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Multiplication And Division Of Integers

Product of Integers (Same Signs)

  • A product is an answer to a multiplication problem.
  • Product of two positive integers is always positive.

Positive integer × Positive integer = Positive integer

  • Product of two negative integers is always positive.

Negative integer × Negative integer = Positive integer

  • For example: Multiply \(-10\) with \(-2\).

\(-10\,(-2)=20\)

  • The product of \(-10\) and \(-2\) is positive \(20\).
  • Examples: 

\(-5\,(-5)=25\)

\(-11\,(-3)=33\)

Illustration Questions

Find:​ \(-15\,(-15)=\,?\)

A \(-225\)

B \(1\)

C \(225\)

D \(0\)

×

Product of two negative integers is always a positive number.

So, \(-15\,(-15)=225\)

Hence, option (C) is correct.

Find:​ \(-15\,(-15)=\,?\)

A

\(-225\)

.

B

\(1\)

C

\(225\)

D

\(0\)

Option C is Correct

Product of Integers (Different Signs)

  • When we multiply two integers with different signs, the product will be a negative integer.

Negative integer × Positive integer = Negative integer

  • For example: Multiply \(4\) with \(-3\).

\(4\,(-3)=-12\)

  • The product of \(4\) and \(-3\) is \(-12\).
  • Examples:
  1. \(8\,(-3)=-24\)
  2. \(-5\,(2)=-10\)
  3. \(12\,(-4)=-48\)

Illustration Questions

Find: ​\(20\,(-11)=\,?\)

A \(220\)

B \(-110\)

C \(9\)

D \(-220\)

×

Product of a negative integer and a positive integer is a negative number.

So, \(20\,(-11)=-220\)

Hence, option (D) is correct.

Find: ​\(20\,(-11)=\,?\)

A

\(220\)

.

B

\(-110\)

C

\(9\)

D

\(-220\)

Option D is Correct

Quotients of Integers (Same Signs)

  • When dividend and divisor both are of same signs, the quotient is a positive integer.
  • When a negative integer is divided by another negative integer, the quotient is always a positive integer.

Negative integer ÷ Negative integer = Positive integer

  • Similarly, Positive integer ÷ Positive integer = Positive integer
  • For example: Divide \(-39\) by \(-3\).

\(-39\div(-3)=13\)

  • The quotient of \(-39\) by \(-3\) is positive \(13\).
  • Examples:
  1. \(-144\div(-12)=12\)
  2. \(-75\div(-25)=3\)
  3. \(-60\div(-5)=12\)

Illustration Questions

Find: \(-99\div(-9)=\,?\)

A \(-108\)

B \(11\)

C \(-90\)

D \(-11\)

×

  • When a negative integer is divided by another negative integer, the quotient is always a positive integer.

So, \((-99)\div(-9)=11\)

Hence, option (B) is correct.

Find: \(-99\div(-9)=\,?\)

A

\(-108\)

.

B

\(11\)

C

\(-90\)

D

\(-11\)

Option B is Correct

Quotients of Integers (Different Signs)

  • When dividend and divisor both have different signs, the quotient will be a negative integer.

Negative integer ÷ Positive integer = Negative integer

Positive integer ÷ Negative integer = Negative integer

  • Whenever integers with different signs are divided, the quotient is always negative.
  • For example: Divide \(-56\) by \(8\).

\(-56\div8=-7\)

  • The quotient of \(-56\) by \(8\) is negative \(7.\)
  • Examples:
  1. \(-105\div15=-7\)
  2. \(39\div(-13)=-3\)
  3. \(-49\div7=-7\)

Illustration Questions

Find: \(-120\div8=\,?\)

A \(-15\)

B \(-112\)

C \(-128\)

D \(15\)

×

When dividend and divisor both have different signs, the quotient will be a negative integer.

So, \(-120\div8=-15\)

Hence, option (A) is correct.

Find: \(-120\div8=\,?\)

A

\(-15\)

.

B

\(-112\)

C

\(-128\)

D

\(15\)

Option A is Correct

Temperature as Integers

  • Integers can be used to represent temperatures.
  • Zero is used as a reference point.

(1) A temperature above zero is represented as a positive integer.

  \(15°F\)  above zero \(=15°F\)

(2) A temperature below zero is represented as a negative integer.

  \(15°F\)  below zero \(=-15°F\)

  • Smaller the integer, colder the temperature is and greater the integer, warmer the temperature is.

\(-15°F\)  is colder than \(-13°F\).

\(15°F\)  is warmer than \(13°F\).

  • Examples:

\(10°F\)  above zero \(=10°F\)

\(5°F\)  below zero \(=-5°F\)

  • For example: The temperature of Sunday was \(2°F\). The temperature of Monday was \(5°F\) colder than the temperature of Sunday. Now, we want to find out the temperature of Monday.

The temperature of Sunday is shown in the figure.

\(\to\) The temperature of Monday was \(5°F\) colder than the temperature of Sunday, and Sunday's temperature was \(2°F\). So, the temperature of Monday,

\(=2-5\)

\(=-3°F\)

  • Hence, the temperature of Monday was \(-3°F\).

Illustration Questions

A man is standing on a diving board which is \(5\) feet above the surface of a swimming pool. He took a dive of \(9\) feet from the diving board. What is the location of the man after the dive?

A \(4\) feet below the pool's surface

B \(12\) feet above the pool's surface

C \(12\) feet below the pool's surface

D \(6\) feet above the pool's surface

×

Any location above the surface represents a positive integer and below the surface represents a negative integer.

A man standing \(5\) feet above the surface of the swimming pool, is represented by \(5.\)

image

He took a dive of \(9\) feet from there.

Now, the new location of the man is:

\(5-9=-4\)

image

Location of the man is \(4\) feet below the surface of the pool.

Hence, option (A) is correct.

A man is standing on a diving board which is \(5\) feet above the surface of a swimming pool. He took a dive of \(9\) feet from the diving board. What is the location of the man after the dive?

A

\(4\) feet below the pool's surface

.

B

\(12\) feet above the pool's surface

C

\(12\) feet below the pool's surface

D

\(6\) feet above the pool's surface

Option A is Correct

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