Let's solve the problem step-by-step.
1. Let \( D \) be the number of marbles Dan has.
2. Leah has 28 more marbles than Dan. Therefore, if \( L \) represents the number of marbles Leah has, we can express this relationship as:
[tex]\[
L = D + 28
\][/tex]
3. According to the problem, one third of Leah's marbles is equal to \(\frac{4}{5}\) of Dan's marbles. We can write this as the following equation:
[tex]\[
\frac{L}{3} = \frac{4}{5} D
\][/tex]
4. Substitute \( L \) from step 2 into the equation in step 3:
[tex]\[
\frac{D + 28}{3} = \frac{4}{5} D
\][/tex]
5. To clear the fractions, multiply both sides of the equation by 15 (the least common multiple of 3 and 5):
[tex]\[
15 \cdot \frac{D + 28}{3} = 15 \cdot \frac{4}{5} D
\][/tex]
This simplifies to:
[tex]\[
5 (D + 28) = 12 D
\][/tex]
6. Distribute and simplify the equation:
[tex]\[
5D + 140 = 12D
\][/tex]
7. Isolate \( D \) by subtracting \( 5D \) from both sides of the equation:
[tex]\[
140 = 7D
\][/tex]
8. Solve for \( D \):
[tex]\[
D = \frac{140}{7} = 20
\][/tex]
So, Dan has 20 marbles.
9. Now, use this value of \( D \) to find out how many marbles Leah has:
[tex]\[
L = D + 28 = 20 + 28 = 48
\][/tex]
Therefore, Leah has 48 marbles.
