We will use the variable
r to represent the number of scones sold to Robbie. We will use the variable
x to represent the increase in scones sold to each person (since the increase is the same, we can use the same variable each time).
Scones sold to Cameron =
r +
x.
Scones sold to Louis =
the scones sold to Cameron +
x =
r +
x +
x =
r + 2
x.
Scones sold to Tom = scones sold to Louis +
x =
r + 2
x +
x =
r + 3
x.
Scones sold to Charlie = scones sold to Tom +
x =
r + 3
x +
x =
r + 4
x.
We add all of these together to get our total of 60:
r + (
r +
x) + (
r + 2
x) + (
r + 3
x) + (
r + 4
x) = 60
Combine our like terms and we have 5
r + 10
x = 60 (there are 4
r's and 10 total
x's).
We have another equation involved here. We know that Robbie + Cameron, or
r +
r +
x = 2
r +
x, is 3/7 of the total of the other three boys (this total given by
r + 2
x +
r + 3
x +
r + 4
x = 3
r + 9
x). We can write this as:
2
r +
x = 3/7(3
r + 9
x)
We can cancel the 7. Fraction bars are the same as division, so we cancel by multiplying: 7(2
r +
x) = 3(3
r + 9
x), which simplifies to 14
r + 7
x = 9
r + 27
x after using the distributive property.
In order to work with a system of equations (two equations at the same time) it's easier if they look the same. Let's get
r and
x on one side of the equation. We will start by cancelling 9
r. Since it's positive, we can cancel or move it by subtracting: 14
r + 7
x - 9
r = 9
r - 9
r + 27
x, which gives us 5
r + 7
x = 27
x. Now we want to move the 27
x. Since it's also positive we will subtract it:
5
r + 7
x - 27
x = 27
x - 27
x, which gives us 5
r -20
x = 0.
Let's align them vertically:
The coefficients of our
r's are the same, so we will eliminate those by subtracting:
Now that our
r's are gone, we have 30
x (10
x - -20
x = 30
x) and 60 on the other side:
30
x = 60
Divide both sides by 30:
Plug this into our first equation,
5
r + 10
x = 60:
Subtract 20 from both sides:
Divide both sides by 5:
This tells us that
r, the scones sold to Robbie, were 8 and
x, the change between the number of scones sold to each person, was 2. That means she sold 8 to Robbie, 10 to Cameron, 12 to Louis, 14 to Tom and 16 to Charlie.