PROBLEM SOLVING USING ORDER OF OPERATIONS

When two or more operations are carried out, which can be performed first.

To avoid this confusion, our mathematicians has found some set of rules followed in order of operations.

Example 1 :

Evaluate :

35 - 10 ÷ 2 x 5 + 3

Solution :

35 - 10 ÷ 2 x 5 + 3

We don't have brackets or parenthesis in this problem.

= 35 - 10 ÷ 2 x 5 + 3 (Division)

= 35 - 5 x 5 + 3 (Multiplication)

= 35 - 25 + 3 (Subtraction)

= 10 + 3 (Addition)

= 13

Example 2 :

Evaluate :

2(3 x 6 - 4) + 7

Solution :

2(3 x 6 - 4) + 7

= 2(3 x 6 - 4) + 7

(Inside the bracket performing multiplication)

= 2(18 - 4) + 7 

(Inside the bracket performing subtraction)

= 2 x 14 + 7 (Multiplication)

= 28 + 7

= 35

Example 3 :

Evaluate :

5 + 4 x 7 + 27 ÷ 9

Solution :

= 5+ 4 x 7 + 27 ÷ 9 (Multiplication comes first)

= 5 + 28 + 27 ÷ 9 (Division)

= 5 + 28 + 3 (Addition)

= 36

Example 4 :

Evaluate :

4 x 3² - (3 + 2)²

Solution :

= 4 x 3² - (3 + 2)² (Bracket)

= 4 x 3²  - (5)²  (Exponents)

= 4 x 9 -  25 (Multiplication)

= 36 - 25 (Subtraction)

= 11

Example 5 :

Evaluate :

(7-3 x 2) ÷ (8 ÷ 4 - 1)

Solution :

= (7- 3 x 2) ÷(8 ÷ 4 - 1)

= (7 - 6) ÷ (8 ÷ 4 - 1)

= (7 - 6) ÷ (2 - 1)

= 1 ÷ 1

= 1

Example 6 :

Evaluate :

19 - [{3 x 7} - {9 ÷ 3}] + 14

Solution :

= 19 - [(3 x 7) - (9 ÷ 3)] + 14

= 19 - [21 - (9 ÷ 3)] + 14

= 19 - [21 - 3] + 14

= 19 - 18 + 14

= 19 - 18 + 14

= 15

Example 7 :

Evaluate :

4 x [(4 x 3) ÷ 2] x 7

Solution :

= 4 x [(4 x 3) ÷ 2] x 7

= 4 x [12 ÷ 2] x 7

= 4 x 6 x 7

= 168

Example 8 :

Evaluate :

5 + [6 + (7 x 2)] ÷ 5

Solution :

= 5 + [6 + (7 x 2)] ÷ 5

= 5 + [6 + 14] ÷ 5

= 5 + 20 ÷ 5

= 5 + 4

= 9

Example 9 :

Evaluate :

5 x 2² + 2 x 3²

Solution :

= 5 x 2² + 2 x 3²

=  5 x 4 + 2 x 9

= 20 + 18

=  38

Example 10 :

Evaluate :

3 - 2² ÷ 2 + 1

Solution :

= 3 - 2² ÷ 2 + 1

= 3 - 4 ÷ 2 + 1

= 3 - 2 + 1

= 1 + 1

= 2

Example 11 :

You buy 2.6 pounds of apples and 1.475 pounds of peaches. You hand the cashier a $20 bill. How much change will you receive?

Solution :

Quantity of apples you have purchased = 2.6 pounds

Quantity of peaches purchased = 1.475 pounds

Cost of 1 pound of apple = $1.23

Cost of 1 pound of peaches = $1.88

Amount as change = 20 - [2.6(1.23) + 1.475(1.88)]

= 20 - [3.198 + 2.773]

= 20 - 5.971

= 14.029

So, she get the change $14.029

Example 12 :

A car can travel 22.36 miles on one gallon of gasoline

a) How far can the car travel on 8.5 gallons of gasoline?

b) A hybrid car can travel 33.1 miles on one gallon of gasoline.

How much farther can the hybrid car travel on 8.5 gallons of gasoline?

Solution :

Difference in distance = 8.5[33.1 - 22.36]

Doing the subtraction first,

= 8.5(10.74)

= 91.29

So, hybrid car has covered 91.29 more miles comparing the car.

Example 13 :

Which of the following expressions is equivalent to a perfect square?

A. 3 + 22 × 7         B. (80 + 4) ÷ 4

C. 34 + 18 ÷ 32      D. 32 + 6 × 5 ÷ 3

Solution :

Option A :

= 3 + 22 × 7

Performing the multiplication first,

= 3 + 154

= 157

Is not a perfect square.

Option B :

= (80 + 4) ÷ 4

Performing the bracket first

= 84 ÷ 4

Performing division,

= 21

Is not a perfect square.

Option C :

= 34 + 18 ÷ 3²

Performing the exponent first,

= 34 + (18 / 9)

= 34 + 2

= 36

Is a perfect square.

Option D :

= 3² + 6 × 5 ÷ 3

Performing the exponent first,

= 9 + 6 × 5 ÷ 3

= 9 + 30 ÷ 3

= 9 + 10

= 19

It is not a perfect square.

So, option C is correct.

Example 14 :

Which number is equivalent to the expression below ?

2 x 4² + 3(6 ÷ 2)

A)   25      B)  73     C)  41      D)  105

Solution :

= 2 x 4² + 3(6 ÷ 2)

= 2 x 16 + 3(3)

= 32 + 9

= 41

So, option C is correct.

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