Math 1 · Unit 8 · Lesson 2 · part 3 of 5

The shortcut: multiply by 1 plus the rate

Working out 5% and adding it on, year after year, is slow.

The shortcut: multiply by 1 plus the rate

Working out 5% and adding it on, year after year, is slow. There is a shortcut, and it comes from the distributive property (Unit 1, Lesson 2).

Adding 5% is multiplying by 1.05Take any balance, call it B. Adding 5% means B + 0.05B. That is B times 1, plus B times 0.05. Pull the B out front: B(1 + 0.05) = B(1.05). Keeping all of your money is multiplying by 1, the do-nothing number. Getting 5% more is the extra 0.05. Together: multiply by 1.05.

So every year multiplies by 1.05. After 1 year: 1000(1.05). After 2 years: 1000(1.05)(1.05) = 1000(1.05)². After t years:

A = P(1 + r)^t

Put that next to Unit 7. It is y = a · b^x with money names: a is P, the starting value, the principal. b is 1 + r, the growth factor, the number you multiply by each year. The rate r is the growth rate, the percent. A constant ratio, so an exponential. Exponentials multiply.

Check it against the table

A = 1000(1.05)³. Power first: 1.05³ = 1.157625. Then times 1000: $1,157.63. Same as the table.

Calculator keys: 1.05, power key, 3, equals, times, 1000, equals.

The factor is 1 plus the rate

4% growth: 1 + 0.04 = 1.04.

5.5% growth: 1 + 0.055 = 1.055.

12% growth: 1 + 0.12 = 1.12.

Going back: a factor of 1.08 is 1 + 0.08, a rate of 8%.

Where to go next

Next: Reading an expression, and moving P and rNow read an expression somebody else wrote. All the partsCompound Interest, Interest on Interest