QUOTIENTS OF RATIONAL NUMBERS

In mathematics, a rational number is any number that can be expressed in the form p/q where 'q' is not equal to zero.

The quotient of the rational number p/q is nothing but the result that we get when we divide 'p' by 'q'. 

The rules for the sign of quotient are summarized below.

Let p and q be rational numbers.

Example 1 : 

Divide 2/3 by -7/6 and find the quotient. 

Solution :

Step 1 :

Take the reciprocal of the second rational number.

-7/6 ---- reciprocal ----> -6/7

Step 2 : 

Multiply the first rational number 2/3 by -6/7

(2/3) x (-6/7)  

Step 3 : Simplify

(2/1) x (-2/7)

Step 4 : Multiply 

(2/1) x (-2/7)  =  -4/7    Positive times negative equals      negative

So,

2/3 ÷ -7/6  =  -4/7

Example 2 : 

Divide -2 by 8/3 and find the quotient. 

Solution :

Step 1 :

Take the reciprocal of the second rational number.

8/3 ---- reciprocal ----> 3/8

Step 2 : 

Multiply the first rational number 2 by 3/8

(-2) x (3/8)  

Step 3 : Simplify

(-1) x (3/4)

Step 4 : Multiply 

(-1) x (3/4)  =  -3/4    Positive times negative equals  negative

So,

-2 ÷ 8/3  =  -3/4

Example 3 : 

Divide 9/5 by 3 and find the quotient.

Solution :

Step 1 :

Take the reciprocal of the second rational number.

3 ---- reciprocal ----> 1/3

Step 2 : 

Multiply the first rational number 9/5 by 1/3

(9/5) x (1/3)  

Step 3 : Simplify

(3/5) x (1/1)

Step 4 : Multiply 

(3/5) x (1/1)  =  3/5    Positive times positive equals   positive

So,

9/5 ÷ 3  =  3/5

Example 4 : 

A diver needs to descend to a depth of 100 feet below sea level. She wants to do it in 5 equal descents. How far should she travel in each descent ?

Solution :

To find how far she should travel in each descent, we have to divide 100 by 5. 

Step 1 :

Take the reciprocal of the divisor 5.

5 ---- reciprocal ----> 1/5

Step 2 : 

Multiply 100 by 1/5

(100) x (1/5)  

Step 3 : Simplify

(20) x (1/1)

Step 4 : Multiply 

(20) x (1/1)  =  20    

So, she should travel 20 feet in each descent.

Example 5 : 

Over 5 months, Carlos wrote 5 checks for a total of $323.75 to pay for his cable TV service. His cable bill is the same amount each month. What was the change in Carlos’s bank account each month to pay for cable?

Solution :

Step 1 :

Use a negative number to represent the withdrawal from his account each month

Step 2 : 

Find -323.75 / 5 

Step 3 : 

Determine the sign of the quotient.

The quotient will be negative because the signs are different.

Step 4 :

Divide. 

-323.75 / 5  =  -64.75 

So, Carlos withdrew $64.75 each month to pay for cable TV.

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