Final answer:
To solve this problem, we need to use a combination of ratios and algebra. By setting up and solving a system of equations, we can find the number of stickers that each person has. By subtracting the number of stickers Marc has from the number of stickers Ethan has, we can determine the difference in the number of stickers they have.
Step-by-step explanation:
To solve this problem, we need to use a combination of ratios and algebra. Let's start by assigning variables to the number of stickers that Marc and Ryan have. Let's say Marc has 3x stickers and Ryan has 7x stickers. According to the second ratio, Ryan and Ethan share stickers in the ratio 1:2, so Ryan has 1y stickers and Ethan has 2y stickers.
We can set up two equations to represent the given information:
3x + 7x + 1y + 2y = 192
10x + 3y = 192
To find the values of x and y, we need to solve this system of equations. We can use the substitution or elimination method. Let's use the elimination method in this case:
Multiply the first equation by 10 to get:
30x + 70x + 10y + 20y = 1920
Simplify:
100x + 30y = 1920
Multiply the second equation by 3:
30x + 9y = 576
Subtract the second equation from the first equation:
100x + 30y - (30x + 9y) = 1920 - 576
Simplify:
70x + 21y = 1344
Now we have a new equation:
70x + 21y = 1344
Let's multiply this equation by 10 to eliminate decimals:
700x + 210y = 13440
Multiply the third equation by 100:
1000x + 300y = 19200
Subtract the third equation from the second equation:
1000x + 300y - (700x + 210y) = 19200 - 13440
Simplify:
300x + 90y = 5760
Now we have a new equation:
300x + 90y = 5760
Divide this equation by 30 to simplify:
10x + 3y = 192
This is exactly the same as the second equation we had earlier. This means that the system of equations is consistent and the value of x and y satisfy both equations. Now we can substitute the value of x back into one of the original equations to find the value of y. Let's substitute it into the first equation:
3(8) + 7(8) + 1y + 2y = 192
Simplify:
24 + 56 + 3y + 2y = 192
Combine like terms:
5y + 80 = 192
Subtract 80 from both sides:
5y = 112
Divide both sides by 5:
y = 22.4
The value of y represents the number of stickers that Ethan has. Since we can't have a fraction of a sticker, we round down to the nearest whole number. Ethan has 22 stickers. To find the number of stickers that Marc has, we substitute the value of y into the second equation:
10x + 3(22) = 192
Simplify:
10x + 66 = 192
Subtract 66 from both sides:
10x = 126
Divide both sides by 10:
x = 12.6
Again, we round down to the nearest whole number. Marc has 12 stickers.
To find how many more stickers Ethan has than Marc, we subtract the number of stickers that Marc has from the number of stickers that Ethan has: 22 - 12 = 10 stickers. Therefore, Ethan has 10 more stickers than Marc.