Using matrices to find the folowing word problem of amount invested

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A man has $205,000 invested in three properties. One earns 12%, one 10% and one 8%. His annual income from the properties is $19,700 and the amount invested at 8% is twice that invested at 12%.  I need the percentages of amount invested and the percentages of the annual income.

Sep 29th, 2015

Hello!

Denote the parts of original investment as a, b, c (in thousands of $). Then

a+b+c=205.

Income is 0.12*a + 0.1*b + 0.08*c = 19.7.

Also given that c=2a.

It is a linear system 3 by 3, let's solve it.

1) substitute c=2a into income equation:
0.12*a + 0.1*b + 0.16a = 0.28*a + 0.1*b = 19.7.

2) express b by a from this:
0.1*b = 19.7 - 0.28*a, so b = 197 - 2.8*a.

3) substitute b and c into investment equation:
a + (197-2.8*a) + 2a = 205, or
(1-2.8+2)*a = 205 - 197,
0.2*a = 8, a = 40 (thousands of $) = 40,000$.

4) compute b = 197 - 2.8*a = 197 - 112 = 85 = 85,000$
and c = 2a = 80 = 80,000$.

5) the corresponding parts of income is
0.12*a = 4.8 (=4800$),
0.1*b = 8.5 (=8500$),
0.08*c = 6.4 (=6400$)

Please ask if something is unclear.
Sep 29th, 2015

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