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Ujarak has of a strong toothpaste with a concentration of sodium fluoride. They have a weaker toothpaste with a 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 a concentration of sodium fluoride?
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Q. Ujarak has of a strong toothpaste with a concentration of sodium fluoride. They have a weaker toothpaste with a 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 a concentration of sodium fluoride?
- Set up initial amount and concentration: Set up the initial amount and concentration of the strong toothpaste. of toothpaste with a concentration of sodium fluoride.First expression:
- Set up variable for weaker toothpaste: Set up the variable for the amount of weaker toothpaste to be added and its concentration.Let be the volume in milliliters of the sodium fluoride toothpaste to be added.Second expression: mL mL mL \times \(0.
- Set up expression for total volume and desired concentration: Set up the expression for the total volume and desired concentration of the final blend.The total volume will be , and the desired concentration is .Final expression: $(\(100\)\,\text{mL} + x\,\text{mL}) \times \(0\).\(8\)\% = (\(100\)\,\text{mL} + x\,\text{mL}) \times \left(\frac{\(0\).\(8\)}{\(100\)}\right) = (\(100\)\,\text{mL} + x\,\text{mL}) \times \(0\).\(008\)
- Create equation based on amounts and desired concentration: Create an equation based on the amount of sodium fluoride in the initial strong toothpaste, the amount to be added from the weaker toothpaste, and the desired final concentration.\(\newline\)Equation: \((100\,\text{mL} \times 0.011) + (x\,\text{mL} \times 0.003) = (100\,\text{mL} + x\,\text{mL}) \times 0.008\)
- Distribute concentrations and set up equation: Distribute the concentrations and set up the equation.\(\newline\)\(1.1\)mL + \(0.003x\) mL = \(0.8\)mL + \(0.008x\) mL
- Rearrange equation to isolate variable x: Rearrange the equation to isolate the variable x on one side.\(\newline\)\(1.1\,\text{mL} + 0.003x\,\text{mL} - 0.008x\,\text{mL} = 0.8\,\text{mL}\)\(\newline\)\(1.1\,\text{mL} - 0.005x\,\text{mL} = 0.8\,\text{mL}\)
- Solve for volume of weaker toothpaste: Solve for \(x\), which represents the volume of \(0.3\%\) sodium fluoride toothpaste needed.\(\newline\)\(1.1\)mL \(- 0.005x\) mL \(= 0.8\)mL\(\newline\)\(0.005x\) mL \(= 1.1\)mL \(- 0.8\)mL\(\newline\)\(0.005x\) mL \(= 0.3\)mL\(\newline\)\(0.3\%\)\(0\)\(\newline\)\(0.3\%\)\(1\)mL
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