Question 173808
Let {{{s}}}= amount invested in short-term
Let {{{m}}}= amount invested in medium term
Let {{{l}}}= amount invested in long yerm
Given is:
{{{s + m + l = 10000}}}
{{{s = m}}}
{{{.04s}}}= annual return on short-terms
{{{.05m}}}= annual return on medium-terms
{{{.06l}}}= annual return on long-terms
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The problem is looking for an overall return
of 5.1% on these 3 investments, so, in words
(sum of the returns)/(total investment) = 5.1%
{{{(.04s + .05m + .06l) / 10000 = .051}}}
Multiply both sides by {{{10000}}}
(1) {{{.04s + .05m + .06l = 510}}} 
Multiply both sides by {{{100}}}
(1) {{{4s + 5m + 6l = 51000}}}
(2) {{{s + m + l = 10000}}} (given)
Multiply both sides of (2) by {{{6}}}
(2) {{{6s + 6m + 6l = 60000}}}
Subtract (1) from (2)
{{{2s + m  = 9000}}}
And, since {{{s = m}}},
{{{2m + m = 9000}}}
{{{3m = 9000}}}
{{{m = 3000}}}
{{{s = 3000}}}
{{{s + m + l = 10000}}}
{{{3000 + 3000 + l = 10000}}}
{{{l = 10000 - 6000}}}
{{{l = 4000}}}
He has to invest $4000 in long-term bonds
check answer:
{{{(.04s + .05m + .06l) / 10000 = .051}}}
{{{(.04*3000 + .05*3000 + .06*4000) / 10000 = .051}}}
{{{(120 + 150 + 240) / 10000 = .051}}}
{{{510 / 10000 = .051}}}
{{{.051 = .051}}}
OK