π Rational & Irrational Numbers
Terminating Decimals Β· Repeating Decimals Β· Self-Study Worksheet
Middle School Math Β· 20 Questions
π Chapter Notes
Rational numbers are numbers that can be written as a fraction
pq
where p and q are integers, and q β 0.
Every rational number is either a terminating decimal (ends) or a
repeating decimal (loops forever).
TERMINATE = STOP β only 2s & 5s in denominator
If the denominator (in simplest form) has only factors of 2 and/or 5 β terminating decimal β
Numbers like Ο (pi) and β2 are irrational β they are non-terminating AND non-repeating. They cannot be written as fractions.
IRRATIONAL = "I RATIO-nally can't be written as a fraction!"
Ο, β2, β3, β5 β all irrational. β4 = 2 β rational β
π How to Check: Terminating?
π STEP-BY-STEP METHOD
1
Reduce the fraction to simplest form (find GCD).2
Look at the denominator only.3
Factor the denominator into primes.4
If factors are ONLY 2s and/or 5s β Terminating β
Otherwise β Repeating β
pq
terminates βΊ q = 2m Γ 5n
(m, n β₯ 0)
βοΈ EXAMPLE
380
= 32β΄ Γ 5
β denominator = 2β΄ Γ 5ΒΉ β only 2s & 5s β Terminating β
380 Γ 5Β³5Β³ = 37510000 = 0.0375
380 Γ 5Β³5Β³ = 37510000 = 0.0375
βοΈ Questions β Let's Practice!
Choose the correct answer. Read explanations carefully when wrong!
1
β Very Easy
π± Concept Check
Which of the following is an irrational number?
2
β Very Easy
π± Concept Check
A terminating decimal is a decimal thatβ¦
3
β Very Easy
π± Identify
Is β4 rational or irrational?
4
ββ Easy
πΏ Denominator Check
Which denominator (already in simplest form) gives a terminating decimal?
5
ββ Easy
πΏ Convert
What is 14 as a decimal?
6
ββ Easy
πΏ Identify Terminating
Which fraction is a terminating decimal?
7
ββ Easy
πΏ Simplify First!
Careful! Does 612 give a terminating decimal?
(Hint: simplify first!)
(Hint: simplify first!)
8
βββ Medium
π³ Multiple Choice Select
From the list below, which ONE fraction is NOT a terminating decimal?
8/12, 3/25, 7/14, 5/60
8/12, 3/25, 7/14, 5/60
9
βββ Medium
π³ Fill the blank
380
= 32β΄ Γ 5
Γ 5Β³5Β³
= 375?
= 0.0375
What goes in the ??
What goes in the ??
10
βββ Medium
π³ Find a/18
a18
is a terminating decimal. Which value of a makes this work?
18 = 2 Γ 3Β² β what must a cancel?
18 = 2 Γ 3Β² β what must a cancel?
11
βββ Medium
π³ Tricky Denominator
21140
β is this a terminating decimal?
Remember: simplify first! 140 = 2Β² Γ 5 Γ 7
Remember: simplify first! 140 = 2Β² Γ 5 Γ 7
12
βββ Medium
π³ Classic Trap
9450
β terminating or repeating?
450 = 2 Γ 3Β² Γ 5Β²
450 = 2 Γ 3Β² Γ 5Β²
13
ββββ Hard
π² 18/a terminating
18a
can be expressed as a terminating decimal.
Which of the following is NOT a possible value of a?
Which of the following is NOT a possible value of a?
14
ββββ Hard
π² Both fractions terminate
Both a22
and a30
are terminating decimals.
What is the smallest positive integer value of a?
What is the smallest positive integer value of a?
15
ββββ Hard
π² Count terminating fractions
How many fractions of the form n30
(where 1 β€ n β€ 30, n is a natural number) are terminating decimals?
16
ββββ Hard
π² Reverse: Decimal β Fraction
0.036 = 361000.
In simplest form, what is the denominator?
17
βββββ Expert
π₯ Trick Question
Trick! 13 = 0.3Μ (repeating).
But 13 Γ 3 = 1 = 1.000β¦ (terminating).
Is 0.9Μ (= 0.999β¦) equal to 1?
Is 0.9Μ (= 0.999β¦) equal to 1?
18
βββββ Expert
π₯ Application
k2Β³ Γ 3 Γ 5Β²
is a terminating decimal. The smallest positive integer k is:
19
βββββ Expert
π₯ 2-Condition
Both 7n
and n56
are terminating decimals (n is a natural number, n < 56).
How many values of n are possible?
How many values of n are possible?
20
π Challenge
π₯ Final Boss
p2a Γ 5b
(p, a, b are positive integers, GCD(p, 2) = GCD(p, 5) = 1).
To how many decimal places does this fraction terminate?