Compound Growth and Decay
Apply a repeated percentage change as a multiplier raised to a power, for savings, depreciation and populations, and find how many steps it takes to pass a target.
کارهایی که یادگیرنده بعد از این درس می‌تواند انجام دهد
- Use the multiplier to the power n for n years of growth
- Tell compound growth from simple interest in a word problem
- Find by trial how many whole steps pass before a value first crosses a target
۱ · بخوان
Compound change repeats the same multiplier every step. With compound interest, the interest earned starts earning interest too. For 3% yearly growth, multiply by 1.03 each year, using the new bigger balance every time. Simple interest is different: it pays only on the original amount, so the yearly addition never grows.
1500 pounds at 4% a year for 6 years needs one calculation: 1500 times 1.04^6. The power counts the years, and the answer is about 1897.98 pounds. In general, value = start times multiplier to the number of steps. Calculate the power first, then multiply once instead of taking six separate steps.
Decay works the same way with a multiplier below 1. A van losing 20% a year keeps 0.80 each year, so after 3 years a 12000 pound van is worth 12000 times 0.8^3 = 6144 pounds. The power still counts the steps, and each yearly loss shrinks because it comes off a smaller value.
To find how many steps it takes to pass a target, try whole powers in order. A 640 g sample losing 25% an hour is multiplied by 0.75 each hour. After 4 hours it is 640 times 0.75^4 = 202.5 g, still above 200 g. After 5 hours it is about 151.9 g. So hour 5 is the first below 200 g. Stop at the first crossing.
Raise one multiplier to the number of steps, and try whole powers in order to find when a value passes a target.
The rewritten reading was written for this dataset. Practice questions are original. Video links are from Khan Academy.
۲ · ببین
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