Problem 40 - Chemistry The Science in Context

Question

How many gallons are there in \(108 \mathrm{mL} ?\)

Step-by-Step Solution

Answer

Answer: There are approximately 0.0285 gallons in 108 milliliters.

Step 1: Write down the given volume and conversion factor

The given volume is 108 mL and the conversion factor is 1 gallon = 3785.41 mL.

Step 2: Set up a proportion to convert mL to gallons

We can set up a proportion where we have gallons on one side, and milliliters on the other side, as follows: \(\dfrac{Gallons}{108\:mL} = \dfrac{1\:Gallon}{3785.41\:mL}\)

Step 3: Solve the equation for gallons

To find the value of Gallons, we will cross-multiply: \(Gallons\:*\:3785.41\:mL = 108\:mL\:*\:1\:Gallon\) Now, divide both sides of the equation by 3785.41 mL: \(Gallons = \dfrac{108\:mL\:*\:1\:Gallon}{3785.41\:mL}\)

Step 4: Calculate the final result

Now, we'll do the math to calculate the result: \(Gallons = \dfrac{108}{3785.41} = 0.0285\) So, there are approximately 0.0285 gallons in 108 mL.

Key Concepts

Milliliters to Gallons

Understanding how to convert milliliters to gallons can be incredibly useful, especially when you want to relate smaller liquid measurements to larger volumes. Milliliters (mL) and gallons are units of volume commonly used in different measurement systems. Milliliters are part of the metric system, which is used widely around the world, whereas gallons are primarily used in countries like the United States.
When converting from one unit to another, it is crucial to know the relationship between those units, which is established using a conversion factor. For this specific conversion, we use the fact that 1 gallon is equivalent to 3785.41 milliliters. By using this conversion factor, you'll be able to seamlessly change milliliters to gallons or vice versa.

Conversion Factor

A conversion factor is a numerical value used to change one set of units to another, maintaining the same quantity or value. It is like a bridge between two different measurement systems. When converting from milliliters to gallons, the conversion factor used is 1 gallon equals 3785.41 milliliters.

• The conversion factor has two parts: the unit you are converting from and the unit you are converting to.
• To use the conversion factor correctly, set it up as a fraction or a ratio, which can look like either \( \frac{1\, \text{gallon}}{3785.41\, \text{mL}} \) or \( \frac{3785.41\, \text{mL}}{1\, \text{gallon}} \), depending on the direction of conversion.
• Always ensure that the units cancel appropriately to leave you with the desired unit.

By choosing the right form of the conversion factor,
you can transition from milliliters to gallons without changing the actual amount of liquid.

Mathematical Calculation

Carrying out a mathematical calculation for the conversion of milliliters to gallons involves a series of steps where you apply the conversion factor. Starting with your given measurement in milliliters, you multiply or divide by the factor in the order that causes units to cancel correctly and leaves you with the desired unit — gallons.
Here's what you'll typically do:

• Begin with your known quantity in milliliters.
• Write the conversion set-up as a multiplication that includes the conversion factor, ensuring that milliliters will cancel out.
• Perform the arithmetic operation (either multiplication or division, as appropriate).

This is the straightforward numerical part of the conversion process.
Though it may seem like a simple plug-and-chug, carrying it out accurately is vital for a precise result.

Cross-Multiplication

Cross-multiplication is a useful algebraic tool for solving equations, especially those involving proportions or fractions. In converting milliliters to gallons, cross-multiplication can help solve the proportion we set up between gallons and milliliters.
In the given problem, we set the proportion \( \frac{\text{Gallons}}{108\, \text{mL}} = \frac{1\, \text{Gallon}}{3785.41\, \text{mL}} \).

• To solve for the unknown (Gallons in this case), we cross-multiply the terms. This means you multiply the numerator of one fraction by the denominator of the other, and vice versa.
• This results in the equation: \( \text{Gallons} \times 3785.41 = 108 \times 1 \).
• By dividing both sides of this equation by the known value (3785.41), you isolate the unknown, giving you the final answer in gallons.

Cross-multiplication simplifies finding your result by making it easy to rearrange and solve the equation for the variable in question.