SME
Grade 12 Science · Problem type
Any isotope: age from the daughter-to-parent ratio
Use this for “A rock holds 3 daughter atoms for every parent atom, and the half-life is 1.25 × 10⁹ years. How old is it?”
Diagram
After 2.5 × 10⁹ years, 2 half-lives have passed: 100 × (1/2)² = 25% left.
In the grid 25 of 100 Parent atoms are left; each half-life, half of those still there decay.
This is an example. Type your own values to start a new problem.
p
%
=
100%
÷
(1
+
R
)
=
100%
×
(0.5)
÷
Each daughter atom was once a parent atom, so 1 + R atoms started for each one left: p = 100% ÷ (1 + R). The half-lives in p then give the age.
n
Half-lives passed
worked out (not typed)
Formulas
p
=
100
1 +
R
25
=
100
1 + 3
p
=
100
×
0.5
n
25
=
100
×
0.5
2
n
=
t
T
2
=
2.50 ×
10
9
1.25 ×
10
9
Assumptions
•
Each daughter atom was once a parent atom, and the rock started with no daughter atoms.
•
No atoms got in or out since the rock formed.
•
Nothing in the solar system is older than about 4.57 × 10⁹ years, so no age can be greater.
Step-by-step
WE KNOW
R = 3, T = 1.25 × 10⁹ years
FIND
parent left (p), half-lives passed (n), age (t)
Step 1 · Find parent left (p)
Use p = 100 ÷ (1 + R)
Each
daughter
atom
was
once
a
parent
atom:
the
parent’s
share
of
1
+
R
atoms.
p
=
100
1 + 3
p
=
100
4
p
=
25%
Step 2 · Find half-lives passed (n)
Use p = 100 × 0.5ⁿ
Count
the
halvings:
n
is
the
power
of
2
that
equals
100
p
.
n
=
ln(
100
p
)/ln(2)
n
=
ln(
100
25
)/ln(2)
n
=
1.386/0.6931
n
=
2
Step 3 · Find age (t)
Use n = t ÷ T
Each
half-life
takes
T
:
multiply.
t
=
n
×
T
t
=
2
×
(1.25
×
10
9
)
t
=
2.50 ×
10
9
years
Check
Put the numbers back in. Both sides should match.
25
=
100
1 + 3
✓
25
=
100
×
0.5
2
✓
2
=
2.50 ×
10
9
1.25 ×
10
9
✓
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