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mathsFractionsNumerical Expressions with Fractions

### Fraction: Simplification of Expressions

In this page, handling of fractions, both positive and negative fractions, in numerical expressions is explained.

The precedence order PEMDAS / BODMAS is explained.

For operations of same precedence order, the sequence of operation "simplification from left to right" is explained.

click on the content to continue..

Let us quickly revise what is numerical expression, and precedence order, sequence in simplifying the numerical expressions. This was introduced in whole numbers and reviewed in integers too. It takes very little time to revise.

Consider 2+4+3. This can be evaluated to 9.

This is an example of a numerical expression.

The word "expression" means: collection of numbers and arithmetic operations between them, which together represent a quantity.

2xx4xx3 is an example of a numerical expression. It is evaluated to 24.

a+3x is not a numerical expression. It is not entirely numbers and arithmetic operations. It has letters a and x.

Solved Exercise Problem:

Is 3+4-2+1 a numerical expression?

• No, it has both addition and subtraction
• Yes, addition and subtraction can be part of a numerical expression
• Yes, addition and subtraction can be part of a numerical expression

The answer is "Yes, addition and subtraction can be part of an expression".

Solved Exercise Problem:

Is 3+4xx2-6-:3 a numerical expression?

• No, as it has the addition, subtraction, multiplication, division
• Yes, all arithmetic operations can be part of a numerical expression
• Yes, all arithmetic operations can be part of a numerical expression

The answer is "Yes, all arithmetic operations can be part of a numerical expression"

Solved Exercise Problem:

is 3 a numerical expression?

• No. 3 is only a number. Not a numerical expression
• Yes. technically a number can also be considered a numerical expression
• Yes. technically a number can also be considered a numerical expression

The answer is "Yes. technically a number can also be considered a numerical expression".

Consider 1+2 and 3xx1. Note that when evaluated, both result in identical numerical value 1+2=3 and 3xx1 = 3.

They are two different expressions, evaluating to equal values.

Consider 9-6-:3.

To simplify the expression, the division is performed first. It is

9-6-:3

=9-2

=7

Division has higher precedence over subtraction.

In a numerical expression the precedence order is given as :

•  Brackets / Parentheses

•  Order / Exponent

•  Division and Multiplication

This is abbreviated as BODMAS or PEMDAS.

Consider 20-4-3.

It is wrong to do 20-4-3 !=20-1 =19.

The correct order of simplification is 20-4-3 = 16-3 =13

The two subtractions are in the same precedence level. This is to be handled from left to right sequence.

Consider 36-:6-:3

It is wrong to simplify as 36-:6-:3 !=36-:2 =18

The correct order of simplification is 36-:6-:3 =6 -:3 = 2.

The two divisions are in same precedence level. This is to be handled from left to right sequence. .

Rule of sequence is, when multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence.

Consider 6-:3xx2.

The division and multiplication are of same precedence, so it is simplified from left to right.

6-:3xx2

=2xx2

=4

Consider 6-:(3xx2)

The bracket has higher precedence, and so the expression inside bracket is simplified first.

6-:(3xx2)

=6-:6

=1

The rule of brackets or parentheses in numerical expression is, the subexpression within a bracket or parentheses has the highest precedence

In a numerical expression, the precedence order is:
•  division and multiplication in same level same level of precedence
•  addition and subtraction in same level of precedence.

This is abbreviated as BODMAS (Division, Multiplication, Addition, Subtraction) or PEMDAS (Multiplication, Division, Addition, Subtraction).

When multiple operation of same precedence is to be simplified, the operations are performed from left to right sequence.

All these were studied as part of whole numbers and integers. The same applies for fractions.

•  Precedence order BODMAS / PEMDAS

•  Left to Right sequence for same precedence

Let us see some more expressions with fractions.

Solved Exercise Problem:

Simplify 1- 1/6 -:3

• 17/18
• 17/18
• 1/2

The answer is "17/18". Division has higher precedence over subtraction. So

1-1/6-:3

=1-1/18

=17/18

Solved Exercise Problem:

Simplify 6-:3xx(1/2)

• 4
• 1
• 1

The answer is "1".

The division and multiplication are of same precedence, so it is simplified from left to right.

6-:3xx(1/2)

=2xx(1/2)

=1

Solved Exercise Problem:

Simplify 6-:(3xx1/2)

• 4
• 4
• 1

The answer is "4".

The bracket has higher precedence, and so the expression inside bracket is simplified first.

6-:(3xx1/2)

=6-:(3/2)

=4

Solved Exercise Problem:

Simplify 4+(-2/3)-:(1/3)xx(-2)

• 8
• 8
• 5

The answer is "8".

The division and multiplication are of higher precedence over addition. so (-2/3)-:(1/3)xx(-2) is to be simplified first.

In that, the division and multiplication are of same precedence, so it is simplified from left to right.

4+(-2/3)-:(1/3)xx(-2)

=4+(-2)xx(-2)

=4+4

=8.

Solved Exercise Problem:

Simplify 10/7-(-1/7)-2xx(-3/14).

• 4/7
• 2
• 2

The answer is "2". The multiplication is of higher precedence over subtraction and so 2xx(-3/14) is simplified first. Then the two subtraction are in the same precedence level and so they are simplified in the left to right sequence.

10/7-(-1/7)-2xx(-3/14)

=10/7-(-1/7)-(-3/7)

=11/7-(-3/7)

=14/7

=2

Solved Exercise Problem:

Simplify (-1+3/2-1)xx2+(-3)/2-2/4

• -3
• -3
• -1

The answer is "-3"

(-1+3/2-1)xx2+(-3)/2-2/4

=(-2/2 + 3/2 -2/2)xx2+(-3)/2-1/2

=-1/2 xx 2 +(-3)/2-1/2

=(-2)/2 + (-3)/2-1/2

=(-5)/2 -1/2

=(-6)/2

=-3

Solved Exercise Problem:

What is the value of -3-((-1/3)-2xx(1/2))?

• 13/5
• -5/3
• -5/3

The answer is "-5/3".

-3-((-1/3)-2xx(1/2))

brackets take precedence and within the brackets, the multiplication is higher precedence.
=-3-((-1/3)-1)

brackets have higher precedence
=-3-(-4/3)

=-9/3-(-4/3)

simplifying
=-5/3

Solved Exercise Problem:

What is the value of -3/2-(-1/3)-1/4?

• 19/12
• -17/12
• -17/12

The answer is "-17/12".
-3/2-(-1/3)-1/4

LCM of 2, 3, 4 is 12
=-18/12 - (-4/12) - 3/12

left to right sequence
=-14/12-3/12

=-17/12

Solved Exercise Problem:

Simplify 4/5+3/4-:(-3/2)

• -3/10
• -3/10
• 3/10

The answer is "3/10".

4/5+3/4-:(-3/2)

The division is of higher precedence over addition.
=4/5+(-1/2)

=8/10 + (-5)/10

=3/10

Solved Exercise Problem:

Simplify (10-3-2)xx(-1/5).

• -1
• -1
• -15

The answer is "-1".

(10-3-2)xx(-1/5)

The expression inside bracket is simplified first. The two subtraction are in the same precedence level and so they are simplified in the left to right sequence.
=(7-2)xx(-1/5)
=5xx(-1/5)

=-1`

Simplification of Expressions : BODMAS

•  B - Brackets

•  O - Order (exponents, roots, logarithm)

•  D - Division

•  M - Multiplication

•  S - Subtraction

•  And Left to Right sequence for multiple operations of same precedence.

PEMDAS

•  P - Parentheses

•  E - Exponents (roots and logarithm)

•  M - Multiplication

•  D - Division