Let's solve this problem step-by-step to find out how many marbles Navin had initially and how many he gave to his brother.
### Part a) Calculate how many marbles Navin had at the beginning.
1. Define the initial number of marbles Navin had:
Let [tex]\( n \)[/tex] be the initial number of marbles.
2. Determine how many marbles Navin gave to his brother:
Navin gave [tex]\(\frac{1}{5}\)[/tex] of his marbles to his brother.
Thus, the number of marbles given to his brother is [tex]\( \frac{n}{5} \)[/tex].
3. Calculate the remaining marbles after giving to his brother:
The remaining marbles after giving to his brother is:
[tex]\[
n - \frac{n}{5} = \frac{5n}{5} - \frac{n}{5} = \frac{4n}{5}
\][/tex]
4. Determine how many marbles Navin sold:
Navin sold [tex]\(\frac{5}{8}\)[/tex] of his remaining marbles.
Thus, the number of marbles he sold is:
[tex]\[
\frac{5}{8} \times \frac{4n}{5} = \frac{5 \times 4n}{8 \times 5} = \frac{4n}{8} = \frac{n}{2}
\][/tex]
5. Calculate the marbles remaining after selling:
The marbles remaining after selling is:
[tex]\[
\frac{4n}{5} - \frac{n}{2}
\][/tex]
6. Simplify the expression for the remaining marbles:
Find a common denominator (which is 10) for the fraction subtraction:
[tex]\[
\frac{4n}{5} - \frac{n}{2} = \frac{8n}{10} - \frac{5n}{10} = \frac{3n}{10}
\][/tex]
7. Set up the equation knowing Navin has 12 marbles remaining:
We know that the remaining marbles after all the transactions are 12:
[tex]\[
\frac{3n}{10} = 12
\][/tex]
8. Solve for [tex]\( n \)[/tex]:
[tex]\[
3n = 12 \times 10 \implies 3n = 120 \implies n = \frac{120}{3} = 40
\][/tex]
Thus, Navin had initially [tex]\( 40 \)[/tex] marbles.
### Part b) How many marbles did Navin give his brother?
1. Calculate the number of marbles given to his brother:
Navin gave his brother [tex]\(\frac{1}{5}\)[/tex] of his marbles:
[tex]\[
\frac{1}{5} \times 40 = 8
\][/tex]
So, Navin gave his brother [tex]\( 8 \)[/tex] marbles.
In summary:
- a) Navin had initially [tex]\( 40 \)[/tex] marbles.
- b) Navin gave his brother [tex]\( 8 \)[/tex] marbles.
