State whether the following statement is true or false. Look at several examples of rational numbers in the form

p
q

(q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). If q has the form

2
a

×
5
b

,

p
q

is a terminating decimal.

Question

State whether the following statement is true or false. Look at several examples of rational numbers in the form

p
    q

(q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). If q has the form

2
    a
   
   ×
    5
    b

,

p
    q

is a terminating decimal.

Accepted Answer

A rational number is any number that can be represented in

p
    q

with q greater than 0. Real numbers cannot be represented in the form

p
    q

, where p and q are integers and q is not equal to 0 is an irrational number.If the rational numbers have only powers of 2 or 5, or both, then q must meet the requirement that the rational numbers always have an ending decimal expansion.We'll look at a few examples of rational integers whose decimal representations end in

p
    q

(q is not equal to 0).

2
      5
     
     =
      2

2
          0
         
         ×
          5
          1

=0.4

3
      
       100
     
     =
      3

2
          2
         
         ×
          5
          5

=0.03

27
      
       16
     
     =
      
       27

2
          4
         
         ×
          5
          0

=1.6875

33
      
       50
     
     =
      
       33

2
          1
         
         ×
          5
          2

=0.66

Let's look at the denominators of the rational numbers above.We can see that the denominators of the given rational numbers is of the form

2
    a
   
   ×
    5
    b

, with a and b being whole numbers.As a result, if q has the form

2
    a
   
   ×
    5
    b

,

p
    q

is a terminating decimal.Hence, it is true.Therefore, the correct option is 1.

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