**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

Hence, 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

Hence, -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

Hence, 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

Hence, 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

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

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

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