If an initial quantity continuous decay at an exponential rate , then the final amount after a time is given by the following formula .

(a)

Find proportion of words remains unchanged.

The equation is .

is the proportion of words that remain unchanged.

is the time since two languages diverged.

is the rate of replacement.

Since two languages diverged years ago .

Since rate of replacement is , .

Substitute and in .

.

Proportion of words remains unchanged is .

(b)

Find how many years will only 1% of the words remain unchanged.

Since 1% of the words remain unchanged, .

Substitute in .

Apply logarithm on each side.

.

1% of the words remain unchanged in years.

(a)

Proportion of words remains unchanged is .

(b)

1% of the words remain unchanged in years.