Solving Practical Problems | 实际问题求解

📚 Solving Practical Problems | 实际问题求解

Practical problems in IGCSE mathematics ask you to take a real-life situation, convert it into a mathematical form, solve it, and then interpret the answer in the original context. You will be expected to choose the correct operation, use efficient methods, and give answers with appropriate accuracy and units.

IGCSE 数学中的实际问题要求你将真实情境转化为数学形式、进行求解，并用原情境来解释答案。你需要选择正确的运算方法，使用高效策略，并以适当的精度和单位给出答案。

1. Understanding the Problem | 理解问题

Before starting any calculation, read the question carefully. Identify what you are given, what you need to find, and which units are involved. Highlight or underline key data and the final question.

在开始任何计算之前，请仔细阅读题目。找出已知条件、需要求的量以及涉及的单位。用笔标出关键数据和最终问题。

List the known quantities with their units.

将已知量及其单位列出来。
Write the question in your own words.

用自己的语言复述问题。
Decide whether the answer should be rounded, and to how many significant figures.

判断答案是否需要取近似值，以及保留几位有效数字。

For example, ‘A train travels 240 km at 60 km/h. How long does the journey take?’ You know distance and speed; time is unknown.

例如：’一列火车以 60 km/h 的速度行驶 240 km，需要多长时间？’ 你已知距离和速度，时间是未知量。

2. Formulating Equations | 建立方程

Many practical problems can be solved by translating words into an equation. Let the unknown be x, then express the relationships in the problem using operations and an equals sign.

许多实际问题可以通过将文字转化为方程来解决。设未知数为 x，然后用运算和等号来表达题目中的关系。

Example: A shop sells pens and notebooks. A pen costs £1.20 and a notebook costs £3.50. If someone buys 4 pens and some notebooks, and pays £13.30, how many notebooks are bought?

例如：一家商店出售笔和笔记本。笔每支 1.20 英镑，笔记本每本 3.50 英镑。如果某人买了 4 支笔和若干本笔记本，共付 13.30 英镑，他买了几本笔记本？

4 × 1.20 + 3.50n = 13.30

Solve: 4.80 + 3.50n = 13.30, so 3.50n = 8.50, n = 2.43. Since the answer must be a whole number, check the question: perhaps the total is wrong or the numbers need adjustment.

求解：4.80 + 3.50n = 13.30，所以 3.50n = 8.50，n = 2.43。因为答案必须是整数，请检查题目：可能总额有误或数字需要调整。

3. Percentages in Real Life | 生活中的百分比

Percentage problems appear in discounts, taxes, profit and loss, and rises or falls. The key is to identify the original amount and the percentage change.

百分比问题出现在折扣、税收、盈利亏损以及涨跌中。关键是找出原始数额和百分比变化。

Simple percentage: 15% of £80 = 0.15 × 80 = £12.

简单百分比：80 英镑的 15% = 0.15 × 80 = 12 英镑。
Percentage increase: A price rises by 8%. New price = original × 1.08.

百分比的增加：价格上升 8%，新价格 = 原价 × 1.08。
Percentage decrease: A price falls by 8%. New price = original × 0.92.

百分比的减少：价格下降 8%，新价格 = 原价 × 0.92。

Example: A coat is priced at £80. In a sale it is reduced by 30%. What is the sale price?

例如：一件外套原价 80 英镑，促销时降价 30%，促销价是多少？

80 × (1 − 0.30) = 80 × 0.70 = £56

4. Ratio and Proportion | 比与比例

Ratios are used to share quantities, adjust recipes, or convert between different scales. A proportion states that two ratios are equal.

比用于分配数量、调整食谱或在不同比例尺之间换算。比例表示两个比相等。

Example: A recipe for 4 people needs 300 g of flour. How much flour is needed for 7 people?

例如：一个 4 人份食谱需要 300 克面粉，7 人份需要多少克？

flour per person = 300 ÷ 4 = 75 g; for 7 people = 75 × 7 = 525 g

When sharing, add the parts of the ratio to find the total number of parts, then divide the amount by that total, and multiply by each part.

分配时，将比的所有部分相加得到总份数，然后用总量除以总份数，再乘以每部分。

5. Speed, Distance and Time | 速度、距离与时间

The relationship between speed, distance and time is central to many practical problems. Always check the units: if speed is in km/h, time must be in hours.

速度、距离和时间的关系是许多实际问题的核心。请始终检查单位：如果速度单位是 km/h，时间必须用小时。

speed = distance ÷ time, distance = speed × time, time = distance ÷ speed

Example: A cyclist travels at 15 m/s for 2 hours. First convert 2 hours to seconds: 2 × 3600 = 7200 s. Distance = 15 × 7200 = 108 000 m = 108 km.

例如：一位骑行者以 15 m/s 的速度行驶 2 小时。先将 2 小时转换为秒：2 × 3600 = 7200 秒。距离 = 15 × 7200 = 108 000 米 = 108 千米。

For average speed, divide the total distance by the total time, even if the journey included stops.

对于平均速度，即使行程中有停顿，也要用总距离除以总时间。

6. Area and Volume Problems | 面积与体积问题

Practical area and volume problems often involve finding how much material or paint is needed, or how many tiles fit a floor. Draw a sketch if possible and use the correct formula.

实际的面积与体积问题通常涉及需要多少材料或油漆，或者多少瓷砖能铺满地面。如有可能请画出示意图，并使用正确的公式。

Example: A rectangular room measures 6 m by 4 m. Tiles are 0.5 m by 0.5 m. How many tiles are needed?

例如：一个长方形房间长 6 米、宽 4 米。瓷砖为 0.5 米 × 0.5 米，需要多少块？

Area of room = 6 × 4 = 24 m². Area of one tile = 0.5 × 0.5 = 0.25 m². Number of tiles = 24 ÷ 0.25 = 96.

房间面积 = 6 × 4 = 24 平方米。一块瓷砖面积 = 0.5 × 0.5 = 0.25 平方米。瓷砖数量 = 24 ÷ 0.25 = 96 块。

For volume problems, remember that 1 litre = 1000 cm³ and 1 m³ = 1000 litres.

对于体积问题，请记住 1 升 = 1000 立方厘米，1 立方米 = 1000 升。

7. Money and Simple Interest | 金钱与单利

Financial problems include profit, loss, discounts, and simple interest. For simple interest, use the formula with principal, rate, and time.

金融问题包括盈利、亏损、折扣和单利。对于单利，使用涉及本金、利率和时间的公式。

simple interest = principal × rate × time, where rate is in decimals

Example: Invest £2000 at 5% per year for 3 years. Interest = 2000 × 0.05 × 3 = £300. Total amount after 3 years = £2300.

例如：以每年 5% 的利率投资 2000 英镑，为期 3 年。利息 = 2000 × 0.05 × 3 = 300 英镑。3 年后的总金额 = 2300 英镑。

Profit and loss: profit = selling price − cost price; percentage profit = profit ÷ cost price × 100%.

盈利与亏损：利润 = 售价 − 成本价；利润率 = 利润 ÷ 成本价 × 100%。

8. Scales and Maps | 比例尺与地图

Maps and scale drawings use a ratio to represent real lengths. A scale of 1 : 50 000 means 1 cm on the map represents 50 000 cm on the ground, which is 500 m or 0.5 km.

地图和比例图使用比值来表示真实长度。比例尺 1 : 50 000 表示地图上的 1 厘米代表地面上的 50 000 厘米，即 500 米或 0.5 千米。

Example: On a map with scale 1 : 200 000, two towns are 4.5 cm apart. What is the real distance in km?

例如：在一张比例尺为 1 : 200 000 的地图上，两个城镇相距 4.5 厘米，实际距离是多少千米？

real distance = 4.5 × 200 000 = 900 000 cm = 9 km

Always convert the units carefully before giving the final answer.

在给出最终答案之前，务必要仔细转换单位。

9. Using Graphs to Solve Practical Problems | 用图像解决实际问题

Sometimes a practical problem is best solved by drawing or interpreting a graph. Conversion graphs, travel graphs, and cost graphs are common in IGCSE exams.

有时实际问题最好通过绘制或解读图像来解决。换算图、行程图和成本图在 IGCSE 考试中很常见。

Example: A taxi fare is £3.00 plus £1.50 per mile. Write the formula and draw the graph for distances from 0 to 10 miles.

例如：出租车费用为起步价 3.00 英镑加每英里 1.50 英镑。写出公式并画出 0 到 10 英里距离的图像。

cost = 3.00 + 1.50 × miles

From the graph, you can read distances for a given cost, or costs for a given distance. The y-intercept represents the fixed charge.

从图上可以读出给定费用所对应的路程，或给定路程所对应的费用。y 轴截距表示固定收费。

10. Common Pitfalls and Checking Answers | 常见陷阱与检验答案

Even with a correct method, small mistakes can cost marks. Always check units, rounding, and whether the answer is sensible.

即使方法正确，小错误也可能丢分。始终检查单位、近似值以及答案是否合理。

Check the answer is in the correct units.

检查答案的单位是否正确。
Substitute the answer back into the problem.

将答案代回原题进行验证。
Estimate first: 3.9 × 51 is about 4 × 50 = 200, so an answer of 198.9 is likely correct.

先估算：3.9 × 51 大约为 4 × 50 = 200，因此 198.9 的答案很可能正确。
Check if the answer makes sense in real life: a negative distance or a fraction of a person signals an error.

判断答案是否符合实际：负距离或非整人数表明存在错误。

If your answer is not sensible, reread the question and look for a misread number or an operation error.

如果答案不合理，请重读题目，检查是否看错数字或写错运算。

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Solving Practical Problems | 实际问题求解