A sample of Potassium-40 has a mass of 2020 grams. Potassium-40 has a half life of 1.3 b. y. How much Potassium-40 and Argon 40 would there be after 5.2 b. y.

The idea here is that the ratio that exists between the number of atoms of argon-40 and the number of atoms of potassium-40 will give you the number of half-lives that passed.

As you know, the half-life of a radioactive nuclide tells you the time needed for half of the atoms of said nuclide to undergo radioactive decay.

In your case, you know that potassium-40 has a half-life of

1.25

billion years because that's how long it takes for half of the number of atoms present in the sample to decay to argon-40.

Now, let's say that your sample started with

A

K-40

atoms of potassium-40 and

0

atoms of argon-40.

You can thus say that the sample will contain--keep in mind that the atoms of potassium that decay form argon-40!

After

1

half-life

1

2

⋅

A

K-40

=

A

K-40

2

1

→

atoms of potassium-40

A

K-40

−

A

K-40

2

1

→

atoms of argon-40

After

2

half-lives

1

2

⋅

A

K-40

2

1

=

A

K-40

2

2

→

atoms of potassium-40

A

K-40

−

A

K-40

2

2

→

atoms of argon-40

After

3

half-lives

1

2

⋅

A

K-40

2

2

=

A

K-40

2

3

→

atoms of potassium-40

A

K-40

−

A

K-40

2

3

→

atoms of argon-40

At this point, we can use this pattern to say that after

n

half-lives pass, the sample will contain

A

K-40

2

n

→

atoms of potassium-40

1

−

A

K-40

2

n

→

atoms of argon-40

Now, you know that sample contains

31

atoms of argon-40 for every

1

atom of potassium-40, which means that you have

A

K-40

−

A

K-40

2

n

A

K-40

2

n

=

31

This is equivalent to

A

K-40

−

A

K-40

2

n

A

K-40

2

n

=

31

2

n

−

1

2

n

⋅

2

n

1

=

31

which gives you

2

n

=

32

Since

32

=

2

5

you can say that

2

n

=

2

5

⇒

n

=

5

This means that

5

half lives must pass in order for the sample to contain

31

atoms of argon-40 for every

1

atom of potassium-40.

Consequently, you can say that the age of the rock is

5

half-lives

⋅

1.25 billion years

1

half-life

=

6.25 billion years

−−−−−−−−−−−−−−−

I'll leave the answer rounded to three sig figs, but keep in mind that you have two significant figures for the number of atoms of argon-40 present per atom of potassium-40.