Informative line

# 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:

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

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

#### 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. $$12\,(-4)=-48$$

#### 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. $$-60\div(-5)=12$$

#### 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. $$-49\div7=-7$$

#### 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 $$-3°F$$.

#### 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 $$5.$$

He took a dive of $$9$$ feet from there.

Now, the new location of the man is:

$$5-9=-4$$

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