📚 Fractions, Decimals and Percentages | 分数、小数与百分比
In KS3 Cambridge Mathematics, fractions, decimals and percentages are three connected ways of representing parts of a whole. This article reviews the key skills you need, from simplifying fractions to solving real-life percentage problems. Mastering these conversions and calculations will help you across number, ratio, proportion and statistics topics.
在 KS3 剑桥数学中，分数、小数和百分比是表示整体部分的三种相互关联的方式。本文复习你所需的关键技能，从约分到解决实际百分比问题。掌握这些转换和计算将帮助你在数、比、比例和统计等主题中取得好成绩。
1. Understanding Equivalent Fractions | 理解等值分数
Equivalent fractions have the same value even though they use different numerators and denominators. You can find an equivalent fraction by multiplying or dividing both the top number and the bottom number by the same non-zero value.
等值分数具有相同的值，尽管它们使用不同的分子和分母。你可以通过将分子和分母同时乘以或除以同一个非零数来找到一个等值分数。
1/2 = 2/4 = 3/6 = 5/10
Always check that the multiplier or divisor is the same for both parts; otherwise the value of the fraction changes. This idea is the foundation for simplifying, comparing, adding and subtracting fractions.
始终检查乘数或除数对分子和分母是否相同；否则分数的值会改变。这个思想是约分、比较、加减分数的基础。
2. Converting Between Mixed Numbers and Improper Fractions | 假分数与带分数互化
A mixed number contains a whole number and a proper fraction, such as 2 3/4. An improper fraction has a numerator that is greater than or equal to its denominator, such as 11/4.
带分数包含一个整数和一个真分数，例如 2 3/4。假分数的分子大于或等于分母，例如 11/4。
To convert a mixed number to an improper fraction, multiply the whole number by the denominator, then add the numerator. Write this result over the original denominator.
要将带分数转换为假分数，先将整数乘以分母，然后加上分子。将所得结果写在原分母之上。
2 3/4 = (2 × 4 + 3) / 4 = 11/4
To convert back, divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays the same.
要转换回带分数，用分子除以分母。商是整数部分，余数是新的分子，分母保持不变。
3. Simplifying Fractions | 约分
Simplifying a fraction means writing it in its lowest terms. You do this by dividing the numerator and the denominator by their greatest common factor, or HCF.
约分意味着将一个分数写成最简形式。你可以通过将分子和分母除以它们的最大公因数（HCF）来实现。
18/24 = 18 ÷ 6 / 24 ÷ 6 = 3/4
If you cannot find the HCF immediately, you can divide in stages. For example, 18/24 can become 9/12 by dividing by 2, then 3/4 by dividing by 3.
如果你无法立刻找到最大公因数，可以分步除以公因数。例如，18/24 可以先除以 2 得到 9/12，再除以 3 得到 3/4。
Always check whether the final fraction can be reduced further. A fraction is fully simplified when the only common factor of the numerator and denominator is 1.
始终检查最终分数是否还能继续约分。当分子和分母的唯一公因数是 1 时，分数就是最简形式。
4. Comparing and Ordering Fractions | 比较和排序分数
To compare fractions, you need a common denominator. The easiest method is to use the lowest common multiple of the denominators, called the lowest common denominator.
要比较分数，你需要一个公分母。最简单的方法是使用分母的最小公倍数，称为最小公分母。
Convert each fraction to an equivalent fraction with that common denominator, then compare the numerators. The larger numerator gives the larger fraction.
将每个分数转换为具有该公分母的等值分数，然后比较分子。分子较大的分数也较大。
Compare 2/3 and 3/5: 2/3 = 10/15, 3/5 = 9/15, so 2/3 > 3/5
You can also convert fractions to decimals or percentages to compare them, especially when using a calculator or dealing with awkward denominators.
你也可以将分数转换为小数或百分比来进行比较，尤其是在使用计算器或处理复杂分母时。
5. Adding and Subtracting Fractions | 分数加减法
To add or subtract fractions, the denominators must be the same. If they are not, find the lowest common denominator and rewrite each fraction as an equivalent fraction.
要加减分数，分母必须相同。如果分母不同，找到最小公分母，并将每个分数改写为等值分数。
1/4 + 2/3 = 3/12 + 8/12 = 11/12
Once the denominators are equal, add or subtract the numerators only. Keep the denominator unchanged. Simplify the answer if possible.
一旦分母相同，只对分子进行加减。分母保持不变。如果可能，将答案化简为最简分数。
For mixed numbers, add or subtract the whole number parts and the fractional parts separately. If the fractional part becomes improper, convert it and adjust the whole number.
对于带分数，分别加减整数部分和分数部分。如果分数部分变成假分数，需要转换并调整整数部分。
6. Multiplying and Dividing Fractions | 分数乘除法
Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. Simplify the result if needed.
分数乘法很直接：分子与分子相乘，分母与分母相乘。如果需要，化简结果。
2/5 × 3/4 = (2 × 3) / (5 × 4) = 6/20 = 3/10
For mixed numbers, convert them to improper fractions first, then multiply as usual.
对于带分数，先转换为假分数，然后按照通常的乘法步骤计算。
To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction flips the numerator and the denominator.
除以一个分数等于乘以它的倒数。一个分数的倒数就是将分子和分母互换位置。
3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2 1/10
7. Fractions of Amounts | 求一个数的几分之几
To find a fraction of an amount, divide the amount by the denominator and then multiply the result by the numerator. This method works for any quantity, including money, length and mass.
要求一个数的几分之几，先用这个数除以分母，然后将结果乘以分子。这个方法适用于任何数量，包括钱、长度和质量。
Find 3/4 of 48: 48 ÷ 4 = 12; 12 × 3 = 36
You can also multiply the amount by the fraction directly. Writing the amount as a fraction over 1 can help you see the calculation clearly.
你也可以直接用这个数乘以分数。把这个数写成分母为 1 的分数可以帮助你更清楚地看到计算过程。
3/4 × 48 = (3 × 48) / 4 = 144/4 = 36
8. Converting Fractions to Decimals | 分数转小数
A fraction can be converted to a decimal by dividing the numerator by the denominator. Some conversions terminate, while others repeat.
分数可以通过分子除以分母转换为小数。有些小数是有限的，有些则是循环小数。
3/8 = 3 ÷ 8 = 0.375
to speed up your work. For example, 1/4 = 0.25, 1/3 = 0.333..., 1/2 = 0.5, and 3/4 = 0.75.
常见的转换应该记住以加快计算速度。例如，1/4 = 0.25，1/3 = 0.333...，1/2 = 0.5，3/4 = 0.75。
To convert a decimal to a fraction, write the decimal as a fraction with a denominator of 10, 100 or 1000 depending on the number of decimal places, then simplify.
要将小数转换为分数，根据小数位数将小数写成分母为 10、100 或 1000 的分数，然后化简。
0.45 = 45/100 = 9/20
9. Converting Decimals to Percentages and Back | 小数与百分比互化
A percentage means 'out of 100'. To convert a decimal to a percentage, multiply the decimal by 100 and add the % sign. To convert a percentage to a decimal, divide by 100.
百分比的意思是“每 100 份中的多少”。要将小数转换为百分比，将小数乘以 100 并加上 % 符号。要将百分比转换为小数，除以 100。
0.62 × 100 = 62% and 45% = 45 ÷ 100 = 0.45
To convert a fraction to a percentage, first convert the fraction to a decimal, then multiply by 100. Alternatively, find an equivalent fraction with denominator 100.
要将分数转换为百分比，先将分数转换为小数，然后乘以 100。或者找到一个分母为 100 的等值分数。
3/5 = 0.6 = 60%
These conversions are essential for comparing fractions, decimals and percentages in the same question.
这些转换对于在同一道题中比较分数、小数和百分比至关重要。
10. Solving Real-Life FDP Problems | 解决实际 FDP 问题
Many KS3 exam questions ask you to apply fraction, decimal and percentage skills to real-life contexts such as discounts, test scores, recipes and surveys. Read the problem carefully and identify what the whole amount is before calculating.
许多 KS3 考试题目要求你将分数、小数和百分比的技能应用到实际情境中，例如折扣、考试成绩、食谱和调查。在计算之前，要仔细阅读题目并确定整体量是什么。
For percentage increase or decrease, work out the change first, then divide by the original amount and multiply by 100. Always state the units clearly in your final answer.
对于百分比的增加或减少，先算出变化量，然后除以原始量并乘以 100。最终答案中要清楚地写出单位。
A jacket costing £40 is reduced by 15%. New price = 40 × 0.85 = £34
When a quantity increases, add the percentage to 100%; when it decreases, subtract from 100%. This is often faster than finding the change and then adding or subtracting it separately.
当数量增加时，用 100% 加上百分比；当数量减少时，用 100% 减去百分比。这通常比先求出变化量再单独相加或相减更快。
11. Common Misconceptions and Exam Tips | 常见误区与考试技巧
One common mistake is adding the denominators when adding fractions. Remember that the denominator tells you the size of the parts and must stay the same once a common denominator is found.
一个常见错误是在分数相加时将分母也相加。请记住，分母表示每份的大小，找到公分母后它必须保持不变。
Another common error is forgetting to simplify the final answer. In exams, un-simplified fractions can lose marks, so always look for common factors at the end.
另一个常见错误是忘记化简最终答案。在考试中，未化简的分数可能会丢分，所以最后一定要寻找公因数。
When converting decimals to percentages, do not forget to multiply by 100. Writing 0.45 as 45% is correct, but writing 0.45% is a different value and is often a sign of a misplaced decimal point.
将小数转换为百分比时，不要忘记乘以 100。把 0.45 写成 45% 是正确的，但写成 0.45% 则是另一个值，通常说明小数点位置放错了。
Finally, show clear working for every question. This helps you check for errors and can earn method marks even if the final answer is not fully correct.
最后，每道题都要写出清晰的计算过程。这有助于你检查错误，即使最终答案不完全正确，也能获得步骤分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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