maths > fractions

Fractions : Precedence Order PEMA / BOMA

what you'll learn...

overview

This page redefines the precedence order in arithmetics with PEMA or BOMA. As part of fractions, the revised version is in handling division -- handle division as inverse of multiplication.

recap

Numerical expressions of fractions are simplified with the following rules.

•  Precedence order BODMAS / PEMDAS

•  Left to Right sequence for same precedence

The sequence order specified is not desirable. If the sequence order is to be strictly followed, then some problems require unnecessary computation.

$3+4-3$$3 + 4 - 3$ has to be simplified to $=7-3=4$$= 7 - 3 = 4$, which involves two steps.

$4×7÷4$$4 \times 7 \div 4$ has to be simplified to $28÷4=7$$28 \div 4 = 7$, which involve two steps.

As part of studies on integers, the expressions involving subtraction are analyzed and the rules of simplification are redefined. A subtraction is modified to addition of inverse of subtrahend. This was explained in the course on integers.

As part of studies on fractions, the expressions involving division are analyzed and the rules of simplification are redefined.

division redefined

Numerical expressions involving division are simplified with the redefined rule

•  division is the inverse of multiplication. And, all divisions are converted to multiplication of reciprocal of divisor.

•  Expression is simplified using precedence order PEMA. That is, Parentheses, Exponents (which are not introduced yet), Multiplication, and Addition.

Note, the precedence order PEMA does not have subtraction and division, as

•  the divisions are converted into multiplication of multiplicative inverse.

The advantage of this is explained in this simple example.

$3×4÷3$$3 \times 4 \div 3$

convert the division to multiplication.
$=3×4×\left(\frac{1}{3}\right)$$= 3 \times 4 \times \left(\frac{1}{3}\right)$

No need to follow the sequence order. Since all are multiplication, add any number to any other in the sequence that suits.
$=3×\left(\frac{1}{3}\right)×4$$= 3 \times \left(\frac{1}{3}\right) \times 4$

$=1×4$$= 1 \times 4$

$=4$$= 4$

example

Simplify $32÷11×44÷4$$32 \div 11 \times 44 \div 4$.

Note that it has $44$$44$ which equals product of $11$$11$ and $4$$4$.

This can be solved in two ways

First : $32÷11×44÷4$$32 \div 11 \times 44 \div 4$ $=\frac{32}{11}×44÷4$$= \frac{32}{11} \times 44 \div 4$ $=\frac{1408}{11}÷4$$= \frac{1408}{11} \div 4$ $=128÷4$$= 128 \div 4$ $=32$$= 32$

Second : $32÷11×44÷4$$32 \div 11 \times 44 \div 4$ $=32×\left(\frac{1}{11}\right)×44×\left(\frac{1}{4}\right)$$= 32 \times \left(\frac{1}{11}\right) \times 44 \times \left(\frac{1}{4}\right)$ $=32×\left(\frac{1}{11}\right)×11$$= 32 \times \left(\frac{1}{11}\right) \times 11$ $=32$$= 32$

Note that the flexibility to change the order of operation can be an advantage in calculations.

change starts with you

There are good reasons to adapt to PEMA. Students at 6th or 7th grade level may skip this explanation.

When learning properties of numbers, we learn the commutative property of multiplication. $a×b=b×a$$a \times b = b \times a$. But the division does not follow the commutative property. What this means is, the order of arithmetic operations cannot be modified to simplify the problem. The redefined precedence order PEMA converts the division into multiplication and allows commutative property to be used to ones advantage. $a÷b=a×\left(\frac{1}{b}\right)=\frac{1}{b}×a$$a \div b = a \times \left(\frac{1}{b}\right) = \frac{1}{b} \times a$

Another property of numbers is associative property of multiplication. $a×\left(b×c\right)=\left(a×b\right)×c$$a \times \left(b \times c\right) = \left(a \times b\right) \times c$. But the division does not follow the associative property. What this means is, the sequence of arithmetic operations cannot be modified to simplify the problem. The redefined precedence order PEMA converts the division into multiplication and allows associative property to be used to ones advantage. $a÷\left(b÷c\right)$$a \div \left(b \div c\right)$$=a×\frac{1}{b×\left(1/c\right)}$$= a \times \frac{1}{b \times \left(1 / c\right)}$$=a×\left(\left(\frac{1}{b}\right)×c\right)$$= a \times \left(\left(\frac{1}{b}\right) \times c\right)$$=\left(a×\frac{1}{b}\right)×c$$= \left(a \times \frac{1}{b}\right) \times c$.

change for algebra

There are good reasons in algebra to adapt to PEMA. Students at 6th or 7th grade level may skip this explanation.

Algebra is extensively based on the numerical expressions and the properties of numerical arithmetic. In an algebraic expression of multiple factors, with division in some of them, simplification is possible only if the sequence order is not required to be followed. For example, $\left(x+2\right)÷\left(x+4\right)×\left(x+4\right)$$\left(x + 2\right) \div \left(x + 4\right) \times \left(x + 4\right)$$×\left(x-8\right)÷\left(x+2\right)$$\times \left(x - 8\right) \div \left(x + 2\right)$. This expression is intuitively understood to be $\left(x+2\right)×\frac{1}{\left(x+4\right)}$$\left(x + 2\right) \times \frac{1}{\left(x + 4\right)}$$×\left(x+4\right)×8×\frac{1}{\left(x+2\right)}$$\times \left(x + 4\right) \times 8 \times \frac{1}{\left(x + 2\right)}$. Then it can be simplified into $\left(x+2\right)×\frac{1}{\left(x+2\right)}×$$\left(x + 2\right) \times \frac{1}{\left(x + 2\right)} \times$$\frac{1}{\left(x+4\right)}×\left(x+4\right)×\left(x-8\right)$$\frac{1}{\left(x + 4\right)} \times \left(x + 4\right) \times \left(x - 8\right)$ which equals, $\left(x-8\right)$$\left(x - 8\right)$.

summary

Numerical Arithmetics Precedence Order : PEMA / BOMA
PEMA = Parentheses, Exponents, Multiplication and Addition

BOMA = Brackets, Order, Multiplication and Addition

Note : Division is converted into multiplication by multiplicative inverse of divisor. And, Subtraction is converted into addition withs additive inverse of subtrahend.

Outline