SOLVING PROBLEMS WITH RATIONAL NUMBERS

Problem 1 : 

The area of a rectangular sandbox is 56 2/3 square feet. The length of the sandbox is 8 1/2 feet. What is the width ?

Solution :

We know the formula to find area of the rectangle.

That is, 

Area of the rectangle  =  length x width 

Plug the known values area and length and solve for the unknown value width.

56 2/3  =  8 1/2 x width 

170/3  =  17/2 x width

(170/3) x (2/17)  =  width

20/3  =  width

6 2/3  =  width

Therefore, the width of the rectangle is 6 2/3 feet.

Problem 2 : 

Mr. Webster is buying carpet for an exercise room where the basement of the room is in the shape of rectangle. The length and width of the room are 18 2/5 feet and 12 1/2 feet respectively. Find the total cost of the carpet, if the price of the carpet is $3 per square feet ?

Solution :

To know the total cost of the carpet, first we have to know the area of the basement.

Area of the basement  =  length x width

=  18 2/5 x 12 1/2

=  (92/5) x (25/2)

=  46 x 5

=  230 square feet

So, we need 230 square feet of carpet.

The cost each square feet of carpet  =  $3

Then, the cost of 230 square ft of carpet is

=  3 x 230

=  $690

Therefore, the total cost of the carpet is $690.

Problem 3 : 

Mr. Webster is buying carpet for an exercise room where the basement of the room is in the shape of rectangle. The length and width of the room are 18 2/5 feet and 12 1/2 feet respectively. Find the total cost of the carpet, if the price of the carpet is $3 per square feet ?

Solution :

To know the total cost of the carpet, first we have to know the area of the basement.

Area of the basement  =  length x width

=  18 2/5 x 12 1/2

=  (92/5) x (25/2)

=  46 x 5

=  230 square feet

So, we need 230 square feet of carpet.

The cost each square feet of carpet  =  $3

Then, the cost of 230 square ft of carpet is

=  3 x 230

=  $690

Therefore, the total cost of the carpet is $690.

Problem 4 : 

David is cooking enough lentils for lentil barley soup and lentil salad. The lentil barley soup recipe calls for 3/4 cup of dried lentils. The lentil salad recipe calls for 1 1/2 cups of dried lentils. David has a 1/8 -cup scoop. How many scoops of dried lentils will David need to have enough for the soup and the salad?

Solution :

David needs 3/4 cup of dried lentils for soup and 1 1/2 cups for salad.

Total amount dried lentils that David need is

=  (3/4) + (1 1/2)

=  3/4 + 3/2

=  3/4 + 6/4

=  (3+6) / 4

=  9/4

David needs 9/4 cups of dried lentils for both the soup and the salad.

To find how many 1/8 -cup scoops he needs, divide the total amount of dried lentils into groups of 1/8.

Then, we have 

=  9/4 ÷ 1/8

=  9/4 x 8/1

=  9 x 2

=  18

Hence, David will need 18 scoops of dried lentils to have enough for both the lentil barley soup and the lentil salad.

Problem 5 :

If the numerator of a fraction is increased by 2 and the denominator by 1, it becomes 1. In case, the numerator is decreased by 4 and the denominator by 2, it becomes 1/2. Find the fraction. 

Solution :

Let "x/y" be the required fraction. 

"If the numerator is increased by 2 and the denominator by 1, the fraction becomes 1"

From the above information, we have (x+2) / (y+1) = 1

(x + 2) / (y + 1)  =  1

x + 2  =  y + 1

x - y = -1 ----(1)

"In case the numerator is decreased by 4 and the denominator by 2, the fraction becomes 1/2"

From the above information, we have

(x - 4) / (y - 2)  =  1 / 2

(x - 4) / (y - 2)  =  1 / 2

2(x - 4)  =  y - 2

2x - y  =  6 ----(1)

Solving (1) and (2), we get

x  =  7 and y  =  8

Then 

x / y  =  7 / 8 

So, the required fraction is 7/8.

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