Order of Operations

While solving the questions on order of operations we follow certain rules that indicate the sequence for simplifying expressions that contain more than one fundamental operation.

Steps to solve order of operations:

Step I: Simplify the operations inside grouping symbols.

Step II: Simplify the powers.

Step III: Solve multiplication and division from left to right.

Step IV: Solve addition and subtraction from left to right.

• In simplifying an expression, all the brackets must be removed first in the order and the grouping symbols are parentheses ( ), brackets [ ], braces or curly brackets { }.

• The other grouping symbols include fraction bars, radical symbols, and absolute-value symbols.

• When we simplify expressions involving more than one grouping symbol, first we need to simplify the innermost set. Within each set, then follow the fundamental order of operations.

• Symbols of grouping can be used when translating the expressions from words to math.

The product of 11 and the sum of 5, 8 and 13 is written as 11(5 + 8 + 13).

A. Worked-out problems on simplifying numerical expressions:

1. Evaluate the expression:

(i) 27 ÷ 3² + 4 · 2 – 1

Solution:

27 ÷ 3² + 4 · 2 – 1

= 27 ÷ 9 + 4 · 2 - 1

= 3 + 4 · 2 - 1

= 3 + 8 - 1

= 11 - 1

= 10

Evaluate powers.

Divide 27 by 9.

Multiply 4 by 2.

Add 3 and 8.

Subtract 1 from 11.

(ii) 27 - [5 + {28 - (29 - 7}]

Solution:

27 - [5 + {28 - (29 - 7}]

= 27 – [5 + {28 – 22}]

= 27 – [5 + 6]

= 27 – 11

= 16

Removing the parenthesis. Subtract 7 from 29.

Removing the curly brackets. Subtract 22 from 28.

Removing the brackets. Add 5 and 6.

Subtract 11 from 27.

B. Worked-out problems on grouping symbols:

Evaluate each expression:

(i) 6(12 - 8) - 3(3 + 1)

Solution:

6(12 - 8) - 3(3 + 1)

= 6(4) - 3(4)

= 24 - 12

= 12

Evaluate inside grouping symbols.

Multiply expressions left to right.

Subtract 12 from 24.

(ii) 4[(24 ÷ 3) - (3 + 2)]

Solution:

4[(24 ÷ 3) - (3 + 2)]

= 4[(8) - (5)]

= 4[3]

= 12

Evaluate innermost expression first.

Evaluate expression in grouping symbol.

Multiply.

(iii) (2⁵ ÷ 2)/(15 - 2³)

Solution:

(2⁵ ÷ 2)/(15 - 2³) means (2⁵ ÷ 2) ÷ (15 - 2³).

(2⁵ ÷ 2)/(15 – 2³)

= (32 ÷ 2)/(15 – 2³)

= 16/(15 – 2³)

= 16/(15 – 8)

= 16/7

Evaluate the power in the numerator.

Divide 32 by 2 in the numerator.

Evaluate the power in the denominator.

Subtract 8 from 15 in the denominator.

C. Worked-out problems on evaluating an algebraic expression:

Evaluate: (a² – 2cb) + b³ if a = 8, b = 3 and c = 5.

Solution:

(a² – 2cb) + b³

= (8² - 2 · 5 · 3) + 3³

= (64 – 2 · 5 · 3) + 3³

= (64 - 30) + 3³

= (34) + 3³

= 34 + 27

= 61

Replace a with 8, b with 3 & c with 5.

Evaluate 8².

Multiply 2, 5, and 3.

Subtract 30 from 64.

Evaluate 33.

Add 34 and 27.

D. Real-life word problem using algebraic expressions:

Ron has $600 to invest for 5 years. She finds a bank that will invest her money at a simple interest rate of 5%. Interest I is equal to the principle P (amount invested) times the product of the rate r as a decimal and the time t in years.

a. Write an expression that represents simple interest.

Words

Variables

Expression

principle

P = Principle,

times

t = time,

the product of rate and time

r = rate

(r × t)

b. Find the amount of interest earned after 5 years.

Evaluate Prt for P = 600, r = 0.05, and t = 5.

Prt = 600(0.05)(5)