Let's denote Amit's current age as [tex]\( A \)[/tex] and his father's current age as [tex]\( F \)[/tex].
### Step 1: Establish the relationships
1. Amit is 32 years younger than his father:
[tex]\[
A = F - 32
\][/tex]
2. Four years ago, Amit's age was one-fifth of his father's age at that time:
[tex]\[
A - 4 = \frac{F - 4}{5}
\][/tex]
### Step 2: Substitute [tex]\( A \)[/tex] from the first equation into the second equation
We already have:
[tex]\[
A = F - 32
\][/tex]
Substitute this into the equation for four years ago:
[tex]\[
(F - 32) - 4 = \frac{F - 4}{5}
\][/tex]
### Step 3: Simplify the second equation
Simplify the left side:
[tex]\[
F - 36 = \frac{F - 4}{5}
\][/tex]
### Step 4: Clear the fraction
To eliminate the fraction, multiply every term by 5:
[tex]\[
5(F - 36) = F - 4
\][/tex]
Distribute 5 on the left side:
[tex]\[
5F - 180 = F - 4
\][/tex]
### Step 5: Move terms involving [tex]\( F \)[/tex] to one side
Subtract [tex]\( F \)[/tex] from both sides:
[tex]\[
5F - F - 180 = -4
\][/tex]
This simplifies to:
[tex]\[
4F - 180 = -4
\][/tex]
### Step 6: Solve for [tex]\( F \)[/tex]
Add 180 to both sides:
[tex]\[
4F = 176
\][/tex]
Divide both sides by 4:
[tex]\[
F = 44
\][/tex]
### Step 7: Solve for [tex]\( A \)[/tex]
We use the relationship [tex]\( A = F - 32 \)[/tex]:
[tex]\[
A = 44 - 32
\][/tex]
[tex]\[
A = 12
\][/tex]
### Conclusion
Amit's father's current age is [tex]\( 44 \)[/tex] years old, and Amit's current age is [tex]\( 12 \)[/tex] years old.
