Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(36 \times \frac{3}{4}=\) ______

Short answer

27

Step-by-step solution

Convert Whole Number to Fraction

First, convert the whole number 36 to a fraction by placing it over 1. This gives \( \frac{36}{1} \).

Multiply the Fractions

Now, multiply the fractions \( \frac{36}{1} \) and \( \frac{3}{4} \). Use the formula \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \). Thus, \( \frac{36}{1} \times \frac{3}{4} = \frac{36 \times 3}{1 \times 4} = \frac{108}{4} \).

Simplify the Result

To reduce \( \frac{108}{4} \) to its lowest terms, find the greatest common divisor (GCD) of 108 and 4, which is 4. Divide both the numerator and the denominator by 4: \( \frac{108 \div 4}{4 \div 4} = \frac{27}{1} \).

Convert Back to Whole Number

Since the fraction simplifies to \( \frac{27}{1} \), which is simply 27, convert it to a whole number. Thus, \( \frac{27}{1} = 27 \).

Key concepts

Simplifying Fractions

Simplifying or reducing fractions makes them easier to work with and more understandable. The goal is to express a fraction in its simplest form, where the numerator and denominator are as small as possible but still retain the same value. To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that divides both numbers evenly.
• Divide both the numerator and the denominator by their GCD. This transforms the fraction into its simplest form.

For example, consider the fraction \(\frac{108}{4}\). Here, the GCD is 4. When you divide both 108 and 4 by 4, you simplify \(\frac{108}{4}\) to \(\frac{27}{1}\), which is simply 27 as a whole number.
Simplifying is crucial when dealing with fractions because it ensures you are working with the smallest numbers possible, making calculations easier and solutions clearer.

Mixed Numbers

Mixed numbers are numbers that consist of a whole number and a fractional part. For instance, 3\(\frac{1}{2}\) is a mixed number with 3 as the whole number and \(\frac{1}{2}\) as the fractional part. When dealing with operations like multiplication, it's often necessary to convert mixed numbers into improper fractions.

• Multiply the whole number by the denominator of the fraction part.
• Add the result to the numerator. This sum becomes the new numerator.
• Place this new numerator over the original denominator.

So, for 3\(\frac{1}{2}\), you'd calculate: 3 times 2 plus 1 equals 7, thus making the improper fraction \(\frac{7}{2}\). This conversion simplifies the process of multiplying fractions by avoiding separate multiplications for whole and fractional parts.
Once your calculations are complete, you might convert back to a mixed number for a more understandable final answer.

Converting Fractions

Converting fractions involves changing one form of a fraction to another, depending on what is most useful for the calculation at hand. One common conversion is changing a whole number to a fraction.

• Any whole number can be written as a fraction by placing it over one.

For example, the number 36 can be written as \(\frac{36}{1}\). This is essential when multiplying fractions, as it allows you to uniformly apply multiplication rules.
Conversely, you might convert an improper fraction back to a whole number or a mixed number after performing operations. For instance, after multiplying to get \(\frac{27}{1}\), you can simply write it as 27, recognizing it as a whole number.
Overall, understanding how to convert between these forms ensures accuracy and ease in various mathematical processes.