The ratio of Maya’s beads to Kayla’s beads was 12:7. After Maya bought another 28 beads and Kayla gave away 3/2 beads, 5/7 of Kayla’s beads were left.

a) How many more beads did Maya have than Kayla in the beginning?

b) Find the new ratio of Maya’s beads to Kayla’s beads.

How did Kayla give away 3/2 beads?

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To find the number of beads Maya and Kayla had initially, we can solve the following system of equations:

Let x be the common ratio between Maya's beads and Kayla's beads.

From the given information, we have the following equations:

Maya's beads = 12x
Kayla's beads = 7x

After Maya bought another 28 beads, her new total number of beads is 12x + 28.

After Kayla gave away 3/2 beads, her new total number of beads is (7x - 3/2).

Given that 5/7 of Kayla's beads were left, we can set up the following equation:

(7x - 3/2) = (5/7)*(7x)

Simplifying the equation:

7x - 3/2 = 5x

Subtracting 5x from both sides:

2x - 3/2 = 0

Adding 3/2 to both sides:

2x = 3/2

Dividing both sides by 2:

x = 3/4

a) To find the number of beads Maya had more than Kayla initially, we can substitute x = 3/4 into the equation:

Maya's beads - Kayla's beads = 12x - 7x = 5x

5*(3/4) = 15/4 = 3 3/4 beads

Therefore, Maya had 3 3/4 more beads than Kayla initially.

b) To find the new ratio of Maya's beads to Kayla's beads after the changes, we add the changes to the initial bead counts:

Maya's beads = 12x + 28 = 12*(3/4) + 28 = 21 beads
Kayla's beads = (7x - 3/2) = 7*(3/4) - 3/2 = 3/2 beads

The new ratio is 21:3/2, which can be simplified to 42:3 (by multiplying both terms by 2):

The new ratio of Maya's beads to Kayla's beads is 42:3.

To solve this problem, we need to follow these steps:

Step 1: Understand the given information.

The given information states that the ratio of Maya's beads to Kayla's beads was 12:7. This ratio refers to the original number of beads each person had.

Step 2: Determine the number of beads for each person.

To find the number of beads for each person in the beginning, we need to set up a proportion based on the given ratio. Let's assign variables to represent the number of beads for Maya and Kayla:

Let the number of Maya's beads be 12x.
Let the number of Kayla's beads be 7x.

Step 3: Account for additional beads and beads given away.

Maya bought another 28 beads, so her new total number of beads is 12x + 28.
Kayla gave away 3/2 beads. We will subtract this number from her original total number of beads.

Step 4: Determine the remaining beads for Kayla.

According to the problem, 5/7 of Kayla's beads were left. We'll set up an equation to represent this:

(5/7) * (7x - (3/2)) = 7x

Step 5: Solve the equation.

Now, we'll solve the equation to find the value of x, which represents the original number of beads for Kayla.

(5/7) * (7x - (3/2)) = 7x
(5/7) * (7x - 21/2) = 7x [Simplified 3/2 as 21/2]
(5/7) * (14x/2 - 21/2) = 7x [Simplified 7x - 21/2 to 14x/2]
(5/7) * (14x - 21) = 7x
(5/7) * 14x - (5/7) * 21 = 7x
10x - 15 = 7x
10x - 7x = 15
3x = 15
x = 5

Step 6: Calculate the number of beads for each person in the beginning.

Now that we have the value of x, we can find the number of beads for Maya and Kayla:

Maya's initial number of beads = 12x = 12 * 5 = 60 beads
Kayla's initial number of beads = 7x = 7 * 5 = 35 beads

Step 7: Answer the questions.

a) To find the difference in the initial number of beads between Maya and Kayla, subtract the number of beads Kayla had from the number Maya had: 60 - 35 = 25 beads. Therefore, Maya had 25 more beads than Kayla in the beginning.

b) To find the new ratio of Maya's beads to Kayla's beads after the changes, we need to account for the additional 28 beads Maya bought and the number of beads Kayla gave away. The new ratio is (12x + 28) : (7x - (3/2)). Plugging in the value of x we found in step 5: (12 * 5 + 28) : (7 * 5 - (3/2)) = 68 : (35 - (3/2)). Simplify the expression to get the new ratio.