Problem Solving: Work Backwards | 数学解题策略：逆向思维

📚 Problem Solving: Work Backwards | 数学解题策略：逆向思维

When solving a problem, sometimes the most direct way is to start from the end and work backwards. This strategy is especially useful when the final result is known, but the starting value is unknown.

在解决数学问题时，有时最直接的方法是从结果出发，逆向推导。这种策略在已知最终结果、但未知起始数值时尤其有效。

1. Understanding the Strategy | 理解逆向思维

Working backwards means reversing the operations described in the problem. If the original problem says “add 5”, then in reverse you “subtract 5”. If it says “multiply by 3”, then in reverse you “divide by 3”.

逆向思维是指将题目中描述的操作反过来进行。如果原题说“加5”，则逆向操作就是“减5”；如果原题说“乘以3”，则逆向操作就是“除以3”。

For example, consider this puzzle: “I think of a number. I add 8 and get 15. What is the number?”

例如，看这个谜题：“我想了一个数，加上8得到15。这个数是多少？”

Number + 8 = 15
Number = 15 − 8 = 7

The answer is 7. We simply reversed the operation of adding 8 by subtracting 8 from 15.

答案是7。我们只需把“加8”反过来，用15减去8即可得到答案。

2. When to Use This Strategy | 什么时候用逆向思维

Working backwards is useful when:

逆向思维适用于以下情况：

The final result is given in the problem.
中文：题目中已经给出了最终结果。
The starting value is unknown.
中文：起始数值未知。
The problem involves a sequence of operations (add, subtract, multiply, divide).
中文：题目涉及一连串的运算（加、减、乘、除）。
The operations are performed in a clear order.
中文：运算按明确的顺序进行。

If a problem tells you what happened at each step and the final answer, you can reverse each step to find the beginning.

如果题目描述了一步一步的操作以及最终结果，你就可以逐步逆向操作来找回起始值。

3. Example 1: One Operation | 例1：单步运算

Problem: A number is multiplied by 4 and the result is 24. What is the number?

题目：一个数乘以4后得到24。这个数是多少？

Number × 4 = 24
Number = 24 ÷ 4 = 6

Check: 6 × 4 = 24 ✓

检验：6 × 4 = 24，正确。

4. Example 2: Two Operations | 例2：两步运算

Problem: I think of a number. I multiply it by 3, then add 5. The result is 26. What is the number?

题目：我想了一个数，把它乘以3，再加5，结果是26。这个数是多少？

To solve, write the operations in order:

我们先按顺序写出运算过程：

Number → ×3 → +5 → 26

Now reverse the order and reverse each operation:

然后倒推顺序，并把每一步运算反过来：

26 → −5 → 21 → ÷3 → 7

So the number is 7. Check: 7 × 3 = 21; 21 + 5 = 26 ✓

所以这个数是7。检验：7 × 3 = 21；21 + 5 = 26，正确。

5. Using a Flow Chart | 用流程图表示

A flow chart can help you keep track of the steps. Each box shows an operation.

流程图可以帮助你理清步骤。每个方框表示一个运算。

Forward flow:

正向流程：

? → ×3 → 21 → +5 → 26

Backward flow:

逆向流程：

26 → −5 → 21 → ÷3 → 7

When working backwards, remember to reverse the order of operations as well as the operation itself. In the forward direction we first multiplied by 3 and then added 5. In the backward direction we first subtract 5 and then divide by 3.

逆向推导时，不仅要反转每个运算本身，还要反转运算的顺序。正向是先乘3再加5；逆向就要先减5再除以3。

6. Example 3: Money Problem | 例3：购物找零问题

Problem: Sarah spent half of her money on a book. Then she spent $6 on lunch. She has $14 left. How much money did she start with?

题目：莎拉买书花了一半的钱，然后买午餐花了6美元，最后剩下14美元。请问她一开始有多少钱？

Let’s write the forward sequence:

先写出正推过程：

Start → ÷2 → −6 → 14

Reverse it:

倒推：

14 → +6 → 20 → ×2 → 40

So Sarah started with $40. Check: $40 ÷ 2 = $20; $20 − $6 = $14 ✓

所以莎拉一开始有40美元。检验：40 ÷ 2 = 20；20 − 6 = 14，正确。

7. Example 4: Temperature Change | 例4：温度变化问题

Problem: The temperature in a town rose 8°C in the morning, then fell 12°C in the afternoon. The final temperature at night was 5°C. What was the temperature in the morning?

题目：某城镇的气温上午上升了8°C，下午下降了12°C，夜间最终气温是5°C。请问上午的气温是多少？

Forward sequence:

正推过程：

Morning → +8 → −12 → 5

Reverse:

倒推：

5 → +12 → 17 → −8 → 9

So the morning temperature was 9°C. Check: 9 + 8 = 17; 17 − 12 = 5 ✓

因此上午气温为9°C。检验：9 + 8 = 17；17 − 12 = 5，正确。

8. Common Mistake | 常见错误

One common mistake is to forget to reverse the order of operations. For example, in the temperature problem above, some students might first subtract 8 from 5 and then add 12, which gives 9 as well by coincidence. But in many problems the order matters.

一个常见错误是忘记反转运算顺序。比如在上面的温度问题中，有些学生可能会先从5减8再加12，碰巧也得到9。但在许多问题中，顺序非常重要。

Consider this problem: “A number is divided by 4, then multiplied by 2. The final result is 10. What is the number?”

看这道题：“一个数先除以4，再乘以2，最终结果是10。这个数是多少？”

Forward: n → ÷4 → ×2 → 10

正推：n → ÷4 → ×2 → 10

Correct reverse: 10 → ÷2 → 5 → ×4 → 20

正确倒推：10 → ÷2 → 5 → ×4 → 20

If you mistakenly do 10 × 2 = 20, then ÷4 = 5, you get the wrong answer. So always reverse the order too.

如果你错误地先10 × 2 = 20，再÷4 = 5，就得到错误答案。所以一定要同时倒转顺序。

9. Using Inverse Operations | 使用逆运算

Working backwards relies on inverse operations. Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

逆向思维依赖于逆运算。加法与减法互为逆运算，乘法与除法互为逆运算。

Forward Operation \| 正向运算	Inverse Operation \| 逆向运算
+ a	− a
− a	+ a
× a	÷ a
÷ a	× a

When reversing a sequence, apply the inverse operations in the opposite order.

在倒推一个运算序列时，要按照相反的顺序应用对应的逆运算。

10. Practice Problems | 练习题目

Try these problems on your own:

请独立尝试以下练习：

1. A number is multiplied by 5, then 3 is subtracted. The result is 22. What is the number?

1. 一个数乘以5，再减去3，得到22。这个数是多少？

2. Tom doubled his money, then spent $10, and had $30 left. How much did he start with?

2. 汤姆的钱翻倍后，花了10美元，还剩30美元。他一开始有多少钱？

3. The temperature dropped 7°C, then rose 4°C. The final temperature is 12°C. What was the starting temperature?

3. 气温先下降7°C，再上升4°C，最终温度为12°C。起始温度是多少？

Answers:

答案：

1. n = (22 + 3) ÷ 5 = 5

2. n = (30 + 10) ÷ 2 = 20

3. n = 12 − 4 + 7 = 15

11. Summary | 总结

Working backwards is a powerful problem-solving strategy. Follow these steps:

逆向思维是一种强大的解题策略。请遵循以下步骤：

Read the problem carefully and list the operations in order.
中文：仔细阅读题目，按顺序列出所有运算。
Start from the final result and reverse each operation in reverse order.
中文：从最终结果出发，按相反顺序对每一步进行逆运算。
Check your answer by plugging it back into the original problem.
中文：将答案代回原题检验。

Remember: when working backwards, the order of operations is reversed, and each operation is replaced by its inverse.

记住：倒推时，运算顺序要反转，每个运算都要换成它的逆运算。

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Problem Solving: Work Backwards | 数学解题策略：逆向思维