You are given 10,000 to invest in either 5% tax free bonds or 10% taxable bonds. Your total investment income for a year was $675. How much was invested at each rate?

At 5% ... $3500

at 10% ... $6500

Let us suppose, I invested $x in 5% tax free bonds

then $ (10,000-x) is invested in 10% tax free bonds

Invested income in the 5% tax free bond is 5x/100

and invested income in the 10% tax free bond is 10 (10,000-x) / 100

Total investment income for a year was $675

So we can say,

10 (10,000-x) / 100 + 5x/100 = 675

or, 100000 - 10x + 5x = 67500

or, 5x = 100000 - 67500 ... changing sides or, x = 32500/5

= 6500

so, 10000-x = 10000 - 6500 = 3500

Henceforth, the amount of investment at 5% rate is $6500 and at 10% rate is $3500.