Problem 41 - Prealgebra

Question

Distance Traveled If a car can travel \(32 \frac{3}{4}\) miles on a gallon of gas, how far will it travel on 5 gallons of gas?

Step-by-Step Solution

Answer

The car will travel \(163 \frac{3}{4}\) miles on 5 gallons of gas.

Step 1: Understand the Problem

We are given that the car can travel \(32 \frac{3}{4}\) miles on 1 gallon of gas. We need to find out how many miles the car will travel on 5 gallons of gas.

Step 2: Convert Mixed Number to Improper Fraction

Convert \(32 \frac{3}{4}\) to an improper fraction. First, multiply the whole number by the denominator: \(32 \times 4 = 128\)Then, add the numerator of the fractional part: \(128 + 3 = 131\)So, \(32 \frac{3}{4} = \frac{131}{4}\).

Step 3: Calculate the Distance for 5 Gallons

Multiply the improper fraction \(\frac{131}{4}\) by 5 to calculate the total distance for 5 gallons:\[ \frac{131}{4} \times 5 = \frac{131 \times 5}{4} = \frac{655}{4} \]

Step 4: Convert Improper Fraction to Mixed Number

Convert \(\frac{655}{4}\) back to a mixed number to express the result in a more understandable form. Divide the numerator by the denominator:\(655 \div 4 = 163\) remainder \(3\)So, \(\frac{655}{4} = 163 \frac{3}{4}\).

Step 5: State the Final Answer

The car can travel \(163 \frac{3}{4}\) miles on 5 gallons of gas.

Key Concepts

Mixed Numbers

Mixed numbers are numbers that include both a whole number and a fraction. For example, in the problem about the car, we look at a distance of \(32 \frac{3}{4}\) miles. This means you have a whole number 32 plus a fraction \(\frac{3}{4}\). Mixed numbers are great for representing values greater than one but not complete enough to be a whole number.
To work with mixed numbers, you often need to convert them into improper fractions. This makes the calculation process easier. Remember, you can always switch back to mixed numbers for your final answer to keep it reader-friendly.

Improper Fractions

Improper fractions are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). In our example, once we changed \(32 \frac{3}{4}\) to an improper fraction, it became \(\frac{131}{4}\).
To achieve this conversion:

• Multiply the whole number by the fraction’s denominator.
• Add that result to the fraction’s numerator.
• The sum becomes the numerator of the improper fraction, keeping the original denominator the same.

Using improper fractions, especially for multiplication or division, simplifies equations significantly.

Multiplication of Fractions

Multiplication of fractions is a straightforward process. In the exercise, we multiplied the improper fraction \(\frac{131}{4}\) by 5 to find the car's total travel distance. Here's how you do it:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together, in this case, 4 by 1 (since 5 is equivalent to \(\frac{5}{1}\)).

This gives us \(\frac{131 \times 5}{4} = \frac{655}{4}\).
Keeping fractions as they are until the solution's end preserves accuracy before conversion to a mixed number if necessary.

Distance Calculation

Calculating distance involves understanding how a given rate scales over multiple units. Here, we calculate how far a car travels on 5 gallons of gas if it can travel \(32 \frac{3}{4}\) miles per gallon.

• First, translate the mixed number to an improper fraction \(\frac{131}{4}\).
• Second, multiply this improper fraction by the number of gallons.

This multiplication provides the distance over the given gallons, ending up with \(\frac{655}{4}\), then converting this back to a more understandable mixed number gets us \(163 \frac{3}{4}\) miles.
By breaking down the steps, you gain clarity on how different fractions can provide real-world solutions.