📚 Year 8 Higher Maths Unit 5-6: Fractions, Percentages, Ratio and Algebra Explained | 8年级高阶数学单元5-6：分数、百分数、比与代数解析
Units 5 and 6 in the Year 8 Higher course bring together number work and the first real steps of algebra. You need to move fluently between fractions, decimals and percentages, use ratio and proportion to solve sharing problems, and begin to simplify, expand and solve linear expressions and equations. The topics are deliberately linked: a percentage is simply a fraction with denominator 100, a ratio is a comparison of two or more quantities, and algebra is the language that ties all of them together.
8年级高阶课程第5和第6单元把数的运算与代数的第一步真正结合起来。你需要熟练地在分数、小数和百分数之间转换，使用比和比例解决分配问题，并开始化简、展开和求解一次式与方程。这些主题是有意联系在一起的：百分数就是分母为100的分数，比是两个或多个量之间的比较，而代数则是把它们串联起来的语言。
Examination questions rarely test one skill in isolation. A typical question might give you a price, ask for a percentage increase, then ask you to share the result in a given ratio, or to write and solve an equation for an unknown length. This guide works through each skill in turn and then shows how they combine in exam-style problems.
考试题很少单独考查某一项技能。一道典型题目可能给出一个价格，要求计算百分数增长，再要求按给定的比进行分配，或者要求你列出并求解关于某个未知长度的方程。本指南逐项讲解每项技能，然后展示它们在考题中如何综合出现。
1. Equivalent Fractions and Simplifying | 等值分数与约分
)/(5 × 4) = 12/20. The value of the fraction does not change, because you have multiplied by 4/4, which is equal to 1.
等值分数是通过把分子和分母同时乘或除以同一个非零数得到的。例如 3/5 = (3 × 4)/(5 × 4) = 12/20。分数的值不会改变，因为你实际上乘的是 4/4，而它等于 1。
a/b = (a × k)/(b × k), b ≠ 0, k ≠ 0
12/20 ÷ 4/4 = 3/5. Dividing by the HCF in one step gets you straight to the simplest form.
约分时，用分子和分母的最大公因数（HCF）同时除。对于 12/20，12 与 20 的最大公因数是 4，所以 12/20 ÷ 4/4 = 3/5。用最大公因数一步相除，就能直接得到最简形式。
A fraction is in simplest form when the only common factor of the numerator and the denominator is 1. Always check whether your final fraction can be simplified, especially after adding, subtracting, multiplying or dividing fractions.
当分子与分母的唯一公因数是 1 时，分数就是最简形式。一定要检查最终结果是否还能约分，尤其是在分数加减乘除之后。
- List the factors of both numbers and pick the largest shared one.
- Or use prime factorisation and take the lowest power of each shared prime.
- Check the final answer by re-expanding.
- 列出两个数的所有因数，取最大的公共因数。
- 或者用质因数分解，取公共质数的最低次幂。
- 通过重新展开来检验最终答案。
d 126: 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7, so HCF = 2 × 3 × 7 = 42, giving 84/126 = 2/3.
对于 36 和 48：36 = 2² × 3²，48 = 2⁴ × 3，所以 HCF = 2² × 3 = 12，于是 36/48 = 3/4。对于 84 和 126：84 = 2² × 3 × 7，126 = 2 × 3² × 7，所以 HCF = 2 × 3 × 7 = 42，于是 84/126 = 2/3。
by anything non-zero is zero), while 5/0 is undefined, because no number times 0 gives 5.
有两个特殊情况值得记住：0/5 = 0（零除以任何非零数都等于零），而 5/0 是没有定义的，因为没有任何数乘 0 能得到 5。
2. Comparing and Ordering Fractions | 分数的大小比较与排序
he LCM of 4 and 7 is 28, so 3/4 = 21/28 and 5/7 = 20/28. Since 21 > 20, we have 3/4 > 5/7.
比较两个分数时，把它们化为同分母。比较 3/4 与 5/7，4 和 7 的最小公倍数是 28，所以 3/4 = 21/28，5/7 = 20/28。因为 21 > 20，所以 3/4 > 5/7。
his avoids writing out equivalent fractions but gives no common denominator for further work.
更快的替代方法是交叉相乘：比较 a/b 与 c/d 时，比较 a × d 与 c × b。对于 3/4 与 5/7，3 × 7 = 21，5 × 4 = 20，所以 3/4 更大。这样不必写出等值分数，但不会给出后续计算可用的公分母。
When ordering mixed numbers, compare the whole parts first. For 3 2/5 and 3 3/7, the whole parts are equal, so compare 2/5 and 3/7: 14/35 against 15/35, so 3 3/7 is the larger number.
给带分数排序时，先比较整数部分。对于 3 2/5 和 3 3/7，整数部分相同，所以比较 2/5 与 3/7：14/35 与 15/35，因此 3 3/7 更大。
g to decimals is often the fastest method when a calculator is allowed. 7/8 = 0.875 = 87.5% papers, use common denominators or convert everything to percentages with denominator 100.
允许使用计算器时，化为小数往往是最快的方法。7/8 = 0.875 = 87.5%，这样可以轻松地与 0.8 和 0.9 比较。在不可使用计算器的试卷中，使用公分母，或把所有的量都化成以 100 为分母的百分数。
3. Adding and Subtracting Fractions | 分数的加减法
p the denominator: 3/8 + 2/8 = 5/8, and 7/9 − 4/9 = 3/9 = 1/3. Always simplify at the end.
同分母时，分子相加或相减，分母保持不变：3/8 + 2/8 = 5/8，7/9 − 4/9 = 3/9 = 1/3。最后一定要约分。
LCM of the denominators. For 2/3 + 1/4 the LCD is 12, so 2/3 + 1/4 = 8/12 + 3/12 = 11/12.
分母不同时，先求最小公分母（LCD），即各分母的最小公倍数。对于 2/3 + 1/4，LCD 是 12，所以 2/3 + 1/4 = 8/12 + 3/12 = 11/12。
a/b ± c/d = (a × d ± c × b)/(b × d), then simplify
2 = 7/12. Watch the order when the second fraction is larger: 1/4 − 5/6 = 3/12 − 10/12 = −
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