Leah has 28 more marbles than Dan. One third of Leah's marbles is equal to [tex]\frac{4}{5}[/tex] of Dan's marbles. How many marbles does Leah have?



Answer :

To solve the problem, we need to find the number of marbles Leah has given two conditions:

1. Leah has 28 more marbles than Dan.
2. One third of Leah's marbles is equal to four-fifths of Dan's marbles.

Let's denote:
- Leah's number of marbles as \( L \).
- Dan's number of marbles as \( D \).

Step-by-step solution:

Step 1: Establish the equations based on the conditions provided.

From the first condition:
[tex]\[ L = D + 28 \tag{1} \][/tex]

From the second condition:
[tex]\[ \frac{L}{3} = \frac{4}{5} D \tag{2} \][/tex]

Step 2: Solve the equations simultaneously.

Start by substituting \( L \) from equation (1) into equation (2):

[tex]\[ \frac{D + 28}{3} = \frac{4}{5} D \][/tex]

To eliminate the fractions, multiply every term 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]\[ 5(D + 28) = 12D \][/tex]

Expand and simplify:

[tex]\[ 5D + 140 = 12D \][/tex]

Move terms involving \( D \) to one side:

[tex]\[ 140 = 12D - 5D \][/tex]

[tex]\[ 140 = 7D \][/tex]

Solve for \( D \):

[tex]\[ D = \frac{140}{7} \][/tex]

[tex]\[ D = 20 \][/tex]

So, Dan has 20 marbles.

Step 3: Find Leah's number of marbles using equation (1).

[tex]\[ L = D + 28 \][/tex]

Substitute \( D = 20 \):

[tex]\[ L = 20 + 28 \][/tex]

[tex]\[ L = 48 \][/tex]

So, Leah has 48 marbles.

Final Answer: Leah has 48 marbles.