📚 IGCSE Mathematics: Foreign Currency Exchange and Exchange Rate Calculations | IGCSE数学：外币兑换与汇率计算
Foreign currency exchange appears in the Edexcel IGCSE Mathematics syllabus as a real-life application of ratio, proportion and basic arithmetic. You are given an exchange rate and asked to convert an amount from one currency to another, often while considering commission or a "buying rate" and "selling rate". This topic rewards clear methodical working and careful interpretation of which operation to use.
外币兑换是Edexcel IGCSE数学考纲中的典型应用型问题，把比例、正比例和基本算术运算用于真实生活场景。题目一般会给你一个汇率，要求你把一个金额从一种货币转换成另一种货币，有时还要考虑手续费或“买入价”与“卖出价”。这个考点要求步骤清晰、理解准确，并能判断该用乘法还是除法。
1. Understanding Exchange Rates | 理解汇率
of one currency you can get for one unit of another currency. For example, if the rate is £1 = € ill give you 1.15 euros. The rate itself is simply a multiplier that links two currencies.
汇率表示用一单位某种货币可以兑换多少单位的另一种货币。例如，若汇率为 £1 = €1.15，则每兑换1英镑可获得1.15欧元。汇率本质上就是连接两种货币的一个乘法因子。
In IGCSE questions the exchange rate may be written in different forms:
在IGCSE题目中，汇率可能有多种写法：
- £1 = €1.15 (pounds to euros)
- £1 = €1.15 （英镑兑欧元）
- €1 = £0.8696 (euros to pounds, the inverse)
- €1 = £0.8696 （欧元兑英镑，即反向汇率）
- $1 = ¥145.30 (US dollars to Japanese yen)
- $1 = ¥145.30 （美元兑日元）
The first currency is the base currency (one unit of it) and the second is the quoted currency. Always identify the base currency before calculating.
写在“=”左边的是基准货币，右边的是报价货币。计算前一定要先判断哪一种货币是基准货币。
2. Direct and Inverse Exchange Rates | 直接汇率与反向汇率
Suppose you are given that £1 = € k to pounds, you need the inverse rate. Dividing both sides of the relation by 1.15 gives:
假设已知 £1 = €1.15，这个直接汇率可以用于“英镑 → 欧元”的换算。若要把欧元换回英镑，则需要反向汇率。把等式两边同除以1.15得到：
€1 = 1 ÷ 1.15 = £0.8696 (to 4 d.p.)
Therefore to convert €250 to pounds, you would calculate 250 × 0.8696 = £217.39. Notice that £250 × 1.15 = €287.50, and then converting €287.50 back at the inverse rate gives £287.50 × 0.8696 = £250.00, confirming the two operations are inverses.
因此，把250欧元换成英镑，应计算 250 × 0.8696 = £217.39。注意，£250 × 1.15 = €287.50，再按反向汇率换回：€287.50 × 0.8696 = £250.00，说明乘法和除法互为逆运算。
So if you know the rate from currency A to currency B, the inverse rate is simply: 1 ÷ rate.
所以，若已知从A币到B币的汇率，反向汇率就是：1 ÷ 原汇率。
3. Converting Local Currency to Foreign Currency | 本币换外币：用乘法
When converting from the base currency to the second currency, multiply the amount by the exchange rate.
当从基准货币换算到第二种货币时，用金额乘以汇率。
Example: The exchange rate is £1 = €1.15. Convert £300 to euros.
例题：已知汇率为 £1 = €1.15，把300英镑换成欧元。
£300 × 1.15 = €345
The answer is €345. The units work out because both the "£1" and "£300" are in pounds; multiplying pounds by euros-per-pound gives euros.
答案是345欧元。单位也能验证：英镑 × (欧元/英镑) = 欧元。
If the given rate is not in the same direction, e.g. you are given $1 = £0.75 and need to convert £400 to dollars, first find the rate in the required direction by taking the reciprocal: £1 = 1 ÷ 0.75 = $1.3333. Then £400 × 1.3333 = $533.33.
如果题目给的汇率方向不对，例如已知 $1 = £0.75，却需要把400英镑换成美元，则先取倒数得到 £1 = 1 ÷ 0.75 = $1.3333。然后 400 × 1.3333 = $533.33。
4. Converting Foreign Currency to Local Currency | 外币换本币：用除法
When converting from the second currency back into the base currency, divide by the exchange rate.
当从第二种货币换回基准货币时，用金额除以汇率。
Example: The exchange rate is £1 = €1.15. A traveller returns with €460 and wants pounds.
例题：汇率是 £1 = €1.15，一位旅行者带回460欧元，想换成英镑。
€460 ÷ 1.15 = £400
Think of it as "how many lots of €1.15 are in €460?" Because each lot corresponds to £1. Some students prefer to multiply by the inverse rate: 460 × (1 ÷ 1.15) = 460 × 0.8696 = £400.
可以理解成“€460里包含多少个€1.15”，因为每包含一个€1.15就相当于£1。也可以换成乘法：460 × (1 ÷ 1.15) = 460 × 0.8696 = £400。
Always check whether the answer seems reasonable. If £1 = € nd, so converting euros to pounds must give fewer units of currency than you started with.
做完后要检查结果是否合理。如果 £1 = €1.15，说明1欧元价值低于1英镑，那么把欧元换成英镑，得到的英镑数应小于原来的欧元数。
5. Reading Exchange-Rate Tables | 汇率表格的解读
Edexcel questions sometimes present exchange rates in a table of "currency per £1" or "£ per unit".
Edexcel题目有时会用表格给出“每一英镑可换的外币数”或“每单位外币可换的英镑数”。
| Currency | Units per £1 | £ per 1 unit |
| US Dollar ($) | 1.300 | 0.7692 |
| Euro (€) | 1.150 | 0.8696 |
| Japanese Yen (¥) | 145.20 | 0.006887 |
Using the table, converting £200 to US dollars means multiplying by the first column: £200 × 1.300 = $260. Converting $ ack to pounds means dividing by 1.300, or multiplying by the second column: 260 × 0.7692 = £200.
用上表把 £200 换成美元，应乘第一列：200 × 1.300 = $260。把 $260 换回英镑，则除以1.300，或乘第二列：260 × 0.7692 = £200。
If a table only gives "units per £1", you can always generate the other column by taking reciprocals. This skill is essential for cross-currency questions.
如果表中只给了“每英镑兑多少外币”，你可以通过取倒数得到“每单位外币兑多少英镑”。这是解决交叉汇率题目的关键技能。
6. Buying and Selling Rates (Bid–Ask Spread) | 买入价与卖出价（买卖价差）
In real life, a bank or foreign-exchange desk offers two different rates. They sell foreign currency to you at a high rate (so they make profit), and they buy foreign currency from you at a low rate. From the bank's perspective:
在现实中，银行或兑换机构会提供两种不同的汇率。它们把外币卖给你时用较高的卖出价（以赚取利润），而从你手中买入外币时用较低的买入价。站在银行的角度：
- Selling rate (bank sells foreign currency): you receive fewer units per pound.
- 买入价/卖出价: 银行卖出外币时，你每英镑得到较少的外币。
- Buying rate (bank buys foreign currency): you receive fewer pounds when selling foreign currency.
- 买入价（银行买入外币）: 你把手里的外币卖给银行时，得到的英镑较少。
Example: A bank quotes €1 = £0.88 (selling rate) and €1 = £0.84 (buying rate). How many euros can you buy for £500, and how many pounds do you get if you sell €300?
例题：某银行报价：卖出欧元时 €1 = £0.88，买入欧元时 €1 = £0.84。用£500能买多少欧元？卖€300能拿回多少英镑？
To buy euros, the bank sells you euros: each euro costs you £0.88. So you receive £500 ÷ 0.88 = €568.18.
你要买入欧元，银行把欧元卖给你：每欧元价格是£0.88。因此你能得到 500 ÷ 0.88 = €568.18。
To sell euros, the bank buys euros from you at £0.84 per euro: €300 × 0.84 = £252.
你要卖出欧元，银行按每欧元£0.84的买入价购买：300 × 0.84 = £252。
The difference between the buying and selling price per unit is the spread. Here the spread is £0.04 per euro, which represents the bank's profit margin.
同一单位外币的买入价与卖出价之差称为“价差”。本例中价差为每欧元£0.04，这是银行的利润来源。
7. Commission and Fees | 手续费与佣金
Some currency exchange problems include a percentage commission or a fixed fee. The commission is usually charged on the value in the original currency, although in exam questions the wording tells you exactly what to do.
有些外币兑换问题包含百分比佣金或固定费用。佣金通常按原货币金额的一定百分比收取，不过考试题目的表述会明确告诉你具体计算方法。
Example: Thomas wants to convert £600 into US dollars. The exchange rate is £1 = $1.30 and the exchange bureau charges 2% commission. How many dollars does he receive?
例题：Thomas要把£600换成美元。汇率是 £1 = $1.30，兑换机构收取2%佣金。他能拿到多少美元？
First deduct the commission from the pounds: £600 × 2% = £12, so the amount exchangeable is £600 − £12 = £588.
先从英镑中扣除佣金：600 × 2% = £12，可用于兑换的英镑为 600 − 12 = £588。
£588 × 1.30 = $764.40
The order is critical: always subtract commission before multiplying by the exchange rate.
顺序至关重要：必须先扣掉佣金，再乘以汇率。
If the commission is a fixed fee, e.g. £3 per transaction, you would instead calculate (£600 − £3) × 1.30 = $776.10.
如果佣金是固定费用，例如每笔交易收取£3，则应计算 (600 − 3) × 1.30 = $776.10。
8. Cross Exchange Rates | 交叉汇率的计算
A cross exchange rate is the exchange rate between two currencies that is not directly quoted, found by combining two rates that both use a common currency.
交叉汇率是指两种货币之间没有直接报价的汇率，需要借助它们各自与第三种货币的汇率来推算。
Example: Given that £1 = $1.30 and €1 = $1.10, calculate how many euros you get for £1.
例题：已知 £1 = $1.30，€1 = $1.10，求£1能换多少欧元。
Since both rates are expressed in dollars, divide the dollar value of £1 by the dollar value of €1:
因为两种汇率都以美元表示，用“£1的美元价值”除以“€1的美元价值”：
£1 = $1.30 ÷ ($1.10 per €) = €1.1818 (to 4 d.p.)
So approximately €1.18 can be bought for £1. Equivalently, €1 = £1 ÷ 1.1818 = £0.8462.
所以约1英镑能兑换1.18欧元。同理，€1 = £1 ÷ 1.1818 = £0.8462。
The general rule is: if A = kB and C = mB, then A per C = k ÷ m.
一般规律是：若 A = kB 且 C = mB（都等于某种中间货币B），则 1单位C可换的A = k ÷ m。
9. Comparing Currency Exchange Deals | 比较不同兑换方案
Edexcel problem-solving questions often ask you to decide which exchange provider offers the better deal. To compare deals fairly, convert the same starting amount using each option and compare the final amounts of foreign currency received.
Edexcel的解决问题类题目常要求你判断哪家兑换机构更划算。公平比较的标准是：用同样的初始金额，按每种方案换算后，比较最终到手的外币数量。
Example: Kira wants to exchange £500 into euros. Bank A offers £1 = €1.12 plus 2% commission. Bank B offers £1 = €1.10 with no commission. Which bank gives more euros?
例题：Kira想把£500换成欧元。银行A：汇率为 £1 = €1.12，另收2%佣金。银行B：汇率为 £1 = €1.10，不收佣金。哪家银行能给更多欧元？
Bank A: amount after commission = 500 × 0.98 = £490, then 490 × 1.12 = €548.80.
银行A：扣佣金后为 500 × 0.98 = £490，再乘汇率：490 × 1.12 = €548.80。
Bank B: 500 × 1.10 = €550.00.
银行B：500 × 1.10 = €550.00。
Bank B gives €1.20 more, so it is the better deal, even though its raw exchange rate looks worse than Bank A's.
银行B多给€1.20，因此银行B更划算，尽管其表面汇率看起来比银行A差。
Always show both calculations in full; many past-paper mark schemes award method marks for correctly identifying the effect of commission and the final comparison.
一定要完整写出两种计算过程。许多历年真题的评分标准都会为“正确考虑佣金影响”和“最终比较结果”给分。
10. Worked Exam-Style Question | 考试型例题精解
Let us try a complete question in the style of Edexcel IGCSE Mathematics (Foundation/Higher Tier crossover).
下面做一道符合Edexcel IGCSE数学风格的综合题（适用于基础/高级层次）。
Question: Mia is travelling to Japan. The exchange rate is £1 = ¥145.20. She changes £450 into yen. Later she spends ¥38,500 and converts the remaining yen back to pounds. The bank buys yen at £1 = ¥148.00. How many pounds does she receive? Give your answer to the nearest penny.
题目：Mia要去日本旅行。汇率是 £1 = ¥145.20。她把£450换成日元。后来她花掉¥38,500，把剩余日元换回英镑。银行买入日元时采用 £1 = ¥148.00。她最后拿到多少英镑？精确到便士。
Solution:
解答：
Step 1: Convert pounds to yen.
第一步：把英镑换成日元。
£450 × 145.20 = ¥65,340
Step 2: Subtract the amount spent.
第二步：减去花掉的金额。
¥65,340 − ¥38,500 = ¥26,840
Step 3: Convert remaining yen to pounds using the bank's buying rate £1 = ¥00. Since the bank buys yen, it wants more yen per pound, so divide the yen amount by 148.
第三步：按银行买入价 £1 = ¥148.00 把剩余日元换回英镑。银行买入日元时，要求每英镑支付更多日元，所以要用除法。
¥26,840 ÷ 148 = £181.351…
Rounded to the nearest penny, Mia receives £181.35.
四舍五入到便士，Mia最终收到 £181.35。
11. Common Mistakes and Tips | 常见错误与备考建议
Below are the most frequent errors students make in this topic, followed by advice to avoid them.
下面列出同学们在这一专题中最常见的错误，以及相应的规避建议。
- Wrong operation: Multiplying when you should divide. Check the units: if you start with pounds and the rate is "euros per pound", multiply. If you start with euros, divide.
- 运算方向搞反：该乘却除，该除却乘。检查单位：如果你一开始有英镑，而汇率是“每英镑兑欧元”，则用乘法；如果你一开始有欧元，则用除法。
- Using the wrong rate: When there is a buying rate and a selling rate, read the question carefully. Does the bank buy or sell the foreign currency? You should use the bank-selling rate when buying foreign currency and the bank-buying rate when selling it.
- 用错买入价/卖出价：当题目同时给出银行买入价和卖出价时，要仔细读题。你买入外币用银行卖出价，卖出外币用银行买入价。
- Ignoring commission: Deduct commission from the original currency before conversion, not afterwards. Do not multiply the commission percentage by the foreign currency amount.
- 忽略手续费：佣金应在转换前从原始货币中扣除，而不是在转换后扣；更不能用外币金额乘佣金百分比。
- Rounding too early: Keep full precision in intermediate steps and only round at the end. Many students lose accuracy by rounding mid-calculation.
- 过早四舍五入：中间步骤保留足够精确的小数，只在最后一步进行四舍五入，以免累积误差。
- Not showing units: Always write the currency symbol ( £, €, $, ¥ ) for every amount. It helps you track the logic and earns marks in written solutions.
- 不写单位：每个金额都要带货币符号（£、€、$、¥），这样既能检查逻辑，也符合评分标准。
Finally, practise inverse rates until you can compute them instantly: the inverse of a rate r is 1 ÷ r. This single skill will help you solve nearly every currency exchange problem on the paper.
最后，请把“取倒数”练到条件反射：汇率 r 的反向汇率就是 1 ÷ r。这一个技能能帮助你解决试卷上几乎所有的货币兑换问题。
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