Interesting Trapezoid

Looking at all these complex formulas... makes me sad. I got next to zero formulas left in my old head, so I solved it like this.

1. since all rectangles are equal, then each rectangle is 20% of total area.
2. the top-left white triangle is half of 4 rectangles, i.e. 80% / 2 = 40%
3. the bottom-right white triangle is half of 3 rectangles, i.e. 60% / 2 = 30%
4. all white area is 70%, so the difference is 30% .

Call the height of the rectangle y and the base of each rectangle x. The area of the trapezoid is then $\dfrac{3xy}{2}$ .

The area of the large rectangle is $5xy$ . Therefore, the ratio of the shaded to unshaded parts will be $\dfrac{\left(\dfrac{3xy}{2}\right) }{5xy}=\boxed{\dfrac{3}{10}}$

Let $y$ be the hight of the rectangle, and $x$ be the width the of the rectangle. The the area of all five rectangles is equal to $5xy$ .

The area of the top left unshaded portion is equal to $(\frac{1}{2})(4xy)=2xy$

The area of the bottom right unshaded portion is equal to $(\frac{1}{2})3xy=\frac{3xy}{2}$

The area of the shaded potion is equal to $5xy-2xy-\frac{3xy}{2}=\frac{3xy}{2}$

The percentage of the area of the shaded portion compared to the area of the five rectangles is equal to $\frac{\frac{3xy}{2}}{5xy}=\frac{3xy}{10xy}=\frac{3}{10}$

The percentage is 30%

Easiest way is to find the area of the two white triangles. To make things simple I gave each of the 5 small rectangles a base of 1 and a height of 2.

The area of the large rectangle is then 5*2=10

The area of the triangle on the left is 4*2/2=4

The area of the triangle on the right is 3*2/2=3

Leftover area (trapezoid) is 10-7 = 3

3/10 = 30%

I just looked at it. No solution went through my head.

There are 5 same area rectangles. Assume each rectangle's area is 1 unit. Then, total area is 5 * 1 = 5(unit). The area of above triangle (bigger one) is a half of 4 rectangles' area = (1/2) * 4 * 1 = 2 (unit). The area of bottom triangle (smaller one) is a half of 3 rectangles' area = (1/2) * 3 * 1 =1.5 (unit). Therefore, the area of trapezoid is 5 - 2 - 1.5 = 1.5 (unit). The percentage of the shaded figure = 1.5/5 = 30%

Dont start solving it guys! Cut and paste that rectangle in your minds!

Start like this- Imagine the trapezoid occupied only one rectangle's side in the left bottom corner. Now its a paralellogram right? and its area will be the area of one rectangle, i.e 20% of total area

Okay now the piece we left out is actually a triangle with one rectangle's bottom side as the base and the same height. Obviously its area is half that of the normal rectangle.i.e 10% of total area.

Add the two pieces to give 30% as your answer :D

One can say the base is 1 and height is x. The area of bottom right side triangle is 3x/2, Area of the left top triangle is 2x.
Total white region is = 3.5 x, Black region = 5x - 3.5 x = 1.5 x, Percentage area = (1.5x / 5x ) * 100 = 30%

Triangle will have half of the area of a rectangle with same base and height.

So the two triangles together will be half of 7 rectangles they are part of = 3.5 rectangles

Area of shaded trapezoid = total area - area of two triangles = 5 -3.5 =1.5 rectangles = 30% of 5 rectangles.

Let the height of the rectangle be y and the base of each rectangle be x.The area of the trapezoid is then 3xy/2 ...(i) and The area of the large rectangle is 5xy ...(ii). Hence,the ratio of the shaded to unshaded parts will be (i)/(ii) Which is =3/10~30/100~30%. Hence, The Answer is 30%.

Totally agree with Adrian Ly (Sep 26, 2014) That was my solution too.

nice problem

• The shaded area has a top width of 1 and bottom width of 2. Average of 1 $\frac{1}{2}$ .
• The whole region has a width of 5.
• 1 $\frac{1}{2}$ / 5 * 100% = 30%

Used x for the height and 1 for the base of each rectangle. Area of the trapezoid was then 3x/2 and the areas of the rectangles are 5x. Dividing the two leaves 3/10

It's easiest to just pick numbers. Assume each rectangle is a 2 x 4 (meaning the big rectangle is a 10 x 4). Then the area of the trapezoid is $\frac{1}{2}$ (b1 + b2)h = $\frac{1}{2}$ (2 + 4)4 = $\frac{1}{2}$ (6)4 = 12.

The large rectangle's area is 40, so the ratio is 12/40 = 3/10 = 30%.

Pick a different set of numbers, and the ratio will still be the same. Anytime an answer to a question is a percentage/ratio/etc., you can often just throw arbitrary numbers in there and see what happens.

I think it is easier to calculate 2 white-triangle area. Assume the width of each rectangle is 1 and the height is X, then: first white area is 0.5 * 4 * X = 2X second white area is 0.5 * 3 * X = 1.5X and total white area is 3.5X compared to all area which is 5X then white area is 70% and the shaded area will be 30%

This is how I solved this. Let's name the pieces in the upper half (not actually half) as A,B,C and D and the pieces in the lower half as E,F and G. All the pieces are named from left to right.

It is clear that A,D and B,C pairs of pieces form one rectangle each which sums to 2 rectangles. In the lower part it is clear that the 3 rectangles have been cut by a diagonal, so that's 3/2=1.5 rectangles. So the unshaded region is 2+1.5=3.5 which means the shaded region is 5-3.5=1.5 rectangles. SO the percentage of shaded region is (1.5/5)*100=30%.

There you go with pure logic and zero formulae.

Ar(rectangle) = 5x2 = 10; Ar(trapezium) = 1/2 x 2 (2+1) = 3(this part is shaded region) ; Then percentage of shaded region = (3/10)x100 = 30 percentage.

looks like just less than a 1/3 so...... 30% makes sense :)

Let horizontal length = 5l and vertical height = h Area of 2 unshaded triangles = 0.5 3l h + 0.5 4l h = 3.5lh Area of the rectangle = 5lh Percentage of unshaded figure = [(3.5lh)/(5lh)]*100% = 70% Percentage of shaded figure = 100 - 70 = 30%

Let the height of rectangles be 'a' and base of each rectangle be 'b'. Thus,area of bigger unshaded triangle is 1/2 x a x 4b and area of smaller unshaded triangle is 1/2 x a x 3b. So,total unshaded area = 7ab/2. But total area of figure = a x 5b = 5ab thus total unshaded region = 5ab - 7ab/2 = 3ab/2. So % unshaded area = (3ab/2)/5ab X 100 = 30%

I ignored the trapezoids and calculated first the area of the rectangle, setting the base as 10x. Then took the upper unshaded part and treated it as a right triangle with area = 8xh/2 = 4xh, did the same for the lower unshaded part 6xh/2 = 3xh, and subtracted the sum of their areas (7xh) from the area of the rectangle. Cumbersome but I think correct, and it was easy to read the answer as a percent.

Call the height of the rectangle x and the base of the rectangle y. The area of the right triangle is 1/2 x 3/5 y=3/10 xy The area of the left triangle is 1/2 x 4/5 y=2/5 xy The area of the rectangle is xy Therefore, percentage of the figure is shaded: (xy - (3/10 xy + 2/5 xy))/xy = 30%