Question 1112210
let x = the amount of money he lends to the council.


let r be the interest rate of the loan.


the interest on the loan is 63 per year.


x * r = 63


when the interest rate on the loan is increased by .5%, the annual interest is raised to 67.50.


.5% / 100 is equal to .005.


x * (r + .005) = 67.5


you have 2 equations that need to be solved simultaneously.


they are:


x * r = 63
x * (r + .005) = 67.5


simplify the second equation and leave the first equation as is to get:


x * r = 63
x * r + .005 * x = 67.5


since x * r = 63 from the first equation, replace x * r in the second equation by 63 to get:


second equation becomes:


63 + .005 * x = 67.5


subtract 63 from both sides of this equation to get:


.005 * x = 4.5


divide both sides of this equation by .005 to get:


x = 4.5 / .005


this results in x = 900.


go back to your original equation and replace x with 900 and solve for r.


the first original equation is:


x * r = 63


replace x with 900 to get:


900 * r = 63


solve for r to get:


r = 63/900 = .07


in the second original equation, replace r + .005 with .075


the first and second original equations now become:


900 * .07 = 63 becomes 63 = 63, which is true.


900 * .075 = 67.5 becomes 67.5 = 67.5, which is true.


both original equations are true when r = .07.


this confirms the solution is correct, given the original equations are correct.


interest rate percent is 100 * interest rate.


conversely, interest rate is interest rate percent / 100.


you had to work with interest rate, not percent.


that'w why we used .5%/100 = .005, rather than .5%, in the equations.