FREE Basic Math Concept-02




FREE Basic Math Concept-02

Multiplication:

We have learned how to add numbers. If we add same number many times, it’s called multiplication. For instance, If we add 3, by 5 times, i.e.,

3 + 3 + 3 + 3 + 3

By taking 3 common,

3 (1 + 1 + 1 + 1 + 1) = 3 × 5 = 15

 

Important Note: When we take common, it would place ‘1’ from all values from whom the common value has taken.

 

Many signs indicate the relationship as multiplication. For instance,

3 × 5 = 3 (5) = (3) 5

All above expressions show multiplications.

When a number is multiplied by zero, the result is zero.

3 × 0 = 0 + 0 + 0 = 0

 

Important Note: 0 is a magnet in Math. When any number is multiplied by 0, it resulted to 0. Basically, 0 works like a magnate, as a magnate when rub with iron rod, makes the iron magnate.

 

Multiplication of big numbers:

Let’s multiply 15 with 25.

FREE-Basic-Math-Concept-02-image-01

So,
25 × 15 = 375 Answer

 

Multiplication of fractions/decimals:

If we are ask to find the product of 0.25 & 0.15. First convert the decimal to fractions by using the following method.

FREE-basic-math-concept-02-image-02

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Important Note: When we multiply 100 by 100, it would add two zeros to hundred. Similarly, If we multiply 100 by 1,000, it would add 3 zeros to hundred… i.e., number of zeros would add-up.

 

When the two fractional expressions are multiplying, then the nominator of first expression multiplies with the nominator of other, also the denominator of the first expression multiplies with the denominator of the other to form result, i.e.,

FREE-basic-math-image-14

 
 
 
 

It can be further simplified by dividing both nominator and denominator by 2;

FREE-basic-math-image-15

 
 
 
 
 
 
 
 
 
 
 
 

Multiplication with negative numbers:

Now let’s discuss what happens when the negative integers are entertained in process of multiplication. For that purpose, we’ll discuss different aspects of negative integers and positive integers combined.

 

Multiplication of One positive and one negative:

When a positive integer multiplies with a negative integer, it always results to a negative integer. i.e.,

4 × (-5) = -20

Similarly,
(-3) × 4 = -12

 

Multiplication of two negative numbers:

When a negative integer multiplies with another negative integer, it always results to a positive integer. i.e.,

(-3) × (-5) = 15

Similarly,
(-4) × (-4) = 16

 

One positive vs two negatives:

When a positive integer multiplies with two negative integers, it always results to a positive integer. i.e.,

3 × (-4) × (-5) = ?

To do this task, first we need to separately calculate result of the two negatives, which would definitely a positive as we learnt above, i.e.,

3 × (-4) × (-5) = 3 × {(-4) × (-5)} = 3 × {20} = 3 × 20

Now it’s a form where the two positive integers multiply, that result to another positive i.e.,

= 60

 

One negative vs two negatives:

When a negative integer multiplies with two negative integers, it always results to a negative integer. i.e.,

(-3) × (-4) × (-5) = ?

Firstly, we need to solve the two negative integers, that would result to a positive one i.e.,

(-3) × (-4) × (-5) = (-3) × {(-4) × (-5)} = (-3) × {20} = (-3) × 20

Now, it becomes a case where one positive and one negative multiplies, that results to a negative.

(-3) × 20 = -60

 

Basic Rule of Multiplication:

Let’s consider a, b & c are numbers. Then,
a × b = b × a

And,

a × (b × c) = (a × b) × c

Also,

a × (-b) = – ab, in other words, if the two numbers ( one positive and other negative) multiply it results to a negative number, rather than positive.

(-a) × (-b) = ab; in other words, if the two negative numbers multiply it results to a positive number, rather than negative.

If a = b, then, a × c = b × c. In other words, if we multiply both sides of an equation with a number, Left hand side would remain equal to Right hand sight of the equation.

 

Division:

When we break a rod into two, we simply divide it into two, each of which resulted to half of the previous rod length.

FREE-basic-math-concept-02-image-03

 

Similarly, we use it in mathematical form as 12 or 1 ÷ 2, stated as ‘1 is divided by 2’, or simply ‘1 by 2’.

5 ÷ 10 is the same as 1 ÷ 2, because it can further be simplified as,

FREE-basic-math-concept-02-image-04

 
 

Similarly, 3 ÷ 15 can be simplified as 1 ÷ 5.

Important Note: When a number is divided by 0, the result can’t be find, i.e., undefined or infinity (∞).

Mathematically,

FREE-basic-math-concept-02-image-05

 
 

Division of fractions:

When, we divide ( 45 ) by ( 23 ), we should know, the point in mind that, If we multiply the value instead of dividing by the value, then the dividing value would be reversed.

FREE-basic-math-concept-02-image-06

 
 
 
 
 
 

Division of positive and negative combined:

Now let’s discuss what happens when the negative integers are entertained in process of division. For that purpose, we’ll discuss different aspects of negative integers and positive integers combined.

 

Division of positive and negative:

When a positive integer is divided by a negative integer, it always results to a negative number, i.e.,

5 ÷ (-2) = ?

In order to solve it, we need to convert in form of fraction, containing a nominator and a denominator, i.e.,

FREE-basic-math-concept-02-image-07

 
 
 
 
 
 
 
 
 
 

Division of two negatives:

When a negative integer is divided by a negative integer, it always results to a positive number, i.e.,
(-5) ÷ (-2) = ?
Again, in order to solve it, we need to convert in form of fraction, containing a nominator and a denominator, i.e.,

FREE-basic-math-concept-02-image-08

 
 
 

Basic rule of division:

Let’s consider a, b & c are numbers. Then,
a ÷ b ≠ b ÷ a
a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c

If a = b, then a ÷ c = b ÷ c. In other words, if we divide both side of an equation by a same number, the left hand side would remain equal to right hand side of the equation.

FREE-basic-math-concept-02-image-09

 
 
 
 
 
 
 
 
 
 
 
 
 
 

Multiplication and Division Combined:

Let’s learn how to solve expressions containing both multiplication & division. Let’s have a look.
When we need to solve:

FREE-basic-math-concept-02-image-10

 
 
 
 
 

So, we’ve learned that a dividing number can be written on any one side at a time below two numbers that are multiplying with each other. i.e., in above example, 25 & 12 are two numbers that are multiplying with each other. So in this case, 4 (which is dividing) can be written on either sides below the two numbers.

 

Important Note: When the two numbers are multiplying, and another number is dividing, then the dividing number can be placed either side of the two numbers that are multiplying. But, when the two numbers are adding or subtracting and the third number is dividing with any of the two numbers, we can’t write the dividing number below any of the other number, rather it will remain below the number to which it’s multiplying. In this case , we’ll take the L.C.M (Least Common Multiple), which we’ll discuss later in FREE Basic Math Concept-03.

For instance,

FREE-basic-math-concept-02-image-11

 
 
 
 
 
 
 

Now, let’s move on to more conceptual scenario:

(64 ÷ 4) × 28 ÷ 8 = ?

FREE-basic-math-concept-02-image-12

 
 
 
 

Important Note: If it comes multiplication and division altogether, Remember to follow left to right rule in the expression. Whatever comes first from left to right, use it first.

 

For instance, 4 ÷ 8 × 2 = ?
Here we need to follow Left-To-Right Rule, i.e., stat solving this from left and see what come first. As division comes first, so first, we have to divide 4 by 8 and then, multiply the product with 2. i.e.,

FREE-basic-math-image-16

 
 
 
 

In other case, we have to follow PEMDAS,
For instance, 5 + 5 × 5 = ?
If we solve using addition first rather than multiplication, we’ll get wrong answer. i.e.,

FREE-basic-math-image-17

 
 
 

Similarly,
5 + 5 ÷ 5 = ?

FREE-basic-math-image-18

 
 
 

Distribution and Factorizing:

When an integer is multiplies two or more values inside parenthesis such that the values inside parenthesis are adding or subtracting, the integer will multiply with both of the values. For instance,

4 × (3 + 2) = ?

Here, 4 is an integer which is multiplying with three values inside parenthesis. Here the values inside the parenthesis are doing addition operation, so 4 will multiply with each values inside parenthesis. i.e:

FREE-basic-math-concept-02-image-13

 
 
 
 

Alternatively, we can also add the values inside the parenthesis to get quick result if we follow PEMDAS rule. i.e solving parenthesis first.

4 × (3 + 2) = 4 (5) = 20

 

Similarly,

4 × (5 + 8 – 9) = 4 × 5 + 4 × 8 – 4 × 9 = 20 + 32 – 36

= 16

 

Alternatively,

4 × (4) = 16

 

When alternate way not helpful:

Alternative method cannot be used when values having different variables are involved in addition or subtraction process. i.e

4 × (5x + 8y – 9z) ≠ 4 × {(5 + 8 – 9)(x + yz)}                           (Many people who are weak in math commit this mistake)

 

Correct way: 4 × (5x + 8y – 9z) = 4 × 5x + 4 × 8y – 4 × 9z = 20x + 32y – 36z

 

Now, similarly, the reverse case is also true in case of addition or subtraction. i.e:

20 + 32 – 36 = ?

After taking 4 common from all the values,

4 (5 + 8 – 9) = 4 (4) = 16                             {As 20 + 32 – 36 = (4 × 5) + (4 × 8) – (4 × 9)}

 

Similarly,

20x + 32y – 36z = 4 (5x + 8y – 9z)

 

But,

4 × (3 × 5) ≠ (4 × 3) × (4 × 5)

As in above expression, 3 and 5 are no doing addition or subtraction operations, therefore 4 will multiply with either 3 or 5. i.e

Correct way: 4 × (3 × 5) = (4 × 3) × 5 = 3 × (4 × 5) = 60

 
 

Impact of removing parenthesis on values:

This FREE basic math concept is very important for arithmetic. Let’s understand this with examples based scenarios.

Let’s assume we need to simplify the following expression:

4x + (2x + 5y – 8z) = ?

In this expression, if we open the parenthesis, the signs will not change i.e

⇒ 4x + 2x + 5y – 8z

= 6x + 5y – 8z

 

But in other case,

Let’s assume the following expression whose simplified version we need to find:

4x – (2x + 5y – 8z) = ?

In the above expression, if we open the parenthesis, the sign will be changed such that positive sign will become negative, while negative sign will become positive. i.e:

⇒ 4x – 2x – 5y + 8z

= 2x – 5y + 8z

 

Point to ponder:

Note that the sign of any number is always adjacent to the left side of the number. Also note that when no sign is written to the left of a number, it is always +ve sign.

For instance, in above expression, the parenthesis starts with 2 (indicating no sign). This implies that the sign of 2 is positive. And when the parenthesis opens, the positive sign will changed to negative; because the parenthesis has negative sign to its left.

 
 

All Arithmetic Operations combined:

Let’s use all four arithmetic operations.
6 – 3 ÷ 2 + 1 × 5 = ?
According to the rule, we have to use division & multiplication first, and then use addition & subtraction. i.e.,
6 – 3 ÷ 2 + 1 × 5 = 6 – 1.5 + 5 = 4.5 + 5
= 9.5 Answer

 

Now, let’s move on next stage, by including parenthesis as well as exponents.

5 – 3(4 + 23) + 42 ÷ 22

By applying PEMDAS rule:

5 – 3(4 + 23) + 42 ÷ 22 = 5 – 3(4 + 8) + 16 ÷ 4

⇒ = 5 – 3(12) + 16 ÷ 4

⇒ = 5 – 36 + 4

= –27 Answer

 

Similarly,

–3 × 12 + 4 ÷ 8 – (4 – 6) = ?

By applying PEMDAS rule:

–3 × 12 + 4 ÷ 8 – (4 – 6) = –3 × 12 + 4 ÷ 8 – (–2)

⇒ = –36 + 0.5 + 2                                 (As opening parenthesis changes the sign of value inside parenthesis)

= –33.5 Answer

 
 

Now, you are ready to attempt a mini Quiz. Click on ‘Mini Quiz’ button below:


 


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Ayesha
Ayesha
May 23, 2017 8:41 am

Hi,
good work as usual!
I have a confusion please if you help me it will be great!
I am talking about the very last question -3*12+4*8-(4-6)=? in the solution, you have = -36+0.5+2 I am wonder how you got 0.5? I tried to get it but still i am unable. thanks!

Zubair Sattar
Zubair Sattar
January 19, 2018 11:45 am
Reply to  Ayesha

Its 4/8 not 4*8 i.e 1/2 or 0.5