Problem 53 - Prealgebra

Question

Price of Beef If each pound of beef costs \(\$ 4.99,\) how much does \(3 \frac{1}{4}\) pounds cost?

Step-by-Step Solution

Answer

The cost for \(3 \frac{1}{4}\) pounds of beef is \(\$16.22\).

Step 1: Convert Mixed Number to Improper Fraction

The mixed number is given as \(3 \frac{1}{4}\). To convert it to an improper fraction, first multiply the whole number \(3\) by the denominator \(4\), which equals \(12\). Then add the numerator, \(1\), to get \(13\). So \(3 \frac{1}{4} = \frac{13}{4}\).

Step 2: Calculate Cost per Pound

The cost per pound of beef is given as \(\\(4.99\). This means for every pound, you pay \(\\)4.99\).

Step 3: Multiply to Find Total Cost

Multiply the cost per pound by the total pounds of beef. This is done by multiplying \(4.99\) by \(\frac{13}{4}\). First calculate \(4.99 \times 13 = 64.87\). Then divide by \(4\) to find the cost for \(\frac{13}{4}\) pounds: \(\frac{64.87}{4} = 16.2175\).

Step 4: Round to Two Decimal Places

The cost must be rounded to the nearest cent, which is two decimal places. Therefore, \(16.2175\) rounds to \(16.22\).

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. For example, in the exercise above, the mixed number is given as \(3 \frac{1}{4}\). Here, \(3\) is the whole number, and \(\frac{1}{4}\) is the fractional part.

To easily work with mixed numbers in calculations, they should often be converted to improper fractions. This process involves the following steps:

• Multiply the whole number by the denominator of the fraction.
• Add the product to the numerator.
• This sum is the new numerator, while the denominator stays the same.

In the example, \(3 \times 4 + 1 = 13\), forming the improper fraction \(\frac{13}{4}\). This conversion simplifies operations such as multiplication and division.

Rounding Decimals Made Simple

Rounding decimals is a skill that simplifies numerical expressions to a desired level of accuracy. The exercise illustrates rounding to two decimal places, which is crucial when dealing with money.

Here's how to round a decimal like the final cost \(16.2175\) to two decimal places:

• Identify the digit in the second decimal place (the hundredths place), which is \(1\) in this case.
• Look at the digit immediately after it, the thousandths place, which is \(7\).
• If this digit is 5 or higher, round up the hundredths place by one; otherwise, leave it unchanged.

Thus, since \(7\) is greater than \(5\), the number rounds up to \(16.22\). Rounding ensures financial transactions are accurately represented in a simplified form.

Improper Fraction Essentials

An improper fraction has a numerator larger than or equal to its denominator, like \(\frac{13}{4}\) in the exercise. These are essential for simplifying complex arithmetic operations.

Steps to convert a mixed number to an improper fraction have already been outlined, but understanding their use in calculations is also important. Here's why improper fractions are useful:

• They simplify multiplication and division, avoiding the added complexity of mixed numbers.
• They provide a straightforward approach to compare and order fractions.
• In arithmetic operations like addition or subtraction, they streamline combination with other fractions.

Using improper fractions can convert multi-step problems into simpler ones, crucial in subjects like algebra and beyond.