Patrick bought some dictionaries and files for $1 474.
The number of dictionaries he bought was 2/3 the number of files he bought.
The cost of a dictionary was $12 more than the cost of a file.
He spent $22 more on dictionaries than on files.
How much did a dictionary cost?
d = 2/3 f
If dictionaries cost x and files cost y, then
x = y+12
d*x = f*y + 22
d*x + f*y = 1474
Now solve for x
Number of files f
number of dictionaries = 2f/3
cost of a file: x
cost of dictionary : x+12
(2f/3)(x+12) - fx = 22
times 3
2f(x+12) - 3fx = 66
2fx + 24f - 3fx = 66
24f - fx = 66 or
f = 66/(24-x)
we also know that fx + (2f/3)(x+12) = 1474
3fx + 2f(x+12) = 4422
5fx + 24f = 4422
f = 4422/(5x + 24)
66/(24-x) = 4422/(5x + 24)
330x + 1584 = 106128 - 4422x
4752x = 104544
x = 22
f = 66/(24-22) = 33
I defined the cost of a dictionary as x+12
So a dictionary costs 34
check:
cost of dictionary --- 34
cost of file ---- 34
number of files --- 33
number of dict's --- 22
total cost = 33(22) + 22(34) = 1474
looks good to me
Note, this is obviously not the only way.
To find out how much a dictionary costs, let's break down the information given step by step.
Let's assume that the number of dictionaries Patrick bought is represented by the variable "x" and the number of files he bought is represented by the variable "y."
We are given that the number of dictionaries he bought was 2/3 the number of files, so we can write the equation: x = (2/3)y.
We are also given that the cost of a dictionary was $12 more than the cost of a file.
Let's assume the cost of a file is represented by the variable "a." So the cost of a dictionary is a + $12.
Furthermore, we are given that Patrick spent $22 more on dictionaries than on files.
Therefore, we can set up another equation: x(a + $12) = y(a) + $22
Now, let's substitute the value of x from the first equation into the second equation:
((2/3)y)(a + $12) = ya + $22
Expanding the equation:
(2/3)ya + (2/3)($12y) = ya + $22
Simplifying the equation:
(2/3)ya + (8/3)y = ya + $22
Next, let's simplify the equation further by multiplying through by the common denominator, which is 3:
2ya + 8y = 3ya + $66
We can simplify this equation by subtracting 2ya and $66 from both sides:
8y - 3ya = $66
Then, let's factor out the variable "y" on the left side:
y(8 - 3a) = $66
Now, we want to find the value of "a" (the cost of a file), so let's solve for y first.
Dividing by (8 - 3a):
y = $66 / (8 - 3a)
Lastly, to find the cost of a dictionary, substitute the value of y into the first equation:
x = (2/3)y
x = (2/3) * ($66 / (8 - 3a))
x = (2/3) * ($66 / (8 - 3a))
So, the cost of a dictionary is (2/3) * ($66 / (8 - 3a)).