Student’s Edition: Grade 2 Lesson 16-7 – Fractions, Decimals and Percentages | 学生版：二年级第16-7课——分数、小数与百分数

📚 Student’s Edition: Grade 2 Lesson 16-7 – Fractions, Decimals and Percentages | 学生版：二年级第16-7课——分数、小数与百分数

In this IGCSE Mathematics lesson, we focus on fractions, decimals and percentages. These three forms represent parts of a whole and appear throughout the Core and Extended papers.

本节 IGCSE 数学课重点复习分数、小数和百分数。这三种形式都表示整体的一部分，在核心和扩展试卷中反复出现。

1. Understanding Fractions | 分数的基本概念

A fraction is written as a/b, where a is the numerator and b is the denominator. The denominator cannot be zero because division by zero is undefined.

分数写作 a/b，其中 a 是分子，b 是分母。分母不能为零，因为除以零没有定义。

For example, 3/5 means 3 equal parts out of 5. The fraction bar represents division, so 3/5 = 3 ÷ 5 = 0.6.

例如，3/5 表示 5 等份中的 3 份。分数线代表除法，因此 3/5 = 3 ÷ 5 = 0.6。

A proper fraction has a numerator smaller than its denominator, such as 2/7. An improper fraction has a numerator greater than or equal to its denominator, such as 9/4.

真分数的分子比分母小，例如 2/7。假分数的分子大于或等于分母，例如 9/4。

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2. Equivalent Fractions and Simplifying | 等值分数与化简

Equivalent fractions have the same value even though they look different. You can multiply or divide both the numerator and the denominator by the same non-zero number without changing the value.

等值分数虽然形式不同，但数值相同。你可以将分子和分母同时乘以或除以同一个非零数，分数值不变。

2/3 = 4/6 = 10/15

To simplify a fraction, divide the numerator and denominator by their highest common factor (HCF). A fraction is in simplest form when the numerator and denominator have no common factor other than 1.

化简分数时，用分子和分母的最大公因数（HCF）同时相除。当分子和分母除 1 以外没有其他公因数时，分数就是最简形式。

For example, 18/24 simplifies to 3/4 because the HCF of 18 and 24 is 6.

例如，18/24 化简为 3/4，因为 18 和 24 的最大公因数是 6。

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3. Mixed Numbers and Improper Fractions | 带分数与假分数

A mixed number combines a whole number and a proper fraction, such as 2 ¾. An improper fraction has a numerator larger than or equal to its denominator, such as 11/4.

带分数由整数和真分数组成，例如 2 ¾。假分数的分子大于或等于分母，例如 11/4。

Convert a mixed number to an improper fraction by multiplying the whole number by the denominator, adding the numerator, and placing the result over the original denominator.

将带分数化为假分数：整数乘以分母，加上分子，将结果作为分子，分母不变。

2 ¾ = (2 × 4 + 3)/4 = 11/4

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number, and the remainder is the new numerator.

将假分数化为带分数时，用分子除以分母。商是整数部分，余数是新的分子。

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4. Adding and Subtracting Fractions | 分数的加减

To add or subtract fractions, they must have the same denominator. If the denominators differ, find the lowest common multiple (LCM) of the denominators first.

分数加减前必须先通分，使分母相同。如果分母不同，先求分母的最小公倍数（LCM）。

Once the denominators are equal, add or subtract the numerators only. The denominator stays the same, and the result should be simplified if possible.

分母相同后，只对分子进行加减，分母保持不变，最后能化简就化简。

1/3 + 1/4 = 4/12 + 3/12 = 7/12

For mixed numbers, add or subtract the whole number parts and the fractional parts separately, borrowing or carrying when needed.

对于带分数，可以分别加减整数部分和分数部分，必要时进行借位或进位。

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5. Multiplying and Dividing Fractions | 分数的乘除

To multiply fractions, multiply the numerators together and multiply the denominators together. Simplify the result if possible.

分数相乘时，分子乘分子，分母乘分母，最后能化简就化简。

2/3 × 3/5 = 6/15 = 2/5

To divide by a fraction, multiply by its reciprocal. The reciprocal of a/b is b/a, provided a is not zero.

分数相除时，乘以除数的倒数。a/b 的倒数是 b/a，其中 a 不为零。

4/5 ÷ 2/3 = 4/5 × 3/2 = 12/10 = 6/5

When multiplying or dividing mixed numbers, change them to improper fractions first. This avoids errors with whole number parts.

乘除带分数时，先把它们化成假分数，这样可以避免整数部分带来的错误。

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6. Decimals and Place Value | 小数与位值

Decimals use place value to show tenths, hundredths, thousandths, and so on. The first digit after the decimal point is the tenths digit.

小数通过位值表示十分位、百分位、千分位等。小数点后第一位是十分位。

For example, in 0.375, the 3 is in the tenths place, the 7 is in the hundredths place, and the 5 is in the thousandths place.

例如，在 0.375 中，3 在十分位，7 在百分位，5 在千分位。

To compare decimals, align the decimal points and compare digits from left to right. Adding zeros to the right of the decimal does not change the value.

比较小数时，将小数点对齐，从左到右逐位比较。在小数右边补零不会改变小数的值。

For example, 0.45 is greater than 0.405 because in the hundredths place 5 is greater than 0.

例如，0.45 大于 0.405，因为百分位上 5 大于 0。

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7. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数互换

A percentage is a fraction with denominator 100. To convert a fraction to a percentage, first divide the numerator by the denominator, then multiply by 100.

百分数是分母为 100 的分数。把分数化为百分数，先用分子除以分母，再乘以 100。

To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100.

小数化为百分数时乘以 100；百分数化为小数时除以 100。

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/10	0.1	10%
1/3	0.333…	33⅓%

Recurring decimals such as 0.333… can be written with dot notation or converted into fractions using algebraic methods.

循环小数如 0.333… 可以用点标记表示，也可以用代数方法转换成分数。

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8. Percentage Increase and Decrease | 百分数增减

To increase an amount by r%, multiply the original amount by (1 + r/100). To decrease an amount by r%, multiply by (1 – r/100).

将一个量增加 r% 时，原量乘以 (1 + r/100)；减少 r% 时，乘以 (1 – r/100)。

New value = Original value × (1 ± r/100)

For example, increasing £80 by 15% gives £80 × 1.15 = £92.

例如，£80 增加 15% 得到 £80 × 1.15 = £92。

Percentage change is calculated as (change ÷ original value) × 100%. A positive result means an increase, and a negative result means a decrease.

百分比变化用（变化量 ÷ 原值）× 100% 计算。结果为正表示增加，结果为负表示减少。

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9. Solving Word Problems | 应用题策略

Read the problem carefully and identify what the ‘whole’ is. Decide whether you need to add, subtract, multiply, divide, or compare before choosing a method.

仔细阅读题目，确定“整体”是什么。先判断需要加减乘除还是比较，再选择方法。

Convert all numbers to the same form — fractions, decimals, or percentages — before calculating. This reduces careless mistakes.

计算前把所有数转换成同一种形式——分数、小数或百分数，这样可以减少粗心错误。

Check that your answer makes sense in the context. For example, a fraction of a positive quantity cannot be negative.

检查答案是否符合题意。例如，正数量的分数不可能为负。

Use estimation first to catch unreasonable answers. For instance, 25% of 80 should be close to 20, not 200 or 2.

先估算，以便发现不合理答案。例如，80 的 25% 应接近 20，而不是 200 或 2。

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10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Always simplify your final fraction unless the question says otherwise. IGCSE marking schemes often require simplest form.

除非题目另有说明，最终分数一定要化为最简形式。IGCSE 评分标准通常要求最简分数。

When adding fractions, do not add the denominators. Convert to a common denominator first, then add only the numerators.

分数相加时不要把分母直接相加，应先通分，然后只把分子相加。

Be careful with percentage increase and decrease: a 20% increase followed by a 20% decrease does not return to the original value.

注意百分数增减：先增加 20% 再减少 20% 不会回到原值，因为第二次的基数是变化后的量。

Show all steps clearly. Even if the final answer is wrong, method marks can be gained in IGCSE papers.

清楚写出每一步。在 IGCSE 试卷中，即使最后答案错误，也能获得方法分。

Check recurring decimal notation and rounding rules, especially when the question asks for the answer correct to 2 decimal places or 3 significant figures.

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