**Solving problems with rational numbers :**

A number that is expressed in the form a/b is called as rational number.

Here, both "a" and "b" are integers and also b ≠ 0.

In this section, we are going to see, how to solve 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

So, x/y = 7/8

Hence, the required fraction is 7/8

After having gone through the stuff given above, we hope that the students would have understood "Solving problems with rational numbers".

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