SOLVING PROBLEMS WITH RATIONAL NUMBERS WORKSHEET

Question 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 ?

Answer :

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.

Question 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.

Question 3 : 

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

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

Question 4 :

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

Simplify. 

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

Simplify. 

(x - 4) / (y - 2)  =  1/2

2(x - 4)  =  y - 2

2x-y = 6---(1)

Solving (1) and (2), we get

x  =  7

y  =  8

Then,

x/y  =  7/8

So, the required fraction is 7/8.

Question 5 :

The ratio of number of boys and number of girls in a school is found to be 15 : 32. How many boys and equal number of girls should be added to bring the ratio to 2/3 ?

Solution :

Ratio between boys to girls = 15 : 32

Let x be the number of boys and girls.

(15 + x) : (32 + x) = 2/3

(15 + x) / (32 + x) = 2/3

3(15 + x) = 2(32 + x)

45 + 3x = 64 + 2x

3x - 2x = 64 - 45

x = 19

So, the required number of boys and girls is 19. Option a is correct.

Question 6 :

The sum of a rational number and its reciprocal is 13/6. Find the number.

Solution :

Let x be the number and 1/x is its reciprocal.

x + 1/x = 13/6

(x² + 1)/x = 13/6

6(x² + 1) = 13x

6x² + 6 = 13x

6x² - 13x + 6 = 0

(3x - 2)(2x - 3) = 0

x = 2/3 and x = 3/2

So, the required number is 2/3 or 3/2.

Question 7 :

The sum of two numbers is 184. If one third of the one exceeds one seventh of the other by 8, find the smaller number.

Solution :

Let x be one number, then the other number be 184 - x.

1/3 of x = 1/7 of (184 - x) + 8

x/3 = [(184 - x)/7] + 8

x/3 - [(184 - x)/7] = 8

[7x - 3(184 - x)]/21 = 8

7x - 552 + 3x = 8(21)

10x - 552 = 168

10x = 168 + 552

10x = 720

x = 720/10

x = 72

184 - x ==> 184 - 72 ==> 112

So, the smaller number is 72.

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