Problem 10 - Prealgebra

Question

Write your answers as proper fractions or mixed numbers, not as improper fractions. Find the following products. (Multiply.) $$10 \cdot 1 \frac{1}{4}$$

Step-by-Step Solution

Answer

The product is \(12 \frac{1}{2}\).

Step 1: Convert the Mixed Number to an Improper Fraction

First, convert the mixed number to an improper fraction. The mixed number is \(1 \frac{1}{4}\). To convert this, multiply the whole number (1) by the denominator (4) and add the numerator (1). This gives: \(1 \times 4 + 1 = 5\). So, \(1 \frac{1}{4}\) can be rewritten as \(\frac{5}{4}\).

Step 2: Write the Multiplication Expression

Now write the multiplication using the improper fraction from Step 1. The expression is \(10 \cdot \frac{5}{4}\).

Step 3: Multiply the Numbers

To multiply \(10\) by \(\frac{5}{4}\), consider \(10\) as a fraction too: \(\frac{10}{1}\). Now multiply the numerators and the denominators: \[\frac{10 \times 5}{1 \times 4} = \frac{50}{4}\]

Step 4: Simplify the Improper Fraction

Simplify \(\frac{50}{4}\) by dividing both the numerator and the denominator by their greatest common divisor. The GCD of 50 and 4 is 2, so:\[\frac{50 \div 2}{4 \div 2} = \frac{25}{2}\]

Step 5: Convert the Improper Fraction to a Mixed Number

Convert \(\frac{25}{2}\) into a mixed number by dividing 25 by 2. The quotient is the whole number part, and the remainder is the numerator of the fraction part. - 25 divided by 2 is 12 with a remainder of 1. So, \(\frac{25}{2}\) equals \(12 \frac{1}{2}\).

Key Concepts

Mixed Numbers

A mixed number is simply a combination of a whole number and a fraction together. For example, in the number \(1 \frac{1}{4}\), the "1" is the whole number and "\(\frac{1}{4}\)" is the fraction part.
Mixed numbers are useful because they can provide a clearer picture of the size of a number, especially when dealing with quantities greater than one that have a fractional part.
When working with mixed numbers, it's often easier to convert them to improper fractions for mathematical operations like multiplication or division. Let’s explore why.

Improper Fractions

An improper fraction has a numerator that is greater than or equal to the denominator, like \(\frac{5}{4}\).
This is different from a proper fraction where the numerator is smaller than the denominator, such as \(\frac{1}{4}\).
An improper fraction is particularly convenient when doing arithmetic operations because it simplifies the process of multiplication and division. For instance, converting mixed numbers to improper fractions allows us to easily find products using simple multiplication rules.

Simplification Steps

After multiplying fractions, we often end up with an improper fraction that needs to be simplified. Simplifying makes the fraction easier to use and understand.
Simplification involves finding the greatest common divisor (GCD) of the numerator and the denominator, and then dividing both by this number.

• For example, in \(\frac{50}{4}\), the GCD is 2.
• By dividing both the numerator and the denominator by 2, we simplify the fraction to \(\frac{25}{2}\).

This step ensures that the resulting fraction is in its simplest form, making it more comprehensible and ready for any subsequent conversion back to a mixed number or other operations.

Conversion Processes

Conversion is a key part of working with fractions and mixed numbers. It involves changing one form into another depending on the context or requirement of the problem.
After calculating the product as an improper fraction, sometimes we'll need to convert it back into a mixed number for the final answer. Here’s how you do it:

• Take \(\frac{25}{2}\) as an example.
• Divide 25 by 2. The quotient, 12, becomes the whole number.
• The remainder, 1, forms the fraction part \(\frac{1}{2}\).

Thus, \(\frac{25}{2}\) is expressed as the mixed number \(12 \frac{1}{2}\).
This conversion lets us present our final result in a form that combines both whole numbers and fractions, often preferred for their clarity in real-world contexts.