IF A FRACTION IS TERMINATING OR REPEATING WITHOUT DIVIDING WORKSHEET

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Terminating decimals result when the rational number has a denominator which has no prime factors other than 2 or 5.

Problem 1 :

1/2

Solution

Problem 2 :

3/4

Solution

Problem 3 :

3/5

Solution

Problem 4 :

17/50

Solution

Problem 5 :

2/3

Solution

Problem 6 :

2/9

Solution

Problem 7 :

9/40

Solution

Problem 8 :

5/6

Solution

Problem 9 :

7/8

Solution

Problem 10 :

17/80

Solution

Problem 11 :

37/125

Solution

Problem 12 :

5/12

Solution

Problem 13 :

3/20

Solution

Problem 14 :

4/11

Solution

Problem 15 :

11/25

Solution

Problem 16 :

7/9

Solution

Problem 17 :

7/90

Solution

Problem 18 :

7/99

Solution

Problem 19 :

7/999

Solution

Problem 20 :

7/9999

Solution

Answer Key

1) Terminating decimal

2) Terminating decimal

3) Terminating decimal

4) Terminating decimal

5) Non terminating decimal

6) Non terminating decimal

7) Terminating decimal

8) Non terminating decimal

9) Terminating decimal

10) Terminating decimal

11) Terminating decimal

12) Non terminating decimal

13) Terminating decimal

14) Non terminating decimal

15) Terminating decimal

16) Non terminating decimal

17) Non terminating decimal

18) Non terminating decimal

19) Non terminating decimal

20) Non terminating decimal

Convert the following recurring decimals to fractions :

Problem 1 :

0.333….

Solution

Problem 2 :

0.444….

Solution

Problem 3 :

1.0909….

Solution

Problem 4 :

2.0909….

Solution

Problem 5 :

0.5333….

Solution

Problem 6 :

1.7333….

Solution

Problem 7 :

0.9444….

Solution

Problem 8 :

2.0555…

Solution

Problem 9 :

a) Write 1/7 as a repeating decimal. How many digits repeat?

b) Write the fractions

2/7, 3/7, 4/7, 5/7 and 6/7

in decimal form. What patterns do you see? Explain how the circle of digits can help you write these fractions as decimals.

Solution

Problem 10 :

0.5454........ x 0.555...........

Solution

Problem 11 :

You run the 40-yard dash in 6 21/25 seconds. Your teammate runs it in the time shown. Who is faster and by how much?

Solution

Answer Key

1) 0.333… = 1/3

2) 0.444… = 4/9

3) 1.0909… = 108/99

4) 2.0909… = 207/99

5) 0.5333… = 48/90

6) 1.7333… = 156/90

7) 0.9444… = 85/90

8) 2.0555… = 185/90

a) 1/7 = 0.142857142857 ................

The repeating digits are 142857. So six digits are repeating.

b) Other fractions and patterns

2/7: 0.285714 (starts at '2' in the cycle).
3/7: 0.428571 (starts at '4' in the cycle).
4/7: 0.571428 (starts at '5' in the cycle).
5/7: 0.714285 (starts at '7' in the cycle).
6/7: 0.857142 (starts at '8' in the cycle).

Patterns and the "Circle of Digits"

Pattern: All fractions (1/7 to 6/7) use the exact same six digits (1, 4, 2, 8, 5, 7) in their repeating cycle, just starting at a different point.

10) the answer is 10/33.

11) you are faster by 0.06 second.

Problem 1 :

Which of these are rational?

a. 8 b. -8 c. 2 1/3 d. -3 1/4

e. √3 f. √400 g. 9.176 h. π - π

Solution

Problem 2 :

Show that these numbers are rational:

a. 0.7..... b. 0.41..... c. 0.324......

Solution

Problem 3 :

a. Why is 0.527 a rational number?

b. 0.9 is a rational number. In fact, 0.9 € Z. Give evidence to support this statement.

Solution

Problem 4 :

Explain why these statements are false:

a. The sum of two irrationals is irrational.

b. The product of two irrationals is irrational.

Solution

Problem 5 :

a. Explain why 1.3 is a rational number.

b. True or false. √4000 € Q ?

Solution

Problem 6 :

a. True or false: 1/√7 € Q ?

b. Show that 0.41 is a rational number.

Solution

Problem 7 :

Is the sum of 3√2 and 4√2 rational or irrational

Solution

Problem 8 :

Jacob is working on his math homework. He decides that the sum of the expression 1/3 + 6√5/7 must be rational because it is fraction. Is Jacob correct ?

Solution

Problem 9 :

State whether 7 - √2 is rational or irrational

Solution

Problem 10 :

Is the product of √16 and 4/7 rational or irrational ?

Solution

Answer Key

a) 8

8 is a integer and it can be written as 8/1. This can be written in the form of p/q. So, it is rational.

b) -8

This can be represented as -8/1. So, it is rational.

c) 2 1/3

This mixed fraction can be converted as improper fraction as 7/3. So, it is rational.

d) -3 1/4 = - 13/4

It is rational.

e) √3

√3 = 1.732 ..........

It cannot be converted as fraction, so it is not a rational number.

f) √400

√400 = 20

It can be represented as 20/1, then it is rational.

g) 9.176

Since it is recurring decimal, it can be converted as fraction. Then, it is rational.

h) π - π

= 0

2) a) 0.7777...... = 7/9

b) x = 41/99

c) x = 36/111

Yes, 0.527 is a rational Number. Since it is ending decimal, it can be represented as fraction.

b. 9/10

4) a) false

b) false

5) a) 13/10

b) it is irrational.

6) a) true

b) rational number

7) Multiplying rational number by irrational number will be a irrational number.

8) It must be an irrational number, because adding, subtracting, multiplying or dividing by irrational by rational number will product rational number as result.

9) Irrational

10) rational

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IF A FRACTION IS TERMINATING OR REPEATING WITHOUT DIVIDING WORKSHEET

Answer Key

Answer Key

Answer Key

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