A bike has an original price of 300 dollars. Its price is reduced by 30%. Then, two weeks later, its price is reduced by an additional 20%. What is the final sale price of the bike?
One might be tempted to say that the bike’s price is discounted 30% + 20% = 50% from its original price, but the key to solving double percent-change questions is to realize that each percentage change is dependent on the last. For example, in the problem we just looked at, the second percent decrease is 20 percent of a new, lower price—not the original amount. Let’s work through the problem carefully and see. After the first sale, the price of the bike drops 30 percent: The second reduction in price knocks off an additional 20 percent of the sale price, not the original price: The trickiest of the tricky percentage problems go a little something like this:
A computer has a price of 1400 dollars. Its price is raised 20%, and then lowered 20%. What is the final selling price of the computer?
If this question sounds too simple to be true; it probably is. The final price is not the same as the original. Why? Because after the price was increased by 20 percent, the reduction in price was a reduction of 20 percent of a new, higher price. Therefore, the final price will be lower than the original. Watch and learn: Now, after the price is reduced by 20%: Double percent problems can be more complicated than they appear. But solve it step by step, and you’ll do fine. Exponential Growth and Decay These types of word problems take the concept of percent change even further. In questions involving populations growing in size or the diminishing price of a car over time, you need to perform percent-change operations repeatedly. Solving these problems would be time-consuming without exponents. Here’s an example:
If a population of 100 grows by 5% per year, how 美国GREat will the population be in 50 years?
To answer this question, you might start by calculating the population after one year: Or use the faster method we discussed in percent increase: After the second year, the population will have grown to: And so on and so on for 48 more years. You may already see the shortcut you can use to avoid having to do, in this case, 50 separate calculations. The final answer is simply: In general, quantities like the one described in this problem are said to be growing exponentially. The formula for calculating how much an exponential quantity will grow in a specific number of years is: Exponential decay is mathematically equivalent to negative exponential growth. But instead of a quantity growing at a constant percentage, the quantity shrinks at a constant percentage. Exponential decay is a repeated percent decrease. That is why the formulas that model these two situations are so similar. To calculate exponential decay: The only difference between the two equations is that the base of the exponent is less than 1, because during each unit of time the original amount is reduced by a fixed percentage. Exponential decay is often used to model population decreases, as well as the decay of physical mass. Let’s work through a few example problems to get a feel for both exponential growth and decay problems.Simple Exponential Growth Problems英语作文
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