mixing $40/lb tea with $50/lb tea to get 20 lbs of $42/lb;

[kaymarie]

I need help with these.

How many pounds of tea, worth $40 a pound, should be mixed with herbal tea, worth $50 a pound, to produce 20 pounds of a blend worth $42 a pound?

To make pickles, cucumbers are soaked in a salt water solution called brine. How many liters of a 2% brine solution must be added to 30 liters of a 10% brine solution to dilute it to an 8% solution?


Thanks so much

 

[soroban]

Re: Mixture word problems

Hello, kaymarie!

How many pounds of tea, worth $40 a pound, should be mixed with herbal tea,
worth $50 a pound, to produce 20 pounds of a blend worth $42 a pound?

Let \(\displaystyle x\) = amount of regular tea.
Then \(\displaystyle 20-x\) = amount of herbal tea.


\(\displaystyle x\) lbs of tea at $40/lb has a value of: .\(\displaystyle 40x\) dollars.

\(\displaystyle 20-x\) lbs of herbal tea at $50/lb has a value of: .\(\displaystyle 50(20-x)\) dollars.

. . The mixture has a total value of: .\(\displaystyle 40x + 50(20-x)\) dollars. .[1]


But we know that the mixture is 20 lbs at $42/lb. .Its value is: .\(\displaystyle (20)(42) \:=\:840\) dollars. .[2]


We just described the value of the mixture in two ways.

There is our equation! . . \(\displaystyle 40x + 50(20-x) \:=\:840\)

To make pickles, cucumbers are soaked in a salt water solution called brine.
How many liters of a 2% brine solution must be added to 30 liters of a 10% brine solution
to dilute it to an 8% solution?

We will equate the amount of salt at each stage.


We have 30 liters which is 10% salt.
. . It contains: .\(\displaystyle 0.10 \times 30 \:=\:0.3\) liters of salt.

We add \(\displaystyle x\) liters which is 2% salt.
. . It contains: .\(\displaystyle 0.02 \times x \:=\:0.02x\) liters of salt.

The mixture contains a total of: .\(\displaystyle 0.3 + 0.02x\) liters of salt. .[1]


We know that the mixture will be \(\displaystyle 30 + x\) liters which is 8% salt.
. . It will contain: .\(\displaystyle 0.08(30 + x)\) liters of salt. .[2]


We just described the final amount of salt in two ways.

There is our equaton! . . \(\displaystyle 0.3 + 0.02x \;=\;0.08(30+x)\)