miercuri, 22 februarie 2017

Percents/ Rule of Three / Problem

In the figure bellow it is represented the outline of a rectangular room with an area of 48 square meters. We note ABCD this rectangle.
It is known that the width is three quarters (¾ ) of the length of the room.

Inside the room is a fireplace (a stove), represented in the figure by the square MNPD with the one meter side.
In the room it is installed hardwood floor except shaded area MNPD where is the stove.
a) find the length of the room;
b) knowing that losses represent 10% of the floor area that will be covered with parquet flooring, show that it is enough to buy 51.7 square meter of needed parquet flooring,
c) the parquet flooring is sold in packs each containing 2,5 square meters of parquet flooring. The price of each box with parquet flooring is 43 $. Find the minimum amount of money required to purchase needed parquet flooring.

Hypothesis (data problem or what you know): Area of the rectangle is 48 m2 and the width is 3/4 of the length of the room.

We note the rectangle ABCD
L - the length of the rectangle ABCD
 l - the width of the rectangle ABCD Formula of the area of the rectangle is A = L x l Replacing the value of the area from hypothesis:
L x l = 48  

l = (3/4) x∙ L = (3 L) / 4 ;(2)
We have two equations that are true at the same time, this means that they form a system of two equations with two unknown. Replace in equation (1) it's expression from equation (2) and obtain: ∙

 (3L) /4 x L = 48 ,
L x L = 48x4 / 3 =16x4=64=8x8

 L = 8 m
b) The width will be equal to (3/4) x∙ L = (3/4)  x 8 =∙ 2 x 3 = 6 m.
In paragraph b) it is said that only the free surface is covered with parquet flooring.... so the shaded area of the square occupied by the fireplace MNPD must be subtracted from the total area of the rectangle ABCD. Area  MNPD is a square where MD =  1m. Area MNPD = 1 square meter.
So the surface that will be covered of the flooring is

Area ABCD – Area MNPD = 48-1 = 47 square meter.
47 square meter of parquet flooring.

The problem said that losses represent 10% of the floor area that will be covered with parquet flooring.
It follows that the losses represent 10% of 47 square meter = 10/100 x47 = 4.7 square meter.

Because there are losses (losses occur when cutting the floorboards) when buying flooring to be sure that it remains sufficient parquet flooring to cover room have to be purchased an amount equal to 47 + 4.7 = 51.7 square meter.
 c) The parquet flooring is sold in packs and each pack contains 2.5 m2 of flooring. To purchase amount of 51.7 m2 should be purchased:

 Using a simple rule of three
2.5 m2 => 1 box

51.7 m2 => X boxes....... X number of boxes is equal to 51.7: 2.5 = 20.68 boxes of flooring.
Because they can only buy whole boxes the solution is
21 boxes of flooring (immediately next whole number decimal number).

This is a minimum number of boxes of flooring that can cover the camera.
To calculate the minimum amount of money we use the simple rule of three:
1 Box =>43 $
21 boxes =>X

X = 21 x 43 = 903 $
Minimum amount of money to purchase needed flooring will be 903 $.

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