About Integer Divsion
The Division Operation for Integers
A set of positive numbers such as 1, 2, 3, … , negative numbers such as -1, -2, -3, … and 0 is known as integers.
To start the division operation in integers we find the absolute values of the two integers first. To make it easier for you, the division of integers is similar to the division process you have already learnt. However, since we deal with different signs in numbers, we have to be careful of the sign of the quotient.
Dividing a positive integer and a negative integer will result in a negative quotient.
If p and q are integers, then \( -\frac{p}{q}=\frac{-p}{q}=\frac{p}{-q} \)
For example, \( 4\div (-2)=-2 \)
(or) \( -4\div 2=-2 \)
Dividing two positive integers will result in a positive quotient.
For example, \( 4\div 2=2 \)
Dividing two negative integers will result in a positive quotient.
For example, \( -4\div (-2)=2 \)
Solved Dividing Integer Examples
Example 1. Find \( -16\div (-2)=? \)
Solution:
Both integers have the same sign. So, the quotient should be positive.
\( -16\div (-2)=8 \)
Therefore, the quotient is 8.
Example 2. Find \( -32\div (-2)=? \)
Solution:
Both integers have the same sign. So, the quotient should be positive.
\( -32\div (-2)=16 \)
Therefore, the quotient is 16.
Example 3. Find \( 148\div 4=? \)
Solution:
Both integers have the same sign. So, the quotient should be positive.
\( 148\div 4=37 \)
Therefore, the quotient is 37.
Example 4. Find \( 70\div (-20)=? \)
Solution:
Both integers have different signs. So, the quotient should be negative.
\( 70\div (-20)=-3.5 \)
Therefore, the quotient is -3.5.
Example 5. Find the quotient for \( 0\div (-4)=? \)
Solution:
0 divided by any number is 0 itself.
\( 0\div (-4)=0 \)
Therefore, the quotient is 0.
Example 6. Find the quotient for \( (-6)\div 4=? \)
Solution:
Both integers have different signs. So, the quotient should be negative.
\( (-6)\div 4=-1.5 \)
Therefore, the quotient is -1.5.
Example 7. Find the quotient for \( \frac{-50}{5}=? \)
Solution:
Both integers have different signs. So, the quotient should be negative.
\( \frac{-50}{5}=-10 \)
Therefore, the quotient is -10.
Example 8. Find the quotient for \( \frac{-64}{8}=? \)
Solution:
Both integers have different signs. So, the quotient should be negative.
\( \frac{-64}{8}=-8 \)
Therefore, the quotient is -8.
Example 9. Find the value of the expression \( \frac{x}{2y} \), where x = 6 and y = -2.
Solution:
\( \frac{x}{2y}=\frac{6}{2(-2)} \) (Substituting 6 for x and -2 for y)
\( =\frac{6}{-4} \) (Dividing by 2)
\( =-\frac{3}{2} \) (6 and 4 are multiples of 2)
\( ~~~~~=-1.5 \)
So, the value of the expression is \( -1.5 \).
Example 10. Find the value of the expression \( \frac{x+9}{3} \), where x = -6.
Solution:
\( \frac{x+9}{3}=\frac{-6+9}{3} \) (substituting -6 for x)
\( ~~~~~~~=\frac{3}{3} \)
\( ~~~~~~~=1 \)
So, the value of the expression is 1.
Example 11. Find the value of the expression \( -x^3+6\div y \), where x = 6 and y = -2.
Solution:
\( -x^3+6\div y=-6^3+6\div (-2) \) (substituting 6 for x and -2 for y)
\( ~~~~~~~~~~~~~~~~~~~~~=-6^3+6\div (-2) \)
\( ~~~~~~~~~~~~~~~~~~~~~=-(216)+6\div (-2) \)
\( ~~~~~~~~~~~~~~~~~~~~~=-216+ (-3) \)
\( ~~~~~~~~~~~~~~~~~~~~~=-219 \)
So, the value of the expression is -219.
Example 12. A caterpillar was crawling down the trunk of a tree. At 4 pm, it was 67 inches off the ground, and at 6 pm, it was 13 inches off the ground. What is the caterpillar’s rate of descent?
Solution:
\( \text{Hourly change}=\frac{\text{final height – initial height }}{\text{elapsed time}} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~=\frac{13~-~67}{2} \) (Substitute the values and the elapsed time from 4 pm to 6 pm)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~=-27 \) inch/hour
Therefore, the rate of descent of the caterpillar is -27 inches per hour.
Example 13. Izzie is writing a book. She has written 540 pages in total as of now. She had written 320 pages 5 months ago. What is the monthly rate of pages written by her?
Solution:
\( \text{Rate in pages}=\frac{\text{Final number of pages – Initial number of pages }}{\text{number of months}} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~=\frac{540~-~320}{5} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~=\frac{220}{5} \)
\( ~~~~~~~~~~~~~~~~~~~~~~~=44 \)
Therefore, the monthly rate of pages Izzie wrote is 44 per month.
An integer is a whole number (not a fractional number) that can be positive, negative, or zero. In other words, a set of whole numbers and their opposites are known as integers.
When two integers are divided by the same sign, the result is always positive. When two integers are divided by the different signs, the result is always negative. 0 divided by any number results in 0 itself, and division by zero is undefined.
Division is the process of grouping a number or a set of things into equal parts. The number of equal groups is the divisor and the size of the groups is the quotient. If there are any items left to be grouped, that is the remainder.