Question

Ujarak has 100 mL of a strong toothpaste with a 1.1% concentration of sodium fluoride. They have a weaker toothpaste with a 0.3% concentration of sodium fluoride. What volume, in milliliters, of the weaker toothpaste would Ujarak need to add to the strong toothpaste to create a blend with 0.8% components of sodium fluoride?

Answer 1

To create a blend of toothpaste with a 0.8% sodium fluoride concentration, Ujarak needs to add 60 mL of the weaker toothpaste with a 0.3% concentration to the 100 mL of the strong toothpaste with a 1.1% concentration.

Step-by-step explanation:

Ujarak wants to create a blend of toothpaste with a 0.8% concentration of sodium fluoride. To determine the volume of the weaker toothpaste that needs to be added to the strong toothpaste to achieve this concentration, we can use the concept of the concentration of mixtures.

We start by setting up an equation based on the initial concentrations and the desired concentration. Let V represent the volume of the weaker toothpaste to be added.

The amount of fluoride in the strong toothpaste is 100 mL × 1.1%, and in the weaker toothpaste, it is V mL × 0.3%. We want the total amount of fluoride after mixing to be equal to (100 mL + V mL) × 0.8%. So, our equation will be:

100 × 1.1% + V × 0.3% = (100 + V) × 0.8%

Solving this equation will give us the value of V.

100 × 1.1 + V × 0.3 = (100 + V) × 0.8

1.1 + 0.003V = 0.8 + 0.008V

1.1 - 0.8 = 0.008V - 0.003V

0.3 = 0.005V

V = 0.3 / 0.005

V = 60 mL

Therefore, Ujarak needs to add 60 mL of the weaker toothpaste to the strong toothpaste to create a blend with a 0.8% concentration of sodium fluoride.

Answer 2

In order to find the volume of weaker toothpaste that Ujarak would need to add to the strong toothpaste to create a blend with a 0.8% concentration of sodium fluoride, we can use the following formula:

(C1V1 + C2V2) / (V1 + V2) = C3

Where:
C1 = concentration of strong toothpaste (1.1%)
V1 = volume of strong toothpaste (100 mL)
C2 = concentration of weaker toothpaste (0.3%)
V2 = volume of weaker toothpaste (unknown)
C3 = desired concentration of blend (0.8%)

First, we can convert all concentrations from percentages to decimal form:
C1 = 1.1 / 100 = 0.011
C2 = 0.3 / 100 = 0.003
C3 = 0.8 / 100 = 0.008

Then we can substitute these values into the formula and solve for V2:

(0.011 * 100 + 0.003 * V2) / (100 + V2) = 0.008

0.011V2 + 0.11 = 0.008V2 + 0.08

0.03V2 = -0.03

V2 = -0.03 / 0.03 = -1

The solution is negative, that means the resulting concentration is not possible with these inputs, that is the concentration of fluoride in the mix will be higher than 0.8%.
You could try with different volumes until you reach the desired concentration.