Question 44917
x = amount borrowed at 9%
y = amount borrowed at 10%
z = amount borrowed at 12%
{{{x + y + z = 500000}}}
Annual interest charged at 9% = {{{.09*x}}}
Annual interest charged at 10% = {{{.10*x}}}
Annual interest charged at 12% = {{{.12*x}}}
Total annual interest = 52000
{{{.09*x + .10*y + .12*z = 5.2 * 10^4}}}
according to problem, {{{y = 2.5*x}}}
{{{.09*x + .10*2.5*x + .12*z = 5.2 * 10^4}}}
{{{.34*x + .12*z = 5.2 * 10^4}}}
also,
{{{x + y + z = 500000}}}
{{{x + 2.5*x + z = 50 * 10^4}}}
{{{3.5*x + z = 50 * 10^4}}}
multiply both sides of this equation by (-.12)
{{{-.42*x + (-.12)*z = (-.12)*50 * 10^4}}}
{{{-.42*x -.12*z = -6 * 10^4}}}
and the other equation was
{{{.34*x + .12*z = 5.2 * 10^4}}}
add these equations
{{{-.08*x = -.8 * 10^4}}}
divide both sides by (-.08)
{{{x = 10*10^4}}}
{{{x = 100000}}} amount borrowed at 9%
{{{y = 2.5*100000}}}
{{{y = 250000}}} amount borrowed at 10%
{{{z = 500000 - 350000 }}}
{{{z = 150000}}} amount borrowed at 12%
check
{{{3.5*x + z = 50 * 10^4}}}
{{{35*10^4 + 15*10^4 = 50 * 10^4}}}
{{{50*10^4 = 50*10^4}}}
OK