Let's break down the problem step-by-step.

Step 1: Defining Variables

Let the number of marbles in ARB, Jersey, and Darby be represented by the variables $A$, $J$, and $D$, respectively. According to the given ratio, we have: $A : J : D = 2 : 5 : 4$

Let's introduce a common multiplier $x$ to represent the number of marbles: $A = 2x, \quad J = 5x, \quad D = 4x$

Step 2: Condition Involving Jersey and ARB

The problem states there are 12 more marbles in Jersey than in ARB: $J = A + 12$ Substitute $A = 2x$ and $J = 5x$: $5x = 2x + 12$ Solving for $x$: $3x = 12 \quad \Rightarrow \quad x = 4$

Step 3: Calculating Initial Number of Marbles

Now we can find the initial number of marbles in each jar: $A = 2x = 2(4) = 8$ $J = 5x = 5(4) = 20$ $D = 4x = 4(4) = 16$

Step 4: Finding Marbles to be Transferred

The problem asks how many marbles should be transferred from Darby to ARB so that ARB and Darby have the same number of marbles.

Let's say we transfer $y$ marbles from Darby to ARB. The new number of marbles in each jar will be:

For ARB: $A_{\text{new}} = 8 + y$

For Darby: $D_{\text{new}} = 16 - y$

Setting them equal, since they need to have the same number: $8 + y = 16 - y$

Solving for $y$: $2y = 8 \quad \Rightarrow \quad y = 4$

Final Answer

So, 4 marbles should be transferred from Darby to ARB for both jars to have the same number of marbles.

Would you like more details or have any questions?

Related Questions:

1. What is the total number of marbles in all three jars after the transfer?
2. If 3 marbles were added to each jar, what would the new ratio be?
3. How many marbles would each jar have if the ratio changed to 3:4:5?
4. What is the percentage increase in the number of marbles in ARB after the transfer?
5. If 6 marbles were removed from Jersey, how many would each jar have then?

Tip: Always double-check ratios and conditions given in word problems by substituting values to ensure your equations and calculations are correct.