MS2 Algebra · Unit 4
Repeating Decimals & Fractions
①
핵심 암기포인트 (Quick Memory)
📌 분수 → 순환소수 판별법: 분모를 소인수분해했을 때 2와 5 이외의 소인수가 있으면 → 순환소수!
📌 순환소수 → 분수 변환 공식:
• 소수점 이하가 모두 순환: 분자 = 숫자, 분모 = 9의 개수 (e.g. 0.̄3 = 3/9)
• 앞에 안 도는 숫자가 있으면: 큰수 - 작은수 / 9...90...0
📌 방정식 풀이 순서: ① x를 정의 → ② 10ⁿx로 이동 → ③ 빼기 → ④ 약분
📌 순환소수 → 분수 변환 공식:
• 소수점 이하가 모두 순환: 분자 = 숫자, 분모 = 9의 개수 (e.g. 0.̄3 = 3/9)
• 앞에 안 도는 숫자가 있으면: 큰수 - 작은수 / 9...90...0
📌 방정식 풀이 순서: ① x를 정의 → ② 10ⁿx로 이동 → ③ 빼기 → ④ 약분
▶ Part A — Identifying Repeating Decimals
EXAMPLE
Is 7/12 a repeating decimal?
Factor the denominator: 12 = 2² × 3 Since 3 is a prime factor other than 2 and 5 → YES, repeating decimal 7/12 = 0.583
Factor the denominator: 12 = 2² × 3 Since 3 is a prime factor other than 2 and 5 → YES, repeating decimal 7/12 = 0.583
★★★ HARD
Q1.
We can write a/360 as a repeating decimal.
Which values of a make this possible?
HINT: Factor 360 first!
(Select ALL that apply — choose the answer set)
(Select ALL that apply — choose the answer set)
📖 풀이 (Solution)
360 = 2³ × 3² × 5. Since 3 is a factor other than 2 or 5, a/360 is repeating unless the 3² in the denominator cancels with the numerator. For the fraction to remain repeating, a must NOT be a multiple of 9 (which would cancel 3²) AND must NOT make the denominator reduce to only 2s and 5s. Among the options, a = 7 gives 7/360 which keeps the factor of 3 → repeating. a = 72 = 8×9 cancels 3² fully, leaving 72/360 = 1/5 → terminating!
★★☆ MEDIUM
Q2.
Which fraction cannot be expressed as a repeating decimal?
📖 풀이
7/40: 40 = 2³ × 5 → only 2s and 5s → terminating (not repeating). The others all have prime factors beyond 2 and 5 in their denominators.
★★★ HARD
Q3.
How many of the following are repeating decimals?
5/6, 3/8, 11/15, 7/25, 4/9
5/6, 3/8, 11/15, 7/25, 4/9
📖 풀이
6=2×3 ✓ repeating | 8=2³ ✗ terminating | 15=3×5 ✓ repeating | 25=5² ✗ terminating | 9=3² ✓ repeating → 3 fractions are repeating.
▶ Part B — Converting Repeating Decimals → Fractions
EXAMPLE · 0.427 → fraction
Let x = 0.4272727…
1000x = 427.2727… and 10x = 4.2727…
990x = 423 → x = 423/990 = 47/110
★★☆ MEDIUM
Q4.
Convert 0.72 to a fraction in simplest form.
순환마디 길이=2 → 99로 나눔!
📖 풀이
Let x = 0.727272…100x = 72.727272…
99x = 72 → x = 72/99 = 8/11
★★☆ MEDIUM
Q5.
Convert 1.6 to a fraction in simplest form.
📖 풀이
x = 1.6666…10x = 16.666… → 9x = 15 → x = 15/9 = 5/3
★★★ HARD
Q6.
Convert 0.245 to a fraction in simplest form.
앞에 안도는 자리 주의!
📖 풀이
x = 0.24545…1000x = 245.4545… and 10x = 2.4545…
990x = 243 → x = 243/990 = 27/110
★★★ HARD
Q7.
Convert 3.145 to a fraction in simplest form.
📖 풀이
x = 3.14545…1000x = 3145.4545… and 10x = 31.4545…
990x = 3114 → x = 3114/990 = 519/165 = 173/55
★★★ HARD
Q8.
Which equation is the correct first step to convert 0.513?
📖 풀이
The non-repeating part has 1 digit (5), repeating part has 2 digits (13).We need 1000x = 513.1313… and 10x = 5.1313…
So the correct pair is 1000x and 10x.
▶ Part C — Fractions → Repeating Decimals
EXAMPLE · 5/11
Divide: 5 ÷ 11 = 0.454545…
So 5/11 = 0.45 (the repeating block is "45")
★★☆ MEDIUM
Q9.
What is the repeating block (순환마디) of 7/12?
📖 풀이
7 ÷ 12 = 0.58333… = 0.583The repeating block is 3.
★★★ HARD
Q10.
What is the 40th decimal digit of 1/7?
순환마디 길이 먼저!
📖 풀이
1/7 = 0.142857 — repeating block length = 6.40 ÷ 6 = 6 remainder 4.
The 4th digit in "142857" is 8.
▶ Part D — Tricky Mixed Problems
★★★ HARD
Q11.
Which of the following is equivalent to 0.9?
함정! 생각보다 답이 놀랍다
📖 풀이
x = 0.999… → 10x = 9.999… → 9x = 9 → x = 1This is a famous result: 0.̄9 = 1 exactly!
★★★ HARD
Q12.
Simplify: 0.3 + 0.6. Express as a fraction.
📖 풀이
0.̄3 = 1/3, 0.̄6 = 2/31/3 + 2/3 = 1
★★★ HARD
Q13.
If x = 0.142857, what is 7x?
📖 풀이
0.142857 = 1/77 × (1/7) = 1
★★★ HARD
Q14.
Convert 0.285714 to a fraction. What is the denominator in simplest form?
📖 풀이
0.285714 = 285714/999999 = 2/7Denominator = 7.
★★★ HARD
Q15.
What is 0.16 as a fraction in lowest terms?
1자리 안도는 구간 있음!
📖 풀이
x = 0.1666…10x = 1.666… and 100x = 16.666…
90x = 15 → x = 15/90 = 1/6
▶ Part E — Advanced Challenge
★★★ HARD
Q16.
A student writes: "All repeating decimals are fractions, but not all fractions are repeating decimals."
Is this statement TRUE or FALSE?
📖 풀이
TRUE. Every repeating decimal converts to a rational fraction (as we showed).But fractions like 1/4 = 0.25 are terminating, not repeating. So not all fractions produce repeating decimals.
★★★ HARD
Q17.
Which denominator guarantees a repeating decimal for ANY numerator (that doesn't cancel)?
📖 풀이
We need a denominator with a prime factor ≠ 2, 5.• 40 = 2³×5 ✗ | • 125 = 5³ ✗ | • 21 = 3×7 ✓ | • 200 = 2³×5² ✗
★★★ HARD
Q18.
If p/q is in lowest terms and q = 2^a × 5^b × 7^c where c > 0,
is p/q a repeating or terminating decimal?
📖 풀이
Since c > 0, the factor 7 (≠ 2, 5) remains in the denominator after reducing.Therefore p/q is always a repeating decimal.
★★★ HARD
Q19.
Compute: 0.12 × 0.3. Give the answer as a simplified fraction.
📖 풀이
0.12 = 12/99 = 4/330.3 = 1/3
(4/33) × (1/3) = 4/99
★★★ HARD
Q20.
The fraction n/990 in simplest form equals 0.123.
What is the value of n?
역으로 분수 먼저 구하기!
📖 풀이
Convert 0.123: x = 0.12323…1000x = 123.2323… and 10x = 1.2323…
990x = 122 → x = 122/990
GCD(122,990) = 2 → 61/495
But 61/495 × (990/990) means we need n such that n/990 = 61/495
n = 122 → n = 122
✏️ SCRATCH SPACE / 계산 공간