Grade 8 · Math · 20 Problems
Grade 8 Mathematics

Converting Repeating
Decimals to Fractions

순환소수 → 유리수  |  20 Practice Problems
Name
Date
Score
Overall Difficulty:
4 / 5 Dot notation: ȧ = digit repeats  |  ȧḃ = both digits repeat
⬤ Level 1 — Starter   Single Repeating Digit
Q 01
0.3̄ = ?
Hint
Let x = 0.333…
10x = 3.333…
10x − x = 3 → 9x = 3
Answer:
Q 02
0.6̄ = ?
Hint
Let x = 0.666…
10x − x = 6 → 9x = 6
Simplify the fraction.
Answer:
Q 03
0.9̄ = ?
Hint
Let x = 0.999…
10x − x = 9 → 9x = 9
Surprising result!
Answer:
Q 04
0.4̄ = ?
Hint
Let x = 0.444…
9x = 4
x = 4/9
Answer:
Q 05
0.1̄ = ?
Hint
Let x = 0.111…
10x − x = 1
9x = 1 → x = ?
Answer:
◆ Level 2 — Building Up   Two-Digit Repeating Block
Q 06
0.1̄2̄ = ?
Hint
Let x = 0.121212…
100x − x = 12
99x = 12 → simplify.
Answer:
Q 07
0.3̄6̄ = ?
Hint
Let x = 0.363636…
Multiply by 100, subtract x.
99x = 36, simplify to lowest terms.
Answer:
Q 08
0.2̄7̄ = ?
Hint
99x = 27
x = 27/99
GCF(27, 99) = 9 → simplify.
Answer:
Q 09
0.8̄1̄ = ?
Hint
x = 0.818181…
100x − x = 81
99x = 81 → simplify.
Answer:
Q 10
0.4̄5̄ = ?
Hint
99x = 45
GCF(45, 99) = 9
Simplify to get a clean fraction.
Answer:
▲ Level 3 — Getting Tricky   Non-Zero Integer Part or Mixed
Q 11
1.3̄ = ?
Hint
x = 1.333…
10x − x = 13 − 1.333… − wait!
10x = 13.333…, 9x = 12 → x = 12/9
Answer:
Q 12
0.16̄ = ?
Hint
Only the 6 repeats: 0.1666…
Let x = 0.1666…
10x = 1.666…, 100x = 16.666…
Subtract: 90x = 15
Answer:
Q 13
0.13̄ = ?
Hint
Only the 3 repeats: 0.1333…
100x − 10x = 13.333… − 1.333…
90x = 12 → simplify.
Answer:
Q 14
2.4̄5̄ = ?
Hint
x = 2.454545…
100x = 245.4545…
100x − x = 243.0
99x = 243
Answer:
Q 15
0.41̄6̄ = ?
Hint
Repeating block "16" starts after 1 non-repeating decimal place.
1000x − 10x = 416.666… − 4.166…
990x = 412.5? Try: 1000x − 10x.
Answer:
★ Level 4 — Expert   Three-Digit Blocks & Complex Mixed
Q 16
0.1̄2̄3̄ = ?
Hint
x = 0.123123…
1000x − x = 123
999x = 123 → simplify.
Answer:
Q 17
0.1̄4̄2̄8̄5̄7̄ = ?
Hint
6-digit repeat: 142857
1000000x − x = 142857
999999x = 142857
GCF = 142857 → famous result!
Answer:
Q 18
3.1̄4̄2̄ = ?
Hint
x = 3.142142142…
1000x = 3142.142142…
1000x − x = 3139
999x = 3139
Answer:
Q 19
0.58̄3̄ = ?
Hint
"83" repeats, 1 non-repeating digit (5).
100x = 58.3838… and 10x = 5.8383…
Wait — careful about which digits repeat.
1000x − 10x = 990x = 578
Answer:
Q 20
1.2̄3̄4̄ = ?
Hint
x = 1.234234…
1000x = 1234.234234…
1000x − x = 1233
999x = 1233 → simplify.
Answer:
📐 Method Reference
The algebraic elimination method — works for every type of repeating decimal.

General Steps

1
Name the repeating decimal x. Write it out to see the pattern clearly.
2
Multiply x by 10ⁿ where n = length of the repeating block, to shift one full cycle left of the decimal point.
3
If there are non-repeating digits before the block, also multiply by 10ᵐ (m = number of non-repeating decimal places). Subtract the smaller equation from the larger.
4
The repeating tails cancel out. Solve for x and reduce the fraction to lowest terms using GCF.
Pure Repeating
0.d₁d₂…dₙ
= d₁d₂…dₙ / 99…9 (n nines)
Mixed Repeating
0.a…ab₁…bₙ
= (a…ab₁…bₙ − a…a) / 99…900…0
✅ Answer Key
Flip over after completing all 20 problems!
Q 01
0.3̄
1/3
Q 02
0.6̄
2/3
Q 03
0.9̄
1
Q 04
0.4̄
4/9
Q 05
0.1̄
1/9
Q 06
0.1̄2̄
4/33
Q 07
0.3̄6̄
4/11
Q 08
0.2̄7̄
3/11
Q 09
0.8̄1̄
9/11
Q 10
0.4̄5̄
5/11
Q 11
1.3̄
4/3
Q 12
0.16̄
1/6
Q 13
0.13̄
2/15
Q 14
2.4̄5̄
27/11
Q 15
0.41̄6̄
5/12
Q 16
0.1̄2̄3̄
41/333
Q 17
0.1̄4̄2̄8̄5̄7̄
1/7
Q 18
3.1̄4̄2̄
3139/999
Q 19
0.58̄3̄
289/495
Q 20
1.2̄3̄4̄
411/333
Pure Repeating (n digits)
Denominator = 10ⁿ − 1
e.g. 0.3̄ → denom = 9; 0.1̄2̄ → denom = 99
Mixed Repeating (m non-rep, n rep)
Denominator = (10ⁿ − 1) × 10ᵐ
e.g. 0.16̄ → m=1, n=1 → denom = 9 × 10 = 90
Always simplify!
Find GCF of numerator & denominator
and divide both by it.
Fun Fact
0.9̄ = 1 exactly — not an approximation!
1/7 = 0.1̄4̄2̄8̄5̄7̄ (6-digit cycle)