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Fraction: Simplification of Expressions

what you'll learn...

overview

•  Numerical expression,

•  precedence order,

•  sequence order.

"express" means "say"

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

$2×4×3$$2 \times 4 \times 3$ is an example of a numerical expression. It is evaluated to $24$$24$.

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

examples

Is $3+4-2+1$$3 + 4 - 2 + 1$ a numerical expression?
"Yes, addition and subtraction can be part of an expression".

Is 3+4xx2-6-:3` a numerical expression?
"Yes, all arithmetic operations can be part of a numerical expression"

is $3$$3$ a numerical expression?
"Yes. technically a number can also be considered a numerical expression".

what order

Consider $1+2$$1 + 2$ and $3×1$$3 \times 1$. Note that when evaluated, both result in identical numerical value $1+2=3$$1 + 2 = 3$ and $3×1=3$$3 \times 1 = 3$.

They are two different expressions, evaluating to equal values.

Consider $9-6÷3$$9 - 6 \div 3$.

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

$9-6÷3$$9 - 6 \div 3$

$=9-2$$= 9 - 2$

$=7$$= 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.

what sequence?

Consider $20-4-3$$20 - 4 - 3$.

It is wrong to do $20-4-3$$20 - 4 - 3$ $\ne 20-1$$\ne 20 - 1$ $=19$$= 19$.

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

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

Consider $36÷6÷3$$36 \div 6 \div 3$

It is wrong to simplify as $36÷6÷3$$36 \div 6 \div 3$ $\ne 36÷2$$\ne 36 \div 2$ $=18$$= 18$

The correct order of simplification is $36÷6÷3$$36 \div 6 \div 3$ $=6÷3$$= 6 \div 3$ $=2$$= 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.

out of order / sequence

Consider $6÷3×2$$6 \div 3 \times 2$.

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

$6÷3×2$$6 \div 3 \times 2$

$=2×2$$= 2 \times 2$

$=4$$= 4$

Consider $6÷\left(3×2\right)$$6 \div \left(3 \times 2\right)$

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

$6÷\left(3×2\right)$$6 \div \left(3 \times 2\right)$

$=6÷6$$= 6 \div 6$

$=1$$= 1$

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

everything needed

In a numerical expression, the precedence order is:
•  Parentheses or Brackets are at the highest precedence order

•  division and multiplication are at the same level of precedence
•  addition and subtraction are at the 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.

examples

Simplify $1-\frac{1}{6}÷3$$1 - \frac{1}{6} \div 3$
The answer is "$\frac{17}{18}$$\frac{17}{18}$". Division has higher precedence over subtraction. So

$1-\frac{1}{6}÷3$$1 - \frac{1}{6} \div 3$

$=1-\frac{1}{18}$$= 1 - \frac{1}{18}$

$=\frac{17}{18}$$= \frac{17}{18}$

Simplify $6÷3×\left(\frac{1}{2}\right)$$6 \div 3 \times \left(\frac{1}{2}\right)$
The answer is "$1$$1$".

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

$6÷3×\left(\frac{1}{2}\right)$$6 \div 3 \times \left(\frac{1}{2}\right)$

$=2×\left(\frac{1}{2}\right)$$= 2 \times \left(\frac{1}{2}\right)$

$=1$$= 1$

Simplify $6÷\left(3×\frac{1}{2}\right)$$6 \div \left(3 \times \frac{1}{2}\right)$
The answer is "$4$$4$".

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

$6÷\left(3×\frac{1}{2}\right)$$6 \div \left(3 \times \frac{1}{2}\right)$

$=6÷\left(\frac{3}{2}\right)$$= 6 \div \left(\frac{3}{2}\right)$

$=4$$= 4$

Simplify $4+\left(-\frac{2}{3}\right)÷\left(\frac{1}{3}\right)×\left(-2\right)$$4 + \left(- \frac{2}{3}\right) \div \left(\frac{1}{3}\right) \times \left(- 2\right)$
The answer is "$8$$8$".

The division and multiplication are of higher precedence over addition. so $\left(-\frac{2}{3}\right)÷\left(\frac{1}{3}\right)×\left(-2\right)$$\left(- \frac{2}{3}\right) \div \left(\frac{1}{3}\right) \times \left(- 2\right)$ is to be simplified first.

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

$4+\left(-\frac{2}{3}\right)÷\left(\frac{1}{3}\right)×\left(-2\right)$$4 + \left(- \frac{2}{3}\right) \div \left(\frac{1}{3}\right) \times \left(- 2\right)$

$=4+\left(-2\right)×\left(-2\right)$$= 4 + \left(- 2\right) \times \left(- 2\right)$

$=4+4$$= 4 + 4$

$=8$$= 8$.

Simplify $\frac{10}{7}-\left(-\frac{1}{7}\right)-2×\left(-\frac{3}{14}\right)$$\frac{10}{7} - \left(- \frac{1}{7}\right) - 2 \times \left(- \frac{3}{14}\right)$.
The answer is "$2$$2$". The multiplication is of higher precedence over subtraction and so $2×\left(-\frac{3}{14}\right)$$2 \times \left(- \frac{3}{14}\right)$ is simplified first. Then the two subtraction are in the same precedence level and so they are simplified in the left to right sequence.

$\frac{10}{7}-\left(-\frac{1}{7}\right)-2×\left(-\frac{3}{14}\right)$$\frac{10}{7} - \left(- \frac{1}{7}\right) - 2 \times \left(- \frac{3}{14}\right)$

$=\frac{10}{7}-\left(-\frac{1}{7}\right)-\left(-\frac{3}{7}\right)$$= \frac{10}{7} - \left(- \frac{1}{7}\right) - \left(- \frac{3}{7}\right)$

$=\frac{11}{7}-\left(-\frac{3}{7}\right)$$= \frac{11}{7} - \left(- \frac{3}{7}\right)$

$=\frac{14}{7}$$= \frac{14}{7}$

$=2$$= 2$

Simplify $\left(-1+\frac{3}{2}-1\right)×2+\frac{-3}{2}-\frac{2}{4}$$\left(- 1 + \frac{3}{2} - 1\right) \times 2 + \frac{- 3}{2} - \frac{2}{4}$
The answer is "$-3$$- 3$"

$\left(-1+\frac{3}{2}-1\right)×2+\frac{-3}{2}-\frac{2}{4}$$\left(- 1 + \frac{3}{2} - 1\right) \times 2 + \frac{- 3}{2} - \frac{2}{4}$

$=\left(-\frac{2}{2}+\frac{3}{2}-\frac{2}{2}\right)×2+\frac{-3}{2}-\frac{1}{2}$$= \left(- \frac{2}{2} + \frac{3}{2} - \frac{2}{2}\right) \times 2 + \frac{- 3}{2} - \frac{1}{2}$

$=-\frac{1}{2}×2+\frac{-3}{2}-\frac{1}{2}$$= - \frac{1}{2} \times 2 + \frac{- 3}{2} - \frac{1}{2}$

$=\frac{-2}{2}+\frac{-3}{2}-\frac{1}{2}$$= \frac{- 2}{2} + \frac{- 3}{2} - \frac{1}{2}$

$=\frac{-5}{2}-\frac{1}{2}$$= \frac{- 5}{2} - \frac{1}{2}$

$=\frac{-6}{2}$$= \frac{- 6}{2}$

$=-3$$= - 3$

What is the value of $-3-\left(\left(-\frac{1}{3}\right)-2×\left(\frac{1}{2}\right)\right)$$- 3 - \left(\left(- \frac{1}{3}\right) - 2 \times \left(\frac{1}{2}\right)\right)$?
The answer is "$-\frac{5}{3}$$- \frac{5}{3}$".

$-3-\left(\left(-\frac{1}{3}\right)-2×\left(\frac{1}{2}\right)\right)$$- 3 - \left(\left(- \frac{1}{3}\right) - 2 \times \left(\frac{1}{2}\right)\right)$

brackets take precedence and within the brackets, the multiplication is higher precedence.
$=-3-\left(\left(-\frac{1}{3}\right)-1\right)$$= - 3 - \left(\left(- \frac{1}{3}\right) - 1\right)$

brackets have higher precedence
$=-3-\left(-\frac{4}{3}\right)$$= - 3 - \left(- \frac{4}{3}\right)$

$=-\frac{9}{3}-\left(-\frac{4}{3}\right)$$= - \frac{9}{3} - \left(- \frac{4}{3}\right)$

simplifying
$=-\frac{5}{3}$$= - \frac{5}{3}$

What is the value of $-\frac{3}{2}-\left(-\frac{1}{3}\right)-\frac{1}{4}$$- \frac{3}{2} - \left(- \frac{1}{3}\right) - \frac{1}{4}$?
The answer is "$-\frac{17}{12}$$- \frac{17}{12}$".
$-\frac{3}{2}-\left(-\frac{1}{3}\right)-\frac{1}{4}$$- \frac{3}{2} - \left(- \frac{1}{3}\right) - \frac{1}{4}$

LCM of $2,3,4$$2 , 3 , 4$ is $12$$12$
$=-\frac{18}{12}-\left(-\frac{4}{12}\right)-\frac{3}{12}$$= - \frac{18}{12} - \left(- \frac{4}{12}\right) - \frac{3}{12}$

left to right sequence
$=-\frac{14}{12}-\frac{3}{12}$$= - \frac{14}{12} - \frac{3}{12}$

$=-\frac{17}{12}$$= - \frac{17}{12}$

Simplify $\frac{4}{5}+\frac{3}{4}÷\left(-\frac{3}{2}\right)$$\frac{4}{5} + \frac{3}{4} \div \left(- \frac{3}{2}\right)$
The answer is "$\frac{3}{10}$$\frac{3}{10}$".

$\frac{4}{5}+\frac{3}{4}÷\left(-\frac{3}{2}\right)$$\frac{4}{5} + \frac{3}{4} \div \left(- \frac{3}{2}\right)$

The division is of higher precedence over addition.
$=\frac{4}{5}+\left(-\frac{1}{2}\right)$$= \frac{4}{5} + \left(- \frac{1}{2}\right)$

$=\frac{8}{10}+\frac{-5}{10}$$= \frac{8}{10} + \frac{- 5}{10}$

$=\frac{3}{10}$$= \frac{3}{10}$

Simplify $\left(10-3-2\right)×\left(-\frac{1}{5}\right)$$\left(10 - 3 - 2\right) \times \left(- \frac{1}{5}\right)$.
The answer is "$-1$$- 1$".

$\left(10-3-2\right)×\left(-\frac{1}{5}\right)$$\left(10 - 3 - 2\right) \times \left(- \frac{1}{5}\right)$

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.
$=\left(7-2\right)×\left(-\frac{1}{5}\right)$$= \left(7 - 2\right) \times \left(- \frac{1}{5}\right)$
$=5×\left(-\frac{1}{5}\right)$$= 5 \times \left(- \frac{1}{5}\right)$

$=-1$$= - 1$

summary

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