The ratio of Maya’s beads to Kayla’s beads was 12:7. After Maya bought another 28 beads and Kayla gave away 32 beads, 57of 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.

let Maya's original number of beads be 12x

and Kayla's be 7x

Maya bought another 28 , now has 12x+28
Kayla gave away 32, now has 7x-32

but 7x-32 = 57
x = 89/7 , which is not a whole number.

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To answer these questions, we need to analyze the information given and work through the problem step by step.

a) To find the difference in the number of beads Maya had compared to Kayla in the beginning, we first need to determine how many beads each of them had initially.

Let's assume that Maya had 12x beads and Kayla had 7x beads initially, where x is some constant.

According to the given information, after Maya bought 28 more beads, she would have a total of 12x + 28 beads. Also, Kayla gave away 32 beads, so she would have a remaining total of 7x - 32 beads.

We know that in the end, Kayla had 57 beads left. Therefore, we can set up an equation:

7x - 32 = 57

Now, we can solve this equation to find the value of x:

7x = 57 + 32
7x = 89
x = 89 / 7
x = 12.7

Now that we have the value of x, we can determine how many beads each person had initially:

Maya's initial number of beads = 12x = 12 * 12.7 = 152.4
Kayla's initial number of beads = 7x = 7 * 12.7 = 88.9

Therefore, Maya initially had 152.4 beads and Kayla initially had 88.9 beads.

To find how many more beads Maya had than Kayla in the beginning, we subtract Kayla's initial number of beads from Maya's initial number of beads:

152.4 - 88.9 = 63.5

So, Maya initially had 63.5 more beads than Kayla.

b) To find the new ratio of Maya’s beads to Kayla’s beads after the changes, we need to account for the additional beads Maya bought and the beads Kayla gave away.

After Maya bought 28 more beads, her new total of beads is 152.4 + 28 = 180.4

After Kayla gave away 32 beads, her new total of beads is 57.

The new ratio of Maya's beads to Kayla's beads is given by:

180.4 / 57

This ratio cannot be simplified any further. So, the new ratio of Maya’s beads to Kayla’s beads is approximately 3.166:1.