Question

A new mixture of​ self-tanning & moisturizing lotions for everyday use is being developed. This mixture is made by mixing 700 ounces of everyday moisturizing lotion for ​$0.70 per ounce with​ self-tanning lotion worth ​$3 per ounce. If the new​ self-tanning & moisturizing lotion mixture is to cost $ 1.60 per​ ounce, how many ounces of the​ self-tanning lotion should be in the​ mixture?

Answer 1

450 ounces

Explanation:

We know that the cost of the mixture
`C_m` must be the cost of the everyday moisturizing lotion
`C_e` plus the cost of the self-tanning lotion
`C_s`, which means
`C_m=C_e+C_s`.

The cost of any substance will be the cost per ounce of the substance (c), multiplied by the number of ounces (n), which means C=nc, so we have from the previous formula that we have:

`n_m c_m=n_e c_e + n_s c_s`

But we also know that the number of ounces of the mixture must be the sum of the number of ounces of the everyday moisturizing lotion with the number of ounces of the self-tanning lotion, so we have

`n_e c_e + n_s c_s=n_m c_m=(n_e+n_s) c_m=n_e c_m+n_s c_m`

We want to calculate the number of ounces of the​ self-tanning lotion (
`n_s`), so we solve for that variable:

`n_e c_e + n_s c_s=n_e c_m+n_s c_m`

`n_s c_s-n_s c_m=n_e c_m-n_e c_e`

`n_s (c_s-c_m)=n_e(c_m-c_e)`

`n_s=(n_e (c_m-c_e))/((c_s-c_m))`

And substitute our values in this formula, to get:

`n_s=((700\ ounces) (\$1.6-\$0.7))/((\$3-\$1.6))=450\ ounces`