Suppose you invested $10,000 part at 6% annual interest and the rest at 9% annual interest. If you received $684 in interest after one year, how much did you invest at each rate?

Assuming simple interest (i. e. no compounding within first year), then

At 6%, interest = 10000*0.06=$600

At 9% interest = 10000*0.09 = $900

Two ways to find the ratio

method A. let x=proportion at 6%

then

600x+900 (1-x) = 684

Expand and solve

300x=900-684=216

x=216/300=0.72 or 72%

So 10000*0.72=7200 were invested at 6%

10000-7200=2800 were invested at 9%

method B: by proportions

Ratio of investments at 6% and 9%

= 900-684 : 684-600

=216 : 84

= 18 : 7

Amount invested at 6% = 18 / (18+7) * 10000 = 0.72*10000 = 7200

Amount invested at 8% = 7 / (18+7) * 10000=0.28*10000=2800