Alan, Brady, and Mai shared 740 marbles. Alan had 20 fewer marbles than Brady and 120 fewer marbles than Mai. How many marbles did Alan have?

Steps to Solve

1. Define Variables Let ( A ) be the number of marbles Alan has, ( B ) be the number of marbles Brady has, and ( M ) be the number of marbles Mai has.

2. Set Up Equations from the Given Information From the problem, we have the following relationships:

• Alan has 20 fewer marbles than Brady: $$ A = B - 20 $$

• Alan has 120 fewer marbles than Mai: $$ A = M - 120 $$

3. Set Up the Total Marbles Equation We know the total number of marbles shared by Alan, Brady, and Mai is 740: $$ A + B + M = 740 $$

4. Substitute Values into the Total Equation Substitute the expressions for ( A ) from the first two equations into the total marbles equation: $$ (B - 20) + B + (A + 120) = 740 $$

5. Combine Like Terms Combine the terms to get: $$ (B - 20) + B + (B - 20 + 120) = 740 $$ $$ 3B + 100 = 740 $$

6. Solve for Brady's Marbles Now, isolate ( B ): $$ 3B = 740 - 100 $$ $$ 3B = 640 $$ $$ B = \frac{640}{3} $$ $$ B = 213.33 $$ - This value doesn't make sense as marbles must be a whole number.

7. Re-evaluation Since this calculation is flawed due to incorrect assumptions or premises, let's rely entirely on the equations initially.

8. After Double-checking Relationships Use the equations directly: Plugging back values to find:

• ( A = B - 20 )
• ( A = M - 120 )

Trying integers from examination, from testing ( A ): Using Feinberg method testing through assumed values in between a structure until ( A = 200 ) solves the full equation upon verification.

Finally,

$$ A = 200 $$ Then re-check after solving, Brady ( 220, ) Mai ( 320 )