SME
Grade 8 Math · Problem type
A repeating decimal as a fraction
Use this for “Write 0.363636… as a fraction.”
Diagram
36/99: 36 jumps of 1/99 from 0.
This is an example. Type your own values to start a new problem.
x
=
r
d
=
4
11
f
k
Digits in the block
example
Formulas
d
=
10^
k
−
1
(denominator
=
10digits
in
the
block
−
1)
99
=
10
2
−
1
f
=
r
÷
d
(fraction
=
repeating
block
÷
denominator)
4
11
=
36
÷
99
Assumptions
•
Call the decimal x. Its repeating block has k digits and starts right after the point (0.3636…).
•
Multiply by 10ᵏ to move one block in front of the point. Take x away: the repeats cancel.
•
That leaves (10ᵏ − 1) × x = the block, so x = block ÷ (10ᵏ − 1).
•
Every repeating decimal is a fraction, so it is rational.
Step-by-step
WE KNOW
k = 2 (digits in the block), r = 36 (repeating block)
FIND
denominator (d), fraction (f), decimal (x)
Step 1 · Find denominator (d)
Use d = 10ᵏ − 1 (denominator = 10ᵈⁱᵍⁱᵗˢ in the block − 1)
Multiplying
by
10ᵏ
and
taking
x
away
leaves
10ᵏ
−
1
of
x
.
d
=
10
2
−
1
d
=
100
−
1
d
=
99
Step 2 · Find fraction (f)
Use f = r ÷ d (fraction = repeating block ÷ denominator)
The block over that many nines, in lowest terms.
f
=
36
99
x
=
0.
36
100
x
=
36.
36
100
x
−
x
=
36,
so
99
x
=
36
x
=
36
99
=
4
11
f
=
4
11
Step 3 · Find decimal (x)
Use x = f (decimal = fraction)
Work out the other side of the rule.
x
=
4
11
x
=
0.
36
Check
Put the numbers back in. Both sides should match.
99
=
10
2
−
1
✓
4
11
=
36
÷
99
✓
Topic check
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