知识点地图 / 问题解决与数据分析

Percent Change, Discounts & Successive Change

百分率增减・折扣与复合变化

先问“以谁为基数”,再把百分率变成倍率。连续变化用乘法,反求原量用除法。

01 / UNDERSTAND

这部分需要会什么?

  • 用乘数表示增加、减少与折扣
  • 计算连续变化后的净变化率
  • 根据变化后的数量反求原数

核心概念与公式

增加 \(r\%\) 对应乘 \(1+r/100\),减少 \(r\%\) 对应乘 \(1-r/100\)。连续变化的总乘数是各步乘数的乘积,不能简单把各百分率相加。

变化率等于 \((\text{新值}-\text{原值})/\text{原值}\)。若已知新值,要用新值除以总乘数反求原值。百分率之间的绝对差用“百分点”描述;从 20% 到 30% 是增加 10 个百分点,相对原百分率增加 50%。

做题时怎样使用

  1. 每一步都明确本次以哪个量为基准
  2. 把文字增减转成乘数,再相乘
  3. 净变化看总乘数与 1 的差,反求原数则做除法

容易混淆的地方

先涨 20% 再降 20% 不会回到原值,因为第二次以涨价后的数为基准。降价后的价格不能直接再减同一比例来求原价。

02 / CONNECT

用一个例题串起方法

知识点教学例题
一件商品原价 200 元,先涨价 20%,再降价 20%。求最终价格与净变化率。
展开解题思路 +

最终价格为 \(200\times1.2\times0.8=192\) 元。总乘数为 0.96,所以相对原价下降了 \(4\%\)。

03 / PRACTISE

本知识点题库

先独立作答,再查看答案与解析。

78 道题目
找到 78 道题目重置筛选
百分率增减・折扣与复合变化基础

A number \(f\) is 130% greater than a positive number \(g\). A number \(h\) is 20% less than the number \(f\). The number \(h\) is how many times the number \(g\)?

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25年12月美国B卷练习 · 原库模块 1 · Q18作答 ↗
百分率增减・折扣与复合变化基础

The percent increase in mass of a certain red kangaroo from 80 days old to 160 days old was 676%. If this red kangaroo's mass was k grams at 80 days old, which expression represents its mass, in grams, at 160 days old?
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2024年6月北美A卷 · 原库模块 2 · Q20作答 ↗
百分率增减・折扣与复合变化进阶

A business owner plans to purchase the same model of chair for each of the 81 employees. The total budget to spend on these chairs is $14,000,which includes a 7% sales tax. Which of the following is closest to the maximum possible price per chair, before sales tax, the business owner could pay based on this budget?
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Bluebook试题1 · 原库模块 3 · Q10作答 ↗
百分率增减・折扣与复合变化基础

The number a is 60% greater than the positive number b. The number c is 45% less than a. The number c is how many times b?
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PSAT 8/9 P1 · 原库模块 2 · Q21作答 ↗
百分率增减・折扣与复合变化进阶

The value of a toy increased by 193% from the end of 2016 to the end of 2017 and then decreased by 31% from the end of 2017 to the end of 2018. What was the net percentage increase in the value of the toy from the end of 2016 to the end of 2018?
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2025年9月国际A卷 · 原库模块 2 · Q20作答 ↗
百分率增减・折扣与复合变化进阶

The regular price of a shirt at a store is $11.70. The sale price of the shirt is 80% less than the regular price, and the sale price is 30% greater than the store's cost for the shirt. What was the store’s cost, in dollars, for the shirt? (Disregard the $ sign when entering your answer. For example, if your answer is $4.97, enter 4.97)
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Bluebook试题8 · 原库模块 3 · Q16作答 ↗
百分率增减・折扣与复合变化进阶

The value of an autographed baseball increased by 173% from the end of 2013 to the end of 2014 and then decreased by 18% from the end of 2014 to the end of 2015. What was the net percentage increase in the value of the autographed baseball from the end of 2013 to the end of 2015?
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2026年3月国际C卷 · 原库模块 2 · Q16作答 ↗
百分率增减・折扣与复合变化基础

A clothing store buys shirts at a wholesale price of 4.00 dollars each and resells them each at a retail price that is 280% of the wholesale price. At the end of the season, any remaining shirts are marked at a discounted price that is 80% off the retail price. What is the discounted price of each remaining shirt, in dollars?
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2023年5月练习卷(易) · 原库模块 1 · Q10作答 ↗
百分率增减・折扣与复合变化基础

A gift shop buys souvenirs at a wholesale price of 7.00 dollars each and resells them each at a retail price that is 290% of the wholesale price. At the end of the season, any remaining souvenirs are marked at a discounted price that is 80% off the retail price. What is the discounted price of each remaining souvenir, in dollars?
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Bluebook试题4 · 原库模块 3 · Q12作答 ↗
百分率增减・折扣与复合变化挑战

A researcher investigated two species of mites: a predator and its prey. At the start of a week, there was an equal number of the two species. At the end of the week, the number of prey had increased by 2100% of the number of prey at the start of the week, and the number of predators had increased by 450% of the number of predators at the start of the week. The number of predators at the end of the week was p% less than the number of prey at the end of the week. What is the value of p?
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2026年5月国际A卷 · 原库模块 2 · Q22作答 ↗
百分率增减・折扣与复合变化挑战

A researcher investigated two species of mites: a predator and its prey. At the start of a week, there was an equal number of the two species. At the end of the week, the number of prey had increased by 2,700% of the number of prey at the start of the week, and the number of predators had increased by 180% of the number of predators at the start of the week. The number of prey at the end of the week was p% greater than the number of predators at the end of the week. What is the value of p?
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24年12月北美B卷 · 原库模块 2 · Q22作答 ↗
百分率增减・折扣与复合变化基础

The price of an item increased by p% from $18 to $21. What is the value of p?
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25年10月国际C卷练习 · 原库模块 2 · Q3作答 ↗