WRITING AND EVALUATING EXPRESSIONS

In this section, you will learn how to write a mathematical expression from a verbal description and evaluate expressions for the given values of the variables or unknowns.  

Writing Expression from a Verbal Phrase 

If there is any unknown value in the given verbal phrase, replace it by some English alphabet.

Writing Expressions

In each case, write the given verbal phrase as mathematical expression.

Example 1 :

The sum of 5 times a number and 8

Solution :

5x + 8

Example 2 :

2 times the sum of 7 times a number and 4

Solution :

2(7x + 4)

Example 3 :

7 less than 4 times a number

Solution :

4x - 7

Example 4 :

One fifth of sum of 3 times a number and 9

Solution :

(3x + 9)/5

Example 5 :

One seventh of 3 less than 4 times a number

Solution :

(4x - 3)/7

Example 6 :

5 times of a number is decreased by 8

Solution :

5x - 8

Example 7 :

6 times of a number is increased by 15

Solution :

6x + 15

Example 8 :

7 less than 3 times the sum of a number and 6

Solution :

3(x + 6) - 7

Example 9 :

3 times the difference between a number and 8

Solution :

3(x - 8)

Example 10 :

One fourth of 3 times the difference between a number and 6

Solution :

3(x - 6)/4

Evaluating Expressions

Example 11-12 : Evaluate the numerical expression.

Example 11 :

12 + 7 x 10

Solution :

= 12 + 7 x 10

In the numerical expression above, we have two operations : addition and multiplication. According to PEMDAS Rule, we have to do multiplication first, then addition.

= 12 + 70

= 82

Example 12 :

9² - 48 ÷ 12

Solution :

In the given numerical  expression, we have more than two operations. Using PEMDAS Rule, we can carry out the operations one by one as shown below and evaluate.

Example 13-15 : Write a numerical expression for the given verbal phrase and evaluate.

Example 13 :

"The sum of 5 times 9 and 8"

Solution :

= 5x9 + 8

Using PEMDAS Rule,

= 45 + 8

= 53

Example 14 :

"1.5 times the sum of 7 times 4 and 2"

Solution :

= 1.5[7(4) + 2]

Using PEMDAS Rule,

= 1.5[28 + 2]

= 1.5 x 30

= 45

Example 15 :

"12 is multiplied by 3, the result is divided by 18 and 5 is added to the final answer"

Solution :

= [(12 x 3)/18] + 5

Using PEMDAS Rule,

= [36/18] + 5

= 2 + 5

= 7

Example 16 :

Evaluate the following expression for x = 3 and y = 5.

3x + 2y

Solution :

Substitute 3 for x and 5 for y into the given expression.

3x + 2y = 3(3) + 2(5)

= 9 + 10

= 19

Example 17 :

Evaluate the following expression for m = 5 and n = 2.

m² + 2mn²

Solution :

Substitute 5 for m and 2 for n into the given expression.

m² + 2mn² = 5² + 2(5)(2)²

= 25 + 40

= 65

Example 18 :

Evaluate the following expression for s = 5.

s² + 7s - 2

Solution :

Substitute 5 for s into the given expression.

s² + 7s - 2 = 5² + 7(5) - 2

= 25 + 35 - 2

= 58

Example 19 :

Evaluate the following expression for m = 1/3.

18m² + 3m + 7

Solution :

Substitute 1/3 for m into the given expression.

18m² + 3m + 7 = 18(1/3)² + 3(1/3) + 7

= 2 + 1 + 7

= 10

Example 20 :

Evaluate the following expression for m = 13.

m² + m - 54

Solution :

Substitute 13 for s into the given expression.

m² + m - 54 = 13² + 13 -54

= 169 + 13 - 54

= 128

Example 21 :

Evaluate the following expression for x = 3 and y = 5.

x² + y²

Solution :

Substitute 3 for x and 5 for y into the given expression.

x² + y² = 3² + 5²

= 9 + 25

= 34

Example 22 :

Evaluate the following expression for m = 5 and n = 2.

5m² + 2m²n

Solution :

Substitute 5 for m and 2 for n into the given expression.

5m² + 2m²n = 5(5)² + 2(5)²(2)

= 5(25) + 2(25)(2)

= 125 + 100

= 225

Example 23 :

Evaluate the following expression for s = 6.

s² - 3s + 10

Solution :

Substitute 6 for s into the given expression.

s² - 3s + 10 = 6² - 3(6) + 10

= 36 - 18 + 10

= 28

Example 24 :

Evaluate the following expression for x = 2 and y = -5.

x² - 3y²

Solution :

Substitute 2 for x and -5 for y into the given expression.

x² - 3y²  =  2(2)² - 3(-5)²

=  2(4) - 3(25)

= 8 - 75

= -77

Example 25 :

Evaluate the following expression for m = 3.

(2m² + 5m - 7)/2

Solution :

Substitute 3 for m into the given expression.

(2m² + 5m - 7)/2 = [2(3)² + 5(3) - 7]/2

= [2(9) + 15 - 7]/2

= (18 + 15 - 7)/2

= 26/2

= 13

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