Please help a dumb mother trying to help with son's homework

[dbush]

So here's the thing... my son has been home from school sick. I'm trying to help with 2 word problems. He missed the lesson so he has no idea where to start. Any help is appreciated. Here goes...

Nature's Best wants to combine mixed nuts they sell for $3.60/lb. with dried fruit they sell for $2.40/lb to create a trail mix
a. Write an equation to represent the problem
b. using your equation in (a), how much of each snack should they use to make 10 pounds of trail mix that would sell for $3.30/lb?


2nd problem

Paula leaves home driving 40 miles per hour. One hour later, Dan leaves home, driving in the same direction at a speed of 50 miles per hour.

a. Write an equation to represent the problem
b. Using your equation in (a), how long will it take Dan to catch up with Paula?

 

[soroban]

Hello, dbush!

The problems are poorly written . . .

Nature's Best wants to combine mixed nuts at $3.60/lb. with dried fruit at $2.40/lb to create a trail mix

a. Write an equation to represent the problem. . How?

"The problem" has not been stated yet.
No equation is possible.

b. Using your equation in (a), how much of each snack should they use to make
. . 10 pounds of trail mix that would sell for $3.30/lb?

Now we can write an equation . . .


Let \(\displaystyle x\) = number of pounds of mixed nuts.
. . At $3.60 per lb, their value is: .\(\displaystyle 3.60x\) dollars.

Then \(\displaystyle 10-x\) = number of pounds of dried fruit.
. . At $2.40 per lb, their value is: .\(\displaystyle 2.40(10-x)\) dollars.

Hence, the total value of the trail mix is: .\(\displaystyle 3.60x + 2.40(10-x)\) dollars. .[1]


But we know that the mixture is 10 lbs of trail mix worth $3.30 per lb.
. . Hence, the total value of the trail mix is: .\(\displaystyle 10 \times 3.30 \:=\:33\) dollars. .[2]


We just described the total value of the trail mix in two ways.

There is our equation! . \(\displaystyle \hdots\;\;3.60x + 2.40(10-x) \:=\:33\)


Solve for \(\displaystyle x:\;\;3.60x + 24 - 2.40x \:=\:33 \quad\Rightarrow\quad 1.20x \:=\:9\)

. . . . . . . . . \(\displaystyle x \:=\:\frac{9}{1.20} \:=\:7.5\)


Therefore: .\(\displaystyle \begin{Bmatrix}\text{7.5 lbs mixed nuts} \\ \text{2.5 lbs dried fruit\;} \end{Bmatrix}\)

Paula leaves home driving 40 mph.
One hour later, Dan leaves home, driving in the same direction at a speed of 50 mph.

a. Write an equation to represent the problem . Same difficulty

b. Using your equation in (a), how long will it take Dan to catch up with Paula?

Paula has a one-hour headstart.
. . At 40 mph, she is already 40 miles away.
During the next \(\displaystyle t\) hours, she drive another \(\displaystyle 40x\) miles.
. . Hence, she is: .\(\displaystyle 40 + 40t\) miles away. .[1]

During the same \(\displaystyle t\) hours, Dan drives at 50 mph.
. . He will have driven \(\displaystyle 50t\) miles. .[2]


Since Dan catches up to Paula, the two distances are equal.

There is our equation! .\(\displaystyle \hdots\;\;40 + 40t \:=\:50t\)


Solve for \(\displaystyle t:\;\;10t \,=\,40 \quad\Rightarrow\quad t \,=\,4\)


Therefore, it takes 4 hours for Dan to overtake Paula.

 

[dbush]

Well, maybe I am not as dumb as I thought. The fact that there is no problem to solve is exactly what I said to my son. The problems were taken directly from a previous math test on which he missed both of these problems. Your answers make complete sense when there is an actual problem to solve. Thanks for the help!