📚 Percentages | 百分比
Percentages appear everywhere, from shop discounts to exam scores. The term 'per cent' comes from Latin 'per centum', meaning 'by the hundred'. At KS3, learning to work confidently with percentages strengthens your number skills and prepares you for real-world problem solving. A solid understanding of percentages connects fractions, decimals, and ratios, and it opens the door to interpreting data, managing money, and making informed decisions.
百分比无处不在，从商店折扣到考试成绩。“百分之”这一术语源自拉丁语，意思是“每一百”。在KS3阶段，学会熟练运用百分比能够加强你的数字运算能力，并为解决现实世界的问题打下基础。扎实掌握百分比能够将分数、小数和比率联系起来，并为你解读数据、管理金钱以及做出明智决策打开大门。
1. Understanding the Meaning of Percentage | 理解百分比的含义
A percentage is a number or ratio expressed as a fraction with denominator 100. The symbol % is equivalent to 1/100. So 7% means 7 out of 100, or 7/100. Similarly, 30% equals 30/100, which simplifies to 3/10, and 125% f 100 makes it easy to compare different amounts, whether you are looking at test results, survey data, or discounts.
百分比是一种以100为分母的分数或比率。符号%相当于1/100。因此7%表示100份中的7份，也就是7/100。同样地，30%等于30/100（可化简为3/10），125%等于125/100（即5/4）。由于所有百分比都以100为统一的基准，无论是考试成绩、调查数据还是折扣，都能轻松进行比较。
If 45% of pupils in a school walk to school, this means 45 out of every 100 pupils walk. You can imagine taking the school population and splitting it into groups of 100; in each group, 45 pupils walk. This visual helps you estimate percentages even when the total is not a multiple of 100.
如果某所学校45%的学生步行上学，就意味着每100名学生中有45人步行。你可以将全校学生想象成很多组100人，每一组中都有45人步行。即使总人数不是100的倍数，这种可视化的方法也能帮助你估算百分比。
2. Converting between Percentages, Fractions and Decimals | 百分比、分数和小数的互换
e by 100. This is done by moving the decimal point two places to the left. For example, 85% ÷ 100 = 0.85. If there is no decimal point visible, put one after the last digit first: 6% ercentage as a fraction, place the percentage number over 100 and simplify if possible: 65% = 65/100 = 13/20. Mixed number percentages like 12½% can be written as 12.5/100 = 125/1000 = 1/8.
将百分比转换成小数时，除以100，也就是把小数点向左移动两位。例如85% = 85 ÷ 100 = 0.85。如果原本没有小数点，先在最后一位数字后加上小数点：6% = 6.0 ÷ 100 = 0.06。将百分比写成分数时，把百分号前的数字作为分子、100作为分母，能化简的要化简：65% = 65/100 = 13/20。像12½%这样的带分数百分比可以写作12.5/100 = 125/1000 = 1/8。
- ge a decimal to a percentage, multiply by 100 (move the point two places right). 0.47 = 47%
- 将小数化为百分比，乘以100（小数点右移两位）。0.47 = 47%
- o a decimal by dividing numerator by denominator, then multiply by 100. 3/8 = 0.375 = 37.5%
- 将分数化为百分比，先用分子除以分母得到小数，再乘以100。3/8 = 0.375 = 37.5%
- Common equivalents should be memorised, such as 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/10 = 10%.
- 一些常用的等值应当牢记，比如1/2 = 50%、1/4 = 25%、1/5 = 20%、1/10 = 10%。
Decimal to percentage: 0.72 × 100 = 72%
小数化百分比：0.72 × 100 = 72%
3. Finding a Percentage of an Amount | 求一个数的百分之几
The first is to convert the percentage to a decimal and multiply. For example, to find 15% of 80, write 15% = 0.15, then 0.15 × 80 = 12. The second method uses simple building blocks: find 10% by dividing by 10, then 5% is half of that, and 1% is one-tenth of 10%. So for 15% of 80: 10% = 8, 5% = 4, therefore 15% = 8 + 4 = 12.
在不使用计算器的情况下，有两种主要方法可以求出一个数的百分之几。第一种是将百分比转换为小数再相乘。例如求80的15%，把15%写成0.15，然后0.15 × 80 = 12。第二种方法是利用基本单位：除以10得到10%，5%是10%的一半，1%是10%的十分之一。80的15%可以这样算：10% = 8，5% = 4，所以15% = 8 + 4 = 12。
With a calculator, you can key in the percentage directly: 80 × 15% e building-block method is particularly handy for mental maths and for percentages like 17% where you can combine 10%, 5% and two 1%s.
使用计算器时，你可以直接输入80 × 15 %得到12，或者使用小数乘法。基本单位法在心算时尤其方便，对于17%这样的百分比，你可以将10%、5%和两个1%组合起来。
Example: 35% of 220 = 0.35 × 220 = 77
示例：220的35% = 0.35 × 220 = 77
4. Percentage Increase and Decrease | 百分数的增加和减少
y a percentage, the new value is greater than the original. If you increase an amount by r%, you multiply the original by (1 + r/100). For a decrease of r%, multiply by (1 - r/100). For instance, a 20% increase on £45 gives £45 × 1.20 = £54. A 15% decrease on a £200 coat gives £200 × 0.85 = £170.
当某个量以百分比增长时，新的值会比原来大。如果将原值增加r%，就乘以(1 + r/100)。如果减少r%，则乘以(1 - r/100)。例如，45英镑增加20%后为45 × 1.20 = 54英镑。一件200英镑的大衣减少15%后为200 × 0.85 = 170英镑。
the increase or decrease amount first and then add or subtract from the original. For a 12% increase on 250: 12% of 250 = 30, so new value = 250 + 30 = 280. For a 40% decrease on 90: 40% methods give the same result, and choosing the right one depends on which you find easier.
你也可以先算出增加或减少的量，再从原值中加上或减去该量。250增加12%：250的12% = 30，新值 = 250 + 30 = 280。90减少40%：90的40% = 36，新值 = 90 - 36 = 54。两种方法得到的结果相同，你可以选择自己觉得更简单的一种。
Increase: New = Original × (1 + percentage/100)
增加：新值 = 原值 × (1 + 百分比/100)
Decrease: New = Original × (1 - percentage/100)
减少：新值 = 原值 × (1 - 百分比/100)
5. Expressing One Quantity as a Percentage of Another | 一个量占另一个量的百分之几
To write one number as a percentage of another, form a fraction with the 'part' as the numerator and the 'whole' as the denominator, then multiply by 100%. For example, if you score 18 out of 25 on a test, your percentage mark is (18/25) × 100% = 0.72 × 100% = 72%. The formula is: percentage = (part ÷ whole) × 100%.
要把一个数写成另一个数的百分比，就用“部分”做分子、“整体”做分母，形成分数，再乘以100%。例如，如果你在一次考试中得了18分（满分25分），你的百分比分数为 (18/25) × 100% = 0.72 × 100% = 72%。公式为：百分比 = (部分 ÷ 整体) × 100%。
s. If you are comparing 30 cm out of 2 m, first convert 2 m to 200 cm, then (30/200) × 100% = 15%. The answer can be a decimal or fraction of a percent, like 33.3% or 16⅔%.
务必确保部分和整体的单位一致。如果比较30厘米和2米，先将2米转化为200厘米，再计算 (30/200) × 100% = 15%。答案可以是小数的百分比，比如33.3%或16⅔%。
| Part | Whole | Fraction | Percentage |
| 15 | 60 | 15/60 = 1/4 | 25% |
| 7 | 20 | 7/20 | 35% |
6. Percentage Change | 百分比变化
to its original size. The formula is: percentage change = (change / original amount) × 100%. The 'change' ew is the difference between the new value and the original value (new - original). If the price of a movie ticket rose from £8 to £10, the change is +£2 and the percentage increase = (2/8) × 100% = 25%. If it fell from £10 to £7, the change is -£3 and the decrease = (3/10) × 100% = 30%.
百分比变化衡量某个值相对于原始大小增长或缩小了多少。公式为：百分比变化 = (变化量 / 原值) × 100%。“变化量”是新值和原值的差（新值 - 原值）。如果一张电影票的价格从8英镑涨到10英镑，变化量为+2英镑，增长的百分比 = (2/8) × 100% = 25%。如果从10英镑降到7英镑，变化量为-3英镑，下降的百分比 = (3/10) × 100% = 30%。
Always use the original amount as the denominator. A common mistake is to divide by the new amount, which gives a different and incorrect result. Percentage change can be more than 100% if the new value is more than double the original.
始终用原值作为分母。一个常见的错误是除以新值，那样会得到不同的错误结果。如果新值超过原值的两倍，百分比变化可以大于100%。
7. Reverse Percentages | 逆向百分数
amount before a percentage increase or decrease. For example, if a guitar is reduced by 20% in a sale and now costs £240, what was the original price? After a 20% reduction, you pay 80% of the original, so 0.80 × original = 240. Thus original = 240 ÷ 0.8 = £300. Similarly, if a salary after a 5% rise is £31,500, then 1.05 × original = 31,500, so original = 31,500 ÷ 1.05 = £30,000.
逆向百分比问题要求你求出百分数增减之前的原始值。例如，一把吉他在促销中降价20%，现价为240英镑，原价是多少？降价20%后，你支付原价的80%，所以0.80 × 原价 = 240。因此原价 = 240 ÷ 0.8 = 300英镑。同样地，如果加薪5%后的工资是31,500英镑，那么1.05 × 原工资 = 31,500，原工资 = 31,500 ÷ 1.05 = 30,000英镑。
Do not simply subtract 20% of £240 from £240; that would give £ epresents of the original. Always set up an equation: (1 ± r/100) × original = new amount.
千万不要只是从240英镑中减去240的20%，那样会得到192英镑，这是错误的。关键是要弄清楚新值对应原值的百分比。始终建立等式：(1 ± r/100) × 原值 = 新值。
8. Percentages Greater than 100% and Less than 1% | 大于100%和小于1%的百分比
A percentage can be more than 100%. For example, if a company's profit one year is £40,000 and the next year it is £100,000, the second year's profit is 250% of the first. To find 150% of an amount, multiply by 1.5. Percentages less than 1% arise when dealing with very small proportions. For instance, 0.2% means 0.2 out of 100, which is 0.002 as a decimal and 1/500 as a fraction. To find 0.5% of 600, multiply 600 by 0.005 to get 3.
百分比可以大于100%。例如，某公司一年的利润为40,000英镑，下一年为100,000英镑，那么下一年的利润是第一年的250%。要求某个数的150%，可乘以1.5。在处理非常小的比例时，会出现小于1%的百分比。比如，0.2%表示100份中的0.2份，写成小数是0.002，写成分数是1/500。要求600的0.5%，就用600乘以0.005得到3。
Working with percentages outside the 0– e follows exactly the same rules. Just remember to convert them to decimals correctly: 205% = 2.05, 0.05% = 0.0005.
处理超出0–100范围的百分比时，遵循完全相同的规则。只需记住正确地将它们转化为小数：205% = 2.05，0.05% = 0.0005。
9. Percentages in Real-Life Contexts | 实际生活中的百分比应用
ed in discounts, VAT (value-added tax), simple interest, and profit calculations. For a 30% discount on a £50 jacket, you pay 70%, so cost = 50 × 0.70 = £35. If VAT at 20% is added to a base price of £120, the final price = 120 × 1.20 = £144. Simple interest: if you invest £400 for 2 years at 5% per year, interest = 400 × 0.05 × 2 = £40.
百分比广泛应用于折扣、增值税、单利和利润计算中。一件50英镑的夹克打七折（30% off），你支付70%，即50 × 0.70 = 35英镑。如果对120英镑的基本价格加收20%的增值税，最终价格为120 × 1.20 = 144英镑。单利：如果你以年利率5%投资400英镑2年，利息 = 400 × 0.05 × 2 = 40英镑。
Being able to calculate these quickly helps you compare deals. For instance, a 'buy one get one half price' offer on two items each costing £10 means you pay £10 + £5 = £15, which is a 25% saving off the full £20, not a 50% saving. Percentages allow you to see the true discount.
能够快速计算这些，能帮助你比较各种优惠。例如，两件单价10英镑的商品享受“第二件半价”，你支付10 + 5 = 15英镑，这相当于原价20英镑的75%，也就是节省了25%，而不是50%。百分比能让你看清真正的折扣。
10. Common Mistakes and How to Avoid Them | 常见错误及避免方法
One common error is mixing up the base when calculating percentage change. Always use the original amount as the denominator. Another mistake is adding or subtracting percentages without considering whether they are applied in sequence. A 10% increase followed by a 10% decrease does not return you to the original; for an item costing £100, a 10% rise makes it £110, then a 10% decrease on £110 brings it to £99. Always apply percentages to the current value.
一个常见错误是在计算百分比变化时弄混了基数。务必始终用原值做分母。另一个错误是在没有考虑先后顺序的情况下直接加减百分比。先涨10%再降10%并不能回到原值；对于一件100英镑的商品，先涨10%变成110英镑，再降10%只降到99英镑。一定要基于当前值应用百分比。
Students also sometimes forget to convert units when expressing one quantity as a percentage of another, or they misplace the decimal point when moving between decimals and percentages. Double-check your decimal places: 0.5% is 0.005, not 0.5. Writing out the steps and using the fraction‑decimal‑percentage triangle can help prevent these mistakes.
学生们有时还会忘记在求一个量占另一个量的百分之几时转换单位，或者在百分数和小数转换时搞错小数点的位置。务必仔细检查小数位数：0.5%是0.005，而不是0.5。写出解题步骤，并运用分数‑小数‑百分比三角关系图，有助于避免这些错误。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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