I know it is essentially an algebra problem because it makes no sense as a financial problem unless Tina is a bad record keeper and lost track of what loans she made

In that case she is probably accustomed to not knowing what the heck is going on with her loans so she probably would never ask about it. Or it would be very evident when she logged in to the web site where she made the loan.

Make up some variables to represent the quantities that are used in the problem

A = Initial Loan Amount on First Loan

B = Initial loan amount on 2nd loan

A + B = 44980 this is the total

A rate = 1.2% annually, simple interest

B rate = 1.8% simple interest per year

After one year the interest on A is 1.2% of A

The interest on B is 1.8% of B

Total interest = 1.2% x A + 1.8% x B (the x means multiply)

I = 0.012 A + 0.018 B The decimal point is moved 2 places to the left to show it is divided by 100 for the percentage

So all the information given can be written using these two equations:

0.012 A + 0.018 B = 679.64

A + B = 44980

Next solve algebraically for A or B.

This can be done by substitution.

There are several ways to do this.

I think one good way is to do this:

A = 44980 - B

Then substitute that into the other equation writing 44980 where B was

This is good because now it will be an equation with only B in it, not A

0.012 (44980 - B) + 0.018 B = 679.64

Multiply through the 0.012

539.76 - 0.012 B + 0.018 B = 679.64

Combine the two B terms and subtract 539.76 from both sides

0.006 B = 139.88

Divide both sides by .006 (which is dividing by 6 and multiplying by 1000)

B = 23,313.33

A = 44980 - B = 21,666.67

You should probably check that this gives the correct interest in case I made an arithmetic error somewhere.