Problem 47 - Prealgebra

Question

FOOD How many \(\frac{1}{4}\) -pound hamburgers can be made from \(2 \frac{3}{4}\) pounds of ground beef?

Step-by-Step Solution

Answer

11 hamburgers can be made.

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(2 \frac{3}{4}\) into an improper fraction. The whole number part is 2, and the fractional part is \(\frac{3}{4}\). To convert it: \[ 2 \frac{3}{4} = \frac{4 \times 2 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}. \] This means we have \(\frac{11}{4}\) pounds of ground beef.

Step 2: Set Up the Division

To find out how many \(\frac{1}{4}\) -pound hamburgers can be made, divide the total amount of ground beef \(\frac{11}{4}\) by the size of each hamburger \(\frac{1}{4}\). This is expressed as: \[ \frac{11}{4} \div \frac{1}{4}. \]

Step 3: Perform the Division Using Multiplication

Dividing by a fraction is the same as multiplying by its reciprocal. So, \(\frac{11}{4} \div \frac{1}{4}\) becomes \(\frac{11}{4} \times \frac{4}{1}\). Multiply the numerators and the denominators: \[ \frac{11 \times 4}{4 \times 1} = \frac{44}{4}. \]

Step 4: Simplify the Fraction

Simplify \(\frac{44}{4}\) by dividing both the numerator and the denominator by 4: \[ \frac{44 \div 4}{4 \div 4} = \frac{11}{1} = 11. \] Thus, 11 hamburgers can be made.

Key Concepts

Mixed Numbers

Mixed numbers are an important aspect of fractions. They combine a whole number with a fraction, making them useful in many practical scenarios. For example, if you have 2 whole apples and another fraction of an apple, you can express this as a mixed number. In the context of the exercise, the ground beef's weight is a mixed number, specifically, \(2 \frac{3}{4}\).

To convert a mixed number into an improper fraction, you must:

• Multiply the whole number by the denominator of the fraction.
• Add the numerator of the fraction to this product.
• Place this sum over the original denominator.

For instance, if you have \(2 \frac{3}{4}\):

• Multiply 2 (the whole number) by 4 (the denominator): \(2 \times 4 = 8\).
• Add the numerator (3): \(8 + 3 = 11\).
• Place over original denominator: \(\frac{11}{4}\).

Now, you understand how to handle weights or values given as mixed numbers in any mathematical problem.

Improper Fractions

Improper fractions occur when the numerator is larger than the denominator. Unlike proper fractions, they represent a value greater than one. In many calculations, it's preferable to use improper fractions as they are easier to manipulate mathematically.

Once a mixed number is converted into an improper fraction, as shown previously, you can work more comfortably with them in operations like multiplication and division. In our exercise, after conversion, we got \(\frac{11}{4}\), which is an improper fraction. This allows us to proceed with division or any necessary operations without the component parts of whole numbers.

Always remember that while improper fractions might look unfamiliar, they are just a different way to represent how much of a whole you have – often, more than one whole! So, on encountering improper fractions, know they make calculations simpler.

Reciprocal

Understanding reciprocals is crucial for dividing fractions. A reciprocal essentially "flips" a fraction upside down. When dividing fractions, you replace the division with multiplication and use the reciprocal of the fraction you're dividing by.

For example, in the exercise: Simple division \(\frac{11}{4} \div \frac{1}{4}\) switches to multiplication by the reciprocal of \(\frac{1}{4}\), which is \(\frac{4}{1}\). Thus, \(\frac{11}{4} \div \frac{1}{4}\) becomes \(\frac{11}{4} \times \frac{4}{1}\).

Using reciprocals simplifies operations because multiplying is more straightforward than dividing. Whenever you encounter division involving fractions, consider flipping to the reciprocal and switching to multiplication – your calculations will become much easier!

Simplifying Fractions

Simplification is a key step in fraction operations, helping you arrive at the cleanest and most minimal expression of your answer. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

In our case, simplifying \(\frac{44}{4}\) is done by:

• Identifying the GCD of 44 and 4, which is 4.
• Dividing both the numerator and the denominator by 4: \(\frac{44 \div 4}{4 \div 4} = \frac{11}{1} = 11\).

This confirms that 11 is both the simplified and the final answer.

By simplifying fractions, you ensure that your answer is not only accurate but also easy to read and interpret. It’s about making your final result as neat and understandable as possible. So, always simplify when you’re dealing with fractions!