Problem 40 - Prealgebra

Question

Cost of Gasoline If a gallon of gas costs \(353 \frac{9}{10} \mathrm{C},\) how much does \(\frac{1}{2}\) gallon cost?

Step-by-Step Solution

Answer

The cost of \(\frac{1}{2}\) gallon is 176.95 cents.

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(353 \frac{9}{10}\) into an improper fraction. The whole number is 353 and the fraction is \(\frac{9}{10}\). To convert, multiply 353 by 10 and then add 9: \((353 \times 10) + 9 = 3530 + 9 = 3539\). Therefore, \(353 \frac{9}{10} = \frac{3539}{10}\).

Step 2: Calculate the Cost of Half a Gallon

We need to find the cost of \(\frac{1}{2}\) gallon. To do this, multiply the improper fraction \(\frac{3539}{10}\) by \(\frac{1}{2}\):\[\frac{3539}{10} \times \frac{1}{2} = \frac{3539 \times 1}{10 \times 2} = \frac{3539}{20}.\]

Step 3: Convert the Result to a Decimal

Now, convert the fraction \(\frac{3539}{20}\) to a decimal by dividing 3539 by 20: \(3539 \div 20 = 176.95\). So, the cost of \(\frac{1}{2}\) gallon is 176.95 cents.

Key Concepts

Mixed Numbers to Improper Fractions

When you come across a mixed number like \(353 \frac{9}{10}\), it has two parts: the whole number \(353\) and the fractional part \(\frac{9}{10}\). To solve problems involving these kinds of numbers, it's easier to convert them into improper fractions.

• First, take the whole number \(353\) and multiply it by the denominator of the fractional part, which is \(10\).
• This gives us: \(353 \times 10 = 3530\).
• Next, add the numerator of the fraction, which is \(9\), to this product. So we have: \(3530 + 9 = 3539\).

This process transforms the mixed number \(353 \frac{9}{10}\) into the improper fraction \(\frac{3539}{10}\). This step simplifies the arithmetic needed for further calculations, like multiplication.

Converting Fractions to Decimals

Once you have a fraction, like \(\frac{3539}{20}\), converting it into a decimal is a straightforward process. This is done by performing division.

• Take the numerator \(3539\) and divide it by the denominator \(20\).
• The operation goes like this: \(3539 \div 20 = 176.95\).

The resulting decimal, \(176.95\), gives a precise number that can be easier to interpret, especially when dealing with money, as decimals align more directly with pricing.

Cost Calculation Using Fractions

When dealing with costs, fractions can be surprisingly handy. Suppose you know the cost per gallon of a product but need to find the price for a different amount, like half a gallon.

• You've already converted the mixed number price into an improper fraction that represents the cost per gallon: \(\frac{3539}{10}\).
• To find the cost of \(\frac{1}{2}\) gallon, multiply the cost per gallon by \(\frac{1}{2}\).
• This results in the expression: \(\frac{3539}{10} \times \frac{1}{2} = \frac{3539}{20}\), which simplifies to finding the value for half a gallon.

This calculation is practical, transforming larger, complex costs into smaller, manageable amounts using the elegance of fractional math.