Pri 6 Math help?

Question

Jo bought 5 times as many pens as marker pens. She spent $20 altogether. The total cost of pens was $5 more than the total cost of maker pens. A marker pen cost $1 more than a pen. Find the cost of a marker pen.

Answer 1

Let C be the total cost of markers and C + 5 be the total cost of pens. Add them up to find the total cost, $20. Then:

C + (C + 5) = 20
2C + 5 = 20
2C = 15
C = 7.5

So the total cost of markers was $7.50. That means the pens cost a total of $20 - $7.50 = $12.50.

Now, let x be the number of markers bought and 5x be the number of pens. Then the cost of a pen, c, is 12.5 / 5x and the cost of a marker, c + 1, is 7.5 / x. The total cost of her purchase is:

5xc + x (c + 1) = 20

Manipulating this last equation algebraically to find x, we get:

5xc + x (c + 1) = 20
5xc + xc + x = 20
6xc + x = 20
x (6c + 1) = 20
x = 20 / (6c + 1)

We have c = 12.5 / 5x from above. So x = 12.5 / 5c. Now we can plug that into the last equation in our sequence above to find c:

x = 20 / (6c + 1)
12.5 / 5c = 20 / (6c + 1)

Cross-multiplying, we get:

75c + 12.5 = 100c
12.5 = 25c
12.5 / 25 = c
0.50 = c This is the cost of a pen.

c + 1 = 0.50 + 1.00 = 1.50 This is the cost of a marker.

So the cost of a marker is $1.50.

Answer 2

The hard part with this problem is converting the word problem into algebraic equations... this is how you do that...

let P represent cost of pens...

and let M represent cost of marker pens...

P and M are in a 5:1 ratio...

5P + M = $20 (she bought 5 times as many pens as marker pens and the total cost was $20)

5P = M + $5 (the total cost of pens was $5 more than total cost of marker pens)

M = P + $1 (a marker pen costs $1 more than a pen)

5P = M = $5
M = P + $1
-------------------
5P = P + $6

4P = $6
P = $1.50.... but that is only the cost of pen, not marker...

The problem says that a marker pen (M) costs $1 more than a pen...

M = P + $1 = $1.50 + $1 = $2.50

She bought 5 times as many pens as marker pens... so there were 10 pens and 2 marker pens... that is a 5:1 ratio of pens to marker pens...

10 pens ($1.50) + 2 marker pens ($2.50) = $15.00 + $5.00 = $20

OR

You could have written them like this...

5xP + xM = $20 (she bought 5 times as many pens as marker pens and the total cost was $20)

5xP = xM + $5 (the total cost of pens was $5 more than total cost of marker pens)

M = P + $1 (a marker pen costs $1 more than a pen)


Three equations and three unknowns (x, P, M).... mean you can solve for x, P, and M

where "x" represents the number of marker pens, "5x" represents the number of pens (so that you have a 5:1 ratio between pens and marker pens), "P" represents cost of pens...and "M" represents cost of marker pens...