📚 Year 9 Maths Unit 5: Fractions, Decimals and Percentages | 九年级数学第5单元：分数、小数与百分数
This revision guide brings together the essential skills in Unit 5: simplifying fractions, comparing and ordering them, adding, subtracting, multiplying and dividing fractions and mixed numbers, converting between fractions, decimals and percentages, understanding recurring decimals, and applying percentage of an amount, percentage increase, percentage decrease and reverse percentage calculations. Every explanation is paired with a Chinese translation so that you can check your understanding in either language, and each section ends with worked examples and exam-style pointers. Work through the sections in order, then attempt timed practice questions without a calculator to build real fluency.
本复习指南汇总第5单元的核心技能：分数化简、分数比较与排序、分数与带分数的加减乘除、分数与小数及百分数的互化、理解循环小数，以及百分数求值、百分数增减与逆百分数运算。每段讲解都配有中文对照，方便用任一语言检验理解；每节末尾附例题与考点提示。建议按顺序学习，再用限时练习（先不用计算器）检验熟练度。
1. Equivalent Fractions and Simplifying | 等值分数与化简
A fraction is written in the form a/b, where a is the numerator and b is the denominator. The denominator cannot be zero, because division by zero is undefined. Equivalent fractions have the same value but different numerators and denominators; you create them by multiplying or dividing both the numerator and the denominator by the same non-zero number.
分数写作 a/b，其中 a 是分子，b 是分母。分母不能为 0，因为除以 0 无意义。等值分数数值相同，但分子和分母不同；将分子和分母同时乘以或除以同一个非零数，就能得到等值分数。
To simplify a fraction, find the highest common factor, HCF, of the numerator and the denominator, then divide both by it. For example, to simplify 24/36, the HCF of 24 and 36 is 12, so divide top and bottom by 12:
化简分数时，先找出分子与分母的最大公因数 HCF，再同时除以它。例如化简 24/36，24 和 36 的 HCF 是 12，因此分子分母同时除以 12：
24/36 = (24 ÷ 12)/(36 ÷ 12) = 2/3
Unless the question says otherwise, final answers should be given in the simplest form. Marks are frequently lost by leaving an answer such as 10/15 instead of 2/3.
除非题目另有说明，最终答案应为最简形式。把 10/15 当作答案而没化简成 2/3，是常见的失分点。
- 1/2 = 2/4 = 3/6 = 4/8 = 5/10
- 3/5 = 6/10 = 9/15 = 12/20 = 30/50
- 10/15 = 2/3
| Fraction | HCF of numerator and denominator | Simplest form |
| 18/24 | 6 | 3/4 |
| 45/60 | 15 | 3/4 |
| 56/84 | 28 | 2/3 |
2. Comparing and Ordering Fractions | 分数比较与排序
To compare two fractions, rewrite them with a common denominator and compare the numerators. The fraction with the larger numerator is larger, because the denominators are now identical. A second reliable method is to convert each fraction to a decimal and compare the decimal values; this is faster when denominators share no obvious common factor.
比较两个分数时，先通分到相同分母，再比较分子。分母相同时，分子大的分数更大。第二种可靠方法是把每个分数化成小数再比较大小；当分母之间没有明显公因数时，这种方法更快。
Example: order 2/3, 3/5 and 7/10 from smallest to largest. Using a common denominator of 30 gives 20/30, 18/30 and 21/30, so the order is 3/5, 2/3, 7/10. Using decimals gives 0.666..., 0.6 and 0.7, which confirms the same order.
例：把 2/3、3/5、7/10 从小到大排序。取公分母 30，得到 20/30、18/30、21/30，所以顺序是 3/5、2/3、7/10。化成小数则得到 0.666…、0.6、0.7，顺序一致。
A useful benchmark is 1/2. If the numerator is less than half the denominator, the fraction is less than 1/2; if it is more than half, the fraction is greater than 1/2. Cross-multiplication also works: for a/b and c/d, compare ad with cb.
一个常用的参照值是 1/2。如果分子小于分母的一半，该分数小于 1/2；如果大于一半，则该分数大于 1/2。交叉相乘同样可行：比较 a/b 与 c/d 时，只需比较 ad 与 cb 的大小。
a/b > c/d ⇔ ad > cb (b, d > 0)
3. Adding and Subtracting Fractions | 分数加减法
iple, LCM, of the denominators. For 1/3 + 1/4 the LCM of 3 and 4 is 12, so convert: 1/3 = 4/12 and 1/4 = 3/12. Then add the numerators: 4/12 + 3/12 = 7/12.
分数加减时，必须先使分母相同。找出最小公分母 LCD，也就是各分母的最小公倍数 LCM。以 1/3 + 1/4 为例，3 和 4 的 LCM 是 12，所以先转化：1/3 = 4/12，1/4 = 3/12。然后分子相加：4/12 + 3/12 = 7/12。
ctions first. Example: 2 1/3 + 1 1/6 = 2 + 1 + 2/6 + 1/6 = 3 3/6 = 3 1/2. Always simplify the final answer.
带分数运算时，可以先加整数部分再加分数部分，也可以先把带分数化为假分数。例如：2 1/3 + 1 1/6 = 2 + 1 + 2/6 + 1/6 = 3 3/6 = 3 1/2。最后务必化简答案。
a/b + c/d = (ad + cb)/bd
a/b − c/d = (ad − cb)/bd
- 1/2 + 1/3 = 3/6 + 2/6 = 5/6
- 3/4 − 1/6 = 9/12 − 2/12 = 7/12
- 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2
r 4 1/5 − 2 3/5, borrow one whole: 3 6/5 − 2 3/5 = 1 3/5, which avoids negative fractions.
考点提示：带分数减法常需"借位"。例如 4 1/5 − 2 3/5，先借 1：3 6/5 − 2 3/5 = 1 3/5，可避免出现负的分数部分。
4. Multiplying and Dividing Fractions | 分数乘除法
lying can make the work easier: 3/8 × 4/9 = (3 × 4)/(8 × 9) = 12/72 = 1/6. With cross-cancelling you divide the 3 and the 9 by 3, and the 4 and the 8 by 4, leaving 1/2 × 1/3 = 1/6 directly.
分数乘法很直接：分子乘分子，分母乘分母。先约分再相乘可以简化计算：3/8 × 4/9 = (3 × 4)/(8 × 9) = 12/72 = 1/6。若用交叉约分，把 3 与 9 同除以 3，把 4 与 8 同除以 4，直接得到 1/2 × 1/3 = 1/6。
iprocal. The reciprocal of a/b is b/a, provided a is not zero. This is often remembered as 'keep, flip, change': keep the first fraction, flip the second, change ÷ to ×.
除以一个分数等于乘以它的倒数。a/b 的倒数是 b/a，其中 a 不能为 0。这常记作"保持、翻转、变号"：保留第一个分数，翻转第二个分数，把除号变成乘号。
a/b × c/d = ac/bd
a/b ÷ c/d = a/b × d/c = ad/bc
- 2/3 × 9/10 = 18/30 = 3/5
- 5/8 ÷ 15/16 = 5/8 × 16/15 = 80/120 = 2/3
- 3 ÷ 1/4 = 3 × 4 = 12
Note that the numerator of a product can be larger than its denominator; leave answers as improper fractions unless a mixed number is requested or the context (for example a measurement) suits one.
注意：乘积的分子可能大于分母；除非题目要求写成带分数，或情境（如测量）更适合带分数，否则保留假分数也是标准答案。
5. Mixed Numbers and Improper Fractions | 带分数与假分数
action, for example 2 1/3. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator: 2 1/3 = (2 × 3 + 1)/3 = 7/3.
假分数的分子大于或等于分母，例如 7/3。带分数由整数与真分数组成，例如 2 1/3。将带分数化为假分数：用整数乘以分母，加上分子，结果写在原分母之上：2 1/3 = (2 × 3 + 1)/3 = 7/3。
denominator and write the remainder as a fraction: 17/5 = 3 remainder 2, so 17/5 = 3 2/5.
把假分数化回带分数：用分子除以分母，余数写成分数：17/5 = 3 余 2，所以 17/5 = 3 2/5。
| Mixed number | Improper fraction | Decimal |
| 1 1/2 | 3/2 | 1.5 |
| 2 3/4 | 11/4 | 2.75 |
| 3 1/6 | 19/6 | 3.1666... |
actions first is usually safer than trying to multiply whole-number and fraction parts separately.
例题：2 1/2 × 1 3/5 = 5/2 × 8/5 = 40/10 = 4。乘法时先化成假分数通常比分别处理整数与分数部分更稳妥。
6. Converting Fractions to Decimals | 分数化为小数
division, long division or a calculator, but you must be able to do simple cases mentally in an exam.
把分数化为小数，只需用分子除以分母。分数本身就是除法，所以 3/8 表示 3 ÷ 8 = 0.375。可以用短除法、长除法或计算器，但考试中必须能口算常见情况。
For example 7/20 terminates (20 = 2² × 5) giving 0.35, while 5/6 recurs (6 = 2 × 3) giving 0.8333...
若最简分数的分母质因数只有 2 和 5，化成的小数是有限小数；若出现其他质因数，则是循环小数。例如 7/20 是有限小数（20 = 2² × 5），等于 0.35；而 5/6 是循环小数（6 = 2 × 3），等于 0.8333…
- 1/8 = 0.125
- 3/4 = 0.75
- 2/5 = 0.4
- 7/8 = 0.875
- 1/3 = 0.333...
a/b = a ÷ b
7. Recurring Decimals | 循环小数
.. = 0.3 with a dot over the 3, and 1/7 = 0.142857142857... = 0.1 with a dot over the 1 and a dot over the 7. The repeating block for 1/7 has length six.
循环小数指某一位数字或某一组数字无限重复。书写时在循环节的首位和末位数字上方加点：1/3 = 0.333… 记作 0.3 在 3 上加一点；1/7 = 0.142857142857… 记作在 1 和 7 上各加一点。1/7 的循环节长度为 6。
For a two-digit repeating block, multiply by 100 instead: x = 0.727272..., 100x = 72.727272..., 99x = 72, so x = 72/99 = 8/11.
把循环小数化为分数可用"相减技巧"。设 x = 0.333…，则 10x = 3.333…，两式相减得 9x = 3，所以 x = 3/9 = 1/3。若循环节是两位，则乘以 100：设 x = 0.727272…，100x = 72.727272…，99x = 72，所以 x = 72/99 = 8/11。
0.333... = 1/3 0.666... = 2/3 0.1666... = 1/6
0.272727... = 3/11 0.454545... = 5/11
minator of 99...9 with n nines before simplifying. This is why 0.454545... = 45/99 = 5/11.
注意：循环节有 n 位数字，就对应分母为 n 个 9 组成的数，然后再化简。因此 0.454545… = 45/99 = 5/11。
8. Decimals and Percentages | 小数与百分数互化
Percent means 'out of one hundred', so 37% r cent sign. To convert a percentage to a decimal, divide by 100. For example 0.375 = 37.5%, 0.6 = 60%, 7% = 0.07 and 82.5% = 0.825.
百分数表示"每一百中有多少"，所以 37% = 37/100 = 0.37。小数化百分数：乘以 100 并加上百分号；百分数化小数：除以 100。例如 0.375 = 37.5%，0.6 = 60%，7% = 0.07，82.5% = 0.825。
ge, first convert it to a decimal and then multiply by 100. For example 3/8 = 0.375 = 37.5% me conversions are worth memorising because they appear constantly in percentage problems.
分数化百分数：先化成小数，再乘以 100。例如 3/8 = 0.375 = 37.5%。有些互化结果在百分数题目中反复出现，值得直接背下来。
