Math 1 · Unit 7 · Lesson 3 · part 3 of 5

Percents and factors, both ways

Real stories rarely hand you b.

Percents and factors, both ways

Real stories rarely hand you b. They say a percent: grows 10% a year, loses 25% a year. The growth factor is what you multiply by; the growth rate is the percent, how far the factor is from 1. Convert both ways, always starting from 100%.

Percent to factor: growing 10% a year

Each year you keep all 100% and add 10% more: 100% + 10% = 110%. As a decimal, the factor is 1.10. A population of 500 growing 10% a year: y = 500 · 1.10^x.

Percent to factor: losing 25% a year

Each year you keep 100% minus 25%, which is 75%. The factor is 0.75, and the decay rate is 25%. A car worth $20,000 losing 25% a year: y = 20000 · 0.75^x.

Factor to percent

b = 1.05: that is 105%, which is 100% plus 5%. Growth rate 5%.

b = 0.85: that is 85%, which is 100% minus 15%. Decay rate 15%. A medicine with 50 · 0.85^x milligrams left after x hours loses 15% every hour: a = 50 mg is the starting dose, and b = 0.85 means 85% remains each hour.

Start from 100%. Growth: add the percent, then write it as a decimal (10% more is 1.10). Decay: take the percent away (25% less is 0.75). Going back: a factor above 1 minus 100% is the growth rate; 100% minus a factor below 1 is the decay rate.

Say what a and b mean in the storyy = 500 · 1.10^x: a = 500 is the population at the start, b = 1.10 means each year it is 110% of the year before, growing 10%. y = 50 · 0.85^x: a = 50 is the starting dose in milligrams, b = 0.85 means 85% of the medicine remains each hour, decaying 15%.

Where to go next

Next: Sort them: growth or decay?Eight items, equations and stories. All the partsGrowth and Decay