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➗ 分数与小数

Fractions and Decimals

分数与小数是 AMC 8 代数模块的基础，贯穿整个竞赛内容。掌握分数的互化、运算规则，以及循环小数化分数的方法，是代数解题的必备技能。

📚 4 章节 💡 5 道例题 ✏️ 8 道练习 🎯 难度：基础 ⏱ 约30分钟
1
分数的基本概念 Basic Concepts of Fractions
基础AMC高频

1.1 真分数、假分数、带分数 Proper, Improper and Mixed Fractions

真分数：分子小于分母，如 3/5、7/8。

A proper fraction has numerator < denominator, e.g., 3/5, 7/8.

假分数：分子大于或等于分母，如 5/3、8/7、4/4。

An improper fraction has numerator ≥ denominator, e.g., 5/3, 8/7, 4/4.

带分数：由整数和真分数组成，如 2⅓ = 7/3。

A mixed number combines an integer and a proper fraction, e.g., 2⅓ = 7/3.

📝 带分数化假分数 / Converting Mixed Numbers to Improper Fractions
a + b/c = (a×c + b) / c
带分数化假分数：整数部分×分母 + 分子，除以原分母
Multiply the whole number by the denominator, add the numerator, keep the denominator.

举例：

  • 2⅓ = (2×3+1)/3 = 7/3
  • 5¾ = (5×4+3)/4 = 23/4
💡 邓老师提示：AMC 8 的选项有时会同时出现带分数和假分数，注意辨别！有时把假分数化简为带分数会让答案一目了然。

1.2 等值分数与约分、通分 Equivalent Fractions, Simplification & Common Denominators

等值分数：分子分母同时乘以或除以同一个非零整数，分数的值不变。

Multiplying or dividing both numerator and denominator by the same non-zero integer does not change the value of the fraction.

📝 约分与通分 / Simplification & Common Denominators
约分：用分子分母的公因数同时除以它们，使分数化简
通分：把异分母分数化为同分母分数（取各分母的最小公倍数）
Simplify: divide numerator and denominator by their GCD. Common denominator: use LCM of denominators.

举例：

  • 约分：18/24 = (18÷6)/(24÷6) = 3/4 ✓
  • 通分：1/4 和 1/6 → lcm(4,6)=12 → 3/12 和 2/12
⚠️ 注意：约分时必须用最大公因数（GCD）一次约到底，得到最简分数。
Always simplify to lowest terms using the GCD of numerator and denominator.
2
分数运算 Fraction Operations
基础AMC高频

2.1 分数加减法 Adding and Subtracting Fractions

📝 异分母分数加减 / Adding Fractions with Different Denominators
a/b + c/d = (ad + bc) / bd    或先通分再计算
a/b + c/d = (ad + bc) / bd   (or find common denominator first)

举例：1/3 + 1/4 = (4+3)/12 = 7/12

先通分：1/3 = 4/12，1/4 = 3/12，4/12 + 3/12 = 7/12 ✓

2.2 分数乘除法 Multiplying and Dividing Fractions

📝 分数乘法 / Multiplying Fractions
a/b × c/d = (a×c) / (b×d)
Multiply numerators together and denominators together. Cancel common factors first for simplicity.
📝 分数除法 / Dividing Fractions
(a/b) ÷ (c/d) = (a/b) × (d/c) = ad / bc
"除以一个分数 = 乘以它的倒数"（flip the second fraction and multiply）
💡 邓老师提示：分数乘法时，先约分再乘可以大幅简化计算。比如 (8/9)×(3/4) 先把8和4约掉得2/1，再乘3/9约掉得1/3，结果=2×(1/3)=2/3。
Always cancel common factors before multiplying. E.g., (8/9)×(3/4) = 2/3.
3
小数与分数互换 Converting Between Decimals and Fractions
中等AMC高频

3.1 有限小数化分数 Terminating Decimals → Fractions

有限小数：看小数点后有几位，就在1后面添几个0作分母，去掉小数点作分子，然后约分。

For a terminating decimal, count the digits after the decimal point, put that many zeros after 1 as the denominator, remove the decimal point as the numerator, then simplify.

📝 有限小数化分数
0.375 = 375/1000 = 3/8   (三位小数 → 分母1000)
2.75 = 275/100 = 11/4 = 2¾   (两位小数 → 分母100)

3.2 循环小数化分数 Repeating Decimals → Fractions

纯循环小数（从小数点后第一位就开始循环）：

📝 纯循环小数化分数 / Pure Repeating Decimal → Fraction
0.\=3 = 3/9 = 1/3    0.\=2\overline{7}=27/99 = 3/11
分子 = 循环节，分母 = 同样位数的9
Numerator = repeating block; denominator = same number of 9s.

混循环小数（循环节不是从小数点后第一位开始）：

📝 混循环小数化分数 / Mixed Repeating Decimal → Fraction
0.58\overline{3} = (583−58)/900 = 525/900 = 7/12
分子 = 全部数字组成的数 − 不循环部分组成的数；分母 = 9的个数（循环节位数）+ 0的个数（非循环部分位数）
Numerator = (all digits) − (non-repeating part); denominator = (9s for repeating part) followed by (0s for non-repeating part).
💡 邓老师提示：AMC 8 最常考的是 0.\=6 = 6/9 = 2/3、0.\=3 = 3/9 = 1/3、0.\=142857 = 1/7 等常见循环节，需要熟记！
Memorize common repeating decimals: 0.\=6 = 2/3, 0.\=3 = 1/3, 0.\=6 = 2/3, 0.\=142857 = 1/7.
4
例题精讲 Worked Examples
5 题含历年真题
📌 例题 1 分数运算

1/2 + 1/3 + 1/6 的值是多少？ What is the value of 1/2 + 1/3 + 1/6?

解题思路：通分找最小公倍数
lcm(2,3,6) = 6
1/2 = 3/6，1/3 = 2/6，1/6 = 1/6
3/6 + 2/6 + 1/6 = 6/6 = 1 LCM(2,3,6) = 6. Convert: 1/2=3/6, 1/3=2/6, 1/6=1/6. Sum = 6/6 = 1.
📌 例题 2 循环小数化分数

0.\=7（循环小数）化成分数是多少？ Express 0.\=7 as a fraction in simplest form.

解题思路：纯循环小数，分子=循环节，分母=同样位数的9
0.\=7：循环节为"7"（1位），分母=9（1个9）
0.\=7 = 7/9（已经最简） 0.\=7: repeating digit "7" (1 digit) → denominator = 9. Numerator = 7. Result = 7/9.
📌 例题 3 分数除法

(3/4) ÷ (5/8) 的值是多少？ What is the value of (3/4) ÷ (5/8)?

解题思路：除以一个分数 = 乘以它的倒数
(3/4) ÷ (5/8) = (3/4) × (8/5)
约分：3/4 × 8/5，4和8约掉 → 3/1 × 2/5 = (3×2)/(1×5) = 6/5 (3/4) ÷ (5/8) = (3/4) × (8/5). Cancel 4 and 8 → 3/1 × 2/5 = 6/5.
📌 例题 4 小数化分数

0.125 化成分数是多少？ Express 0.125 as a fraction in simplest form.

解题思路：有限小数化分数
0.125 有三位小数 → 分母=1000
0.125 = 125/1000
约分：125÷125=1，1000÷125=8 → 1/8 0.125 has 3 decimal places → denominator = 1000. 125/1000 = divide by 125 → 1/8.
📌 例题 5 带分数运算

2¾ × 1⅔ 的值是多少？ What is the value of 2¾ × 1⅔?

解题思路：先化带分数为假分数，再乘法
2¾ = 11/4，1⅔ = 5/3
11/4 × 5/3 = 55/12（已经最简） Convert: 2¾ = 11/4, 1⅔ = 5/3. Multiply: 11×5 / 4×3 = 55/12.
5
巩固练习 Practice Problems
8 题提交即判

第1题 3/5 + 2/7 = ? What is 3/5 + 2/7?

第2题 4/9 × 3/8 = ? What is 4/9 × 3/8?

第3题 0.\=4（循环小数）化成分数是多少？ Express 0.\=4 as a fraction.

第4题 5⅔ − 2¾ = ?（结果写成最简分数或带分数） Evaluate 5⅔ − 2¾. Give your answer as a simplified fraction or mixed number.

第5题 7/9 ÷ 2/3 = ? What is 7/9 ÷ 2/3?

第6题 下列哪个分数大于 3/5？ Which of the following fractions is greater than 3/5?

第7题 0.625 化成分数是？ What is 0.625 as a fraction in simplest form?

第8题 哪个分数最大：2/3、5/7、7/10？ Which fraction is the largest: 2/3, 5/7, 7/10?

⬅ 上一页 因数与倍数 / Factors & Multiples