annoying math i dont understand

Question

A bank loaned out $18,500, part of it at the rate of 7% annual interest, and the rest at 3% annual interest. The total interest earned for both loans was $1,255.00. How much was loaned at each rate?

$ was loaned at 7% and

$ was loaned at 3%.

Answer 1

$17500 at 7%

$1000 at 3%

A pair equations in two unknowns x and y should work here

Equation 1 for total amount loaned

x + y = 18500

Equation 2 for total interest accumulated at each rate

.07x + .03y = 1255

I will use Equation 1 to set up a substitution to use in Equation 2 and solve the problem

Equation 1

x + y = 18500

y = 18500 - x

Substitute this quantity for y in Equation 2

.07x + .03y = 1255

.07x + .03(18500 - x) = 1255

.07x + 555 - .03x = 1255

Combine like terms

.04x + 555 = 1255

Subtract 555 from both sides of the equation

.04x = 1255 - 555

.04x = 700

Divide both sides by .04

x = 17500

y = 18500 - 17500

y = 1000

Checking in Equation 2

.07x + .03y = 1255

.07(17500) + .03(1000) = 1255

1225 + 30 = 1255

I hope you find this useful if you have any question please send me a message.

Answer 2

We denote $x the part of loan that earns 7%. Then $(18,500 - x) earn 3%. Hence, the total interest is

0.07x + 0.03(18,500 - x) = 1,255.

We solve the above equation for x

0.04x = 1,255 - 0.03*18,500 = 700. Thus x = 700/0.04 = 17,500. Hence, $17,500 was loaned at 7% and 18,500 -17,500 = $1,000 was loaned at 3%.