent rectangles form a bigger rectangle as shown in the attachment.If the area of the bigger attachment is 336 square units,what is the perimeter of the bigger rectangle.(Sorry fig. not accurate)

**Matt Westwood** · October 1st 2009, 10:28 PM

Originally Posted by anshulbshah

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Seven congruent rectangles form a bigger rectangle as shown in the attachment.If the area of the bigger attachment is 336 square units,what is the perimeter of the bigger rectangle.(Sorry fig. not accurate)

You can get an equation relating the lengths of the sides of one of the smaller rectangles straight away, which will help.

Let the sides of the small rects be a and b.

Then $3a = 4b$ .

Then the perimeter is $4b + 2 (a+b) + 3a$ .

The area is given by $(a+b)(3a) = 336$ .

Then I expect you just need to eliminate either a or b from this system of simultaneous equations and the solution should drop into your lap.

**jashansinghal** · October 1st 2009, 10:30 PM

area of 7 rectangles=336 square unit
area of 1 rectangle=48 square units
lets take the breadth of smaller rectangle as B and length as L
[B will be the smaller side as per the figure]

As per the figure
4. B= 3. L
B= 3L/4

area of smaller rectangle=48square units
so,
L.B=48
now we substitute the value of B in this equation
L. 3L/4=48
so
L=8
B=6

as per the figure
area of bigger rectangle=
5L+6B
=> 5X8+6X6
=>40 + 36
= 76 units

**Soroban** · October 1st 2009, 10:33 PM

Hello, anshulbshah!

Seven congruent rectangles form a bigger rectangle as shown in the diagram.
If the area of the bigger rectangle is 336 square units,
what is the perimeter of the bigger rectangle?

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      * - - - - - * - * - - - * - - - * - * - - - - - *
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              x               x               x

Let: . $\begin{array}{ccc}x &=& \text{length of a small rectangle} \\ y &=& \text{width of a small rectangle} \end{array}$

From the top and bottom edges of the rectangle,
. . we see that: . $4y \:=\:3x \quad\Rightarrow\quad y \:=\:\tfrac{3}{4}x$ .[1]

The length of the big rectangle is: $L \,=\,3x$
The width of the big rectanble is: $W \,=\,x + y$
. . Hence, its area is: . $L\cdot W \:=\:(3x)(x+y) \:=\:3x^2 + 3xy$

We are told that the area is 336 units².
. . So we have: . $3x^2 + 3xy \:=\:336$ .[2]

Substitute [1] into [2]: . $3x^2 + 3x\left(\tfrac{3}{4}x\right) \:=\:336 \quad\Rightarrow\quad 3x^2 + \tfrac{9}{4}x^2 \:=\:336$

. . $\tfrac{21}{4}x^2 \:=\:336 \quad\Rightarrow\quad x^2 \:=\:64 \quad\Rightarrow\quad \boxed{x \:=\:8}$

Substitute into [1]: . $y \:=\:\tfrac{3}{4}(8) \quad\Rightarrow\quad \boxed{y \:=\:6}$

The length of the big rectangle is: . $L \:=\:3x \:=\:3(8) \quad\Rightarrow\quad L\:=\:24$
The width of the big rectangle is: . $W \:=\:x+y \:=\:8+6 \quad\Rightarrow\quad W\:=\:14$

Therefore, the perimeter is: . $2L + 2W \;=\;2(24) + 2(14) \;=\;76$ units.

Edit: Too slow again . . .

**Matt Westwood** · October 1st 2009, 10:36 PM

Originally Posted by Soroban

Edit: Too slow again . . .

... too complete as well ...

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