finding amts invested at each rate; finding running times

[aimforthemedic]

I am having a problem with two questions.

On the first question, I end up with a negative number every time no matter what I do to the formula.

1: A total of $32,000 is invested to provide retirement income. Part of the $32,000 is invested in an account paying 9% annual simple interest. How much should be invested in an account paying 11% simple interest so that the total income after one year is $35,360?

On this next question...I don't even know where to start.

2: To qualify for the Big-Foot half-marathon, a runner must have completed 5 half-marathons with an average time of (at most) 1:50, rounded up to the nearest minute. A certain runner has completed 4 half marathons with times: 1:45, 1:55, 2:01, and 1:56 (all times have been rounded up to the nearest minute). What completion times (if any) will qualify this runner for the Big-Foot half-marathon? (Rounding up to the nearest minute means if you run a half marathon in 1:50:01 it is recorded as 1:51)

Can anyone help me out?

 

[soroban]

Re: Need help

Hello, aimforthemedic!

A total of $32,000 is invested to provide retirement income.
Part of the $32,000 is invested in an account paying 9% annual simple interest.
How much should be invested in an account paying 11% simple interest
so that the total income after one year is $35,360?

I bet your set-up is incorrect.


\(\displaystyle \text{Let: }\:x\text{ = amount invested at 9 percent}\)

\(\displaystyle \text{Let: }\,{\bf 32,000 - x}\text{ = amount invested at 11 percent}\)

\(\displaystyle \text{The equation is: }\:0.09x + 0.11(32,000-x) \:=\:35360\)

To qualify for a race, a runner must have completed 5 races
with an average time of (at most) 1:50, rounded up.
A certain runner has completed 4 races with times: 1:45, 1:55, 2:01, and 1:56.
What completion times (if any) will qualify this runner for the race?

First, convert all times to minutes.

. . . \(\displaystyle \begin{array}{ccc}\text{Time} & \text{Minutes} \\ \hline 1\!:\!45 & 105 \\ 1\!:\!55 & 115 \\ 2\!:\!01 & 121 \\ 1\!:\!56 & 116 \\ \hline \text{Total:} & 457 \end{array}\)


\(\displaystyle \text{Let }x\text{ = time of his 5th race}\)

\(\displaystyle \text{The average of his }five\text{ races must be at most }1\!:\!50 \,=\,110\text{ minutes.}\)

\(\displaystyle \text{The inequality is: }\;\frac{x + 457}{5} \:\leq \:110\)