Question

Mrs Tan bought 4 times as many pens as notebooks and each notebook cost $8.20 more than each pen. She spent $26 more on the books than the pens. Mrs Tan spent $394 altogether on the pond and notebooks, how much was the cost of a notebook?

Answer 1

Let x be the cost of 1 pen
then cost of 1 notebook = x + 8.20

Let y be the number of pens Tan buys
then number of notebooks Tan buys = y/4

She spent $26 more on books than on pens which means

Cost of notebooks - Cost of pens = 26
(x + 8.20) * y/4 - xy = 26

Sinplifying it

(xy + 8.20y)/4 - xy = 26
(xy + 8.20y - 4xy)/4 = 26
8.20y - 3xy = 104

She spent $394 which means

Cost of notebooks + Cost of pens = 394
(x + 8.20) * y/4 + xy = 394

Simplifying it

(xy + 8.20y)/4 + xy = 394
(xy + 8.20y + 4xy)/4 = 394
8.20y + 5xy = 1576

Now, we have two equations,

(1) 8.20y - 3xy = 104
(2) 8.20y + 5xy = 1576

Now we need to find a third equation with either x or y as the subject of any of both the previous equations.

Let's make y the subject of (2) equation

8.20y + 5xy = 1576
y(8.20 + 5X) = 1576
(3) y = 1576/(8.20 + 5x)

Let's substitute the new value of y from (3) into (1) because we rearranged (2) to from (3)

8.20y - 3xy = 104
y(8.20 - 3x) = 104
y = 104/(8.20 - 3x)
1576/(8.20 + 5x) = 104/(8.20 - 3x)
1576 * (8.20 - 3x) = 104 * (8.20 + 5x)
12923.2 - 4728x = 852.8 + 520x
12923.2 - 852.8 = 4728x + 520x
12070.4 = 5248x
12070.4/5248 = x
x = 2.3

Now find the value of y by substituting the value of x in either equation, preferably (3)

y = 1576/(8.20 + 5x)
y = 1576/(8.20 + 5 * (2.3))
y = 80

Therefore cost of 1 notebook = x + 8.20 = 2.3 + 8.20 = $10.50