If 25 dimes were moved from Box A to Box B, there would be an equal number of dimes in both boxes. If 100 dimes were moved from Box B to Box A, the ratio of dimes in Box A to Box B would be 7 : 2. What was the original number of dimes in Box A?

If we started with a in A and b in B, then we are told:

a-25 = b+25
(a+100)/(b-100) = 7/2

a = 250

Let's start by assigning variables to the unknown quantities. Let the original number of dimes in Box A be x, and let's also assume the original number of dimes in Box B is y.

According to the first statement, if 25 dimes were moved from Box A to Box B, there would be an equal number of dimes in both boxes. This implies that the number of dimes in Box A after the transfer is x - 25, and the number of dimes in Box B is y + 25.

Next, the second statement states that if 100 dimes were moved from Box B to Box A, the ratio of dimes in Box A to Box B would be 7:2. This means that the number of dimes in Box A after the second transfer is x - 25 + 100 = x + 75, and the number of dimes in Box B is y + 25 - 100 = y - 75.

Given the ratio of 7:2, we can set up the following equation:

(x + 75) / (y - 75) = 7/2

To solve for x, we can cross-multiply:

2(x + 75) = 7(y - 75)
2x + 150 = 7y - 525

Next, simplify the equation:

2x - 7y = -675

Since we have two unknowns, x and y, we need one more equation to solve the system. We can use the information from the first statement, which states that after transferring 25 dimes, there is an equal number of dimes in both boxes:

x - 25 = y + 25

Rearrange this equation:

x - y = 50

Now we have a system of equations:

2x - 7y = -675
x - y = 50

We can solve this system using any method, such as substitution or elimination. Let's solve it using the elimination method:

Multiply the second equation by 2 to make the coefficients of x in both equations match:

2(x - y) = 2(50)
2x - 2y = 100

Now use elimination by subtracting the second equation from the first:

(2x - 7y) - (2x - 2y) = -675 - 100
-5y = -775

Divide both sides by -5:

y = 155

Now substitute this value of y into one of the original equations to find x:

x - 155 = 50
x = 205

Therefore, the original number of dimes in Box A was 205.

To solve this problem, let's work through it step by step.

Let's suppose the original number of dimes in Box A was x. Since 25 dimes were moved from Box A to Box B, we now have (x - 25) dimes in Box A.

According to the problem, if 25 dimes are moved from Box A to Box B, there will be an equal number of dimes in both boxes. So, the number of dimes in Box B will also be (x - 25).

Next, it states that if 100 dimes are moved from Box B to Box A, the ratio of dimes in Box A to Box B will be 7:2. This means that after the movement, Box A will have 7 times as many dimes as Box B. So, we can write the equation:

(x - 25 + 100) / (x - 25) = 7/2

Let's simplify this equation:

(x + 75)/(x - 25) = 7/2

Cross-multiplying, we get:

2(x + 75) = 7(x - 25)

2x + 150 = 7x - 175

Subtracting 2x and adding 175 to both sides:

150 + 175 = 7x - 2x

325 = 5x

Dividing by 5:

x = 65

Therefore, the original number of dimes in Box A was 65.