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。
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
- 每一步都明确本次以哪个量为基准
- 把文字增减转成乘数，再相乘
- 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。
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 \(4\%\)。
本知识点题库
先独立作答，再查看答案与解析。
百分率增减・折扣与复合变化挑战
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?
原库参考解析
百分率增减・折扣与复合变化挑战
The value of a collectible comic book increased by 167% from the end of 2011 to the end of 2012 and then decreased by 16% from the end of 2012 to the end of 2013. What was the net percentage increase in the value of the collectible comic book from the end of 2011 to the end of 2013?
原库参考解析
百分率增减・折扣与复合变化挑战
The manager of an online news service received the report above on the number of subscriptions sold by the service. The manager estimated that the percent increase from 2012 to 2013 would be double the percent increase from 2013 to 2014. How many subscriptions did the manager expect would be sold in 2014?
原库参考解析
百分率增减・折扣与复合变化挑战
he line graph shows the total amount of snow, in inches, recorded each year in Washington, DC, from 2003 to 2015. If p% is the percent decrease in the annual snowfall from 2003 to 2007, what is the value of p ?
原库参考解析
百分率增减・折扣与复合变化挑战
The number a is 70% less than the positive number b. The number c is 80% greater than a. The number c is how many times b ?
原库参考解析
百分率增减・折扣与复合变化挑战
210 is p% greater than 30. What is the value of p ?
原库参考解析
百分率增减・折扣与复合变化挑战
The number a is 70% less than the positive number b. The number c is 60% greater than a. The number c is how many times b ?
原库参考解析
百分率增减・折扣与复合变化挑战
The number a is 110% greater than the number b. The number b is 90% less than 47. What is the value of a?
原库参考解析
百分率增减・折扣与复合变化挑战
The table below shows the distribution of US states according to whether they have a state-level sales tax and a state-level income tax.
To the nearest tenth of a percent, what percent of states with a state-level sales tax do not have a state-level income tax?
原库参考解析
百分率增减・折扣与复合变化挑战
The value of z is 1.13 times 100. The value of z is what percent greater than 100?
原库参考解析
百分率增减・折扣与复合变化进阶
The number of coins in a collection increased from 9 to 90. What was the percent increase in the number of coins in this collection?
原库参考解析
百分率增减・折扣与复合变化基础
The number a is 190% greater than the number b. The number b is 80% less than 24. What is the value of a?
原库参考解析
