How many one-cent stamps did Maria buy?

The word problem gives us an expression that looks like this:

#1.00=0.05n+0.02t+0.01p#

where #n# is the number of five-cent stamps, #t# is the number of two-cent stamps, and #p# is the number of one-cent stamps.

We also know that Maria bought ten times as many one-cent stamps as two-cent stamps. If we write this out as another expression:

#color(blue)(p=10t)#

We then substitute it into the first expression:

#1.00=0.05n+0.02t+0.01color(blue)((10t))#

#1.00=0.05n+0.02t+0.10t#

#1.00=0.05n+0.12t#

Now, we need to figure out how many two and five-cent stamps were purchased. Assuming Maria spent EXACTLY $1, the number of two-cent stamps must give a total such that 0.12 times that number gives a 5 or zero in the remainder. This is so we have an integral value for #n#.

The only multiple of 0.12 that satisfies this, AND results in a value less than $1 is 5, proof below:

#1.00=0.05n+0.12(5)#

#1.00=0.05n+0.6color(red)(0)#

#0.40=0.05n#

#n=8#

Now we have a solution for #n# and #t#, but we really only need #p#. Luckily, we can use that relationship in the problem statement:

#color(blue)(p=10t)#

#p=10(5)#

#color(green)(p=50)#

Plugging in all of the values to check:

#1.00=0.05(8)+0.02(5)+0.01(50)#

#1.00=0.40+0.10+0.50#

#1.00=1.00#

The math checks out.