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・折扣与复合变化进阶
A clothing store buys shirts at a wholesale price of 6.00 dollars each and resells them each at a retail price that is 310% of the wholesale price. At the end of the season, any remaining shirts are marked at a discounted price that is 75% of the retail price. What is the discounted price of each remaining shirt, in dollars?
原库参考解析
百分率增减・折扣与复合变化进阶
A furniture company is selling all tables with a 20% discount off their original price for the month of October. What is the discount, in dollars, on a table with an original price of $130?
原库参考解析
百分率增减・折扣与复合变化挑战
The daily precipitation total at a weather station is recorded by a weather instrument each day. The daily precipitation total recorded on Monday was 19.00 millimeter. The daily precipitation total recorded on Tuesday was a 134.00% increase from the daily precipitation total recorded on Monday. What was the daily precipitation total, in millimeters, recorded on Tuesday?
原库参考解析
百分率增减・折扣与复合变化进阶
A number \( f \) is 120% greater than a positive number g. A number h is 90% less than the number . The number h is how many times the number g?
原库参考解析
百分率增减・折扣与复合变化挑战
At a hotel, guests used the pool for a total of \( t \) hours in March. At the same hotel, guests used the pool for \( 3.61t \) hours in April. What is the percent increase in the number of hours guests used the pool from March to April?
原库参考解析
百分率增减・折扣与复合变化进阶
In 2010, a skating team had a total of 51 skaters, each classified as either advanced or intermediate. From 2010 to 2020, the number of advanced skaters on the team increased by approximately 59%, and the number of intermediate skaters on the team increased by approximately 48%. The total number of skaters on the team increased by approximately 53%. Which equation best represents this situation, where a represents the number of advanced skaters on the team in 2010 and b represents the number of intermediate skaters on the team in 2010?
原库参考解析
百分率增减・折扣与复合变化挑战
In 2010, a skating team had a total of 35 skaters, each classified as either advanced or intermediate. From 2010 to 2020, the number of advanced skaters on the team increased by approximately 39%, and the number of intermediate skaters on the team increased by approximately 47%. The total number of skaters on the team increased by approximately 43%. Which equation best represents this situation, where a represents the number of advanced skaters on the team in 2010 and b represents the number of intermediate skaters on the team in 2010?
原库参考解析
百分率增减・折扣与复合变化挑战
The percent increase in mass of a certain red kangaroo from 90 days old to 160 days old was 438%. If this red kangaroo's mass was k grams at 90 days old, which expression represents its mass, in grams, at 160 days old?
原库参考解析
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 1,900% of the number of prey at the start of the week, and the number of predators had increased by 150% 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?
原库参考解析
百分率增减・折扣与复合变化挑战
The result of increasing the quantity x by 800% is 504. What is the value of x?
原库参考解析
百分率增减・折扣与复合变化挑战
A scientist studied the effects of mixing two chemicals, chemical A and chemical B, by gradually adding more of each chemical to a mixture of the two chemicals. At the start of the study, the mass of chemical A in the mixture was equal to the mass of chemical B in the mixture. From the start of the study to the end of the study, the mass of chemical A in the mixture increased by 2,600% and the mass of chemical B in the mixture increased by 380%. At the end of the study, approximately how many times greater was the mass of chemical A in the mixture than the mass of chemical B in the mixture?
原库参考解析
百分率增减・折扣与复合变化进阶
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?
参考答案
