Multiplying Fractions

A part of a whole is referred to as a fraction, and it has two components: the Numerator and the denominator. The numerator is a number located at the top of the fraction and shows the number of parts we have. The denominator is at the bottom and indicates the total number of equal parts. A fraction is represented in the form:

\(\frac{Numerator}{Denominator}\)

For instance, \(\frac 26\) is a fraction, where 2 and 6 are the numerator and the denominator, respectively.

Unlike adding or subtracting fractions, a common denominator is not required when multiplying fractions. Even if the fractions have different denominators, multiplication can still be performed. The main requirement is that the fractions should be either proper or improper, not in mixed form. To multiply fractions, follow these steps: 

Step 1: Multiply the numerators together.

Step 2: Multiply the denominators together.

Step 3: Simplify the fractions to their lowest form, if necessary. 
 

For example, let us try multiplying \(\frac14\) and \(\frac {2}{3}.\)

Let us multiply the numerators first.

\(1 \times 2 = 2\)

Now, let us multiply the denominators.

\(4 \times 3 = 12\)

The fraction becomes \(\frac{2}{12}.\)

Now, let us reduce the fraction to its lowest terms. 

The final answer would be \(\frac{1}{6}.\)

The difference between multiplication and division of fractions is as follows.

Simplifying fractions means reducing a fraction to its lowest terms. After finding the product of two fractions, we can simplify the result by writing it in its simplest form. This process is called simplification of fractions, as shown in the following example.

Simplify, \(\frac56 \times  \frac34.\)

\(\frac{(3\times 5)}{(4\times6)} = \frac{15}{24}\)

The simplest form of this fraction is \(\frac58.\)

Therefore, \(\frac{5}{8}\) is the simplified answer.

There are certain rules that we need to follow while multiplying fractions. Take a look at them down below:

Rule 1: If we have a mixed fraction, convert it into an improper fraction. Then multiply the numerators. 

Rule 2: Multiply the denominators together.

Rule 3: Simplify the obtained fraction to its lowest form. 

Rule 4: Another method to multiply fractions is to simplify the fractions before multiplying. It can be performed by canceling out common factors between the numerators and denominators. After that, multiply the numerators together, and then multiply the denominators together.  

Fractions that have the same denominator are also known as like fractions. The rules for addition and subtraction of like fractions differ from those of unlike fractions. However, in the case of multiplying like and unlike fractions, the rule remains the same. To multiply fractions, multiply the numerators together, and then the denominators. After that, the obtained fraction is simplified to its lowest terms.

For example, multiply \(\frac32 × \frac12:\)

Here, we will multiply the numerators.

\(3 × 1 = 3\)

Then, the denominators: \(2 × 2 = 4\)

The product we get is \(\frac34.\)

The answer cannot be simplified further, so the result is \(\frac34.\)

The method for multiplying fractions with different denominators is the same as multiplying fractions with same denominators. Multiply the numerators together, then the denominators, and simplify the fraction to its lowest term.

For instance, multiply \(\frac54 × \frac13\)

First, we multiply the numerators: \(5 × 1 = 5\)

Then, the denominators: \(4 × 3 = 12\)

The product we get is \(\frac{5}{12}.\)

This cannot be simplified further, therefore, \(\frac{5}{12}\) is the answer. 

There is another method that we can use to multiply fractions. In this method, we first simplify the fractions and then multiply the numerators before multiplying the denominators. We can understand this method with an example.

Let's multiply \(\frac{4}{12} × \frac{16}{24}\)

Here we will first simplify the fractions.

We can reduce \(\frac{4}{12}\) to \(\frac13\)

\(\frac{1}{3} = \left( \frac{4 \div 4}{12 \div 4} \right) \)

We can reduce \(\frac{16}{24}\) to \(\frac23\)

\(\frac{2}{3} = \left( \frac{16 \div 8}{24 \div 8} \right) \)

Now let us multiply the numerators: \(1 × 2 = 2\)

Then, the denominators: \(3 × 3 = 9\)

Hence, the product is \(\frac{2}{9}.\)

Multiplying fractions with whole numbers is a simple process. We must convert the whole number into a fraction by placing 1 as its denominator. Then multiply the numerators and denominators separately. After obtaining the product, simplify it if necessary. This is represented as: 

\(a \times \frac{b}{c} = \frac{a}{1} \times \frac{b}{c} = \frac{ab}{c} \)

For instance, multiply \(4 × \frac13\)

Step 1: Convert the whole number (4) into a fraction. Therefore, it becomes 4/1.

Step 2: Multiply the numerators. 

\(4 × 1 = 4\)

Step 3: Multiply the denominators. 

\(1 × 3 = 3 \)

Step 4: Write down the product as \(\frac43.\)

Step 5: Since \(\frac43\) is an improper fraction, we change it into mixed fraction. In this case, the numerator is greater. 

So, dividing 4 by 3, we get 1 as the quotient and 1 as the remainder. 

So, write the fraction as \(1 \frac{1}{3} \).

Multiplication of mixed numbers involves converting the mixed fractions into improper fractions and multiplying the numerators and denominators separately. Mixed fractions contain a whole number and a proper fraction. Before multiplication, we need to change it to improper fraction. Take a look at this example.

Multiply \(3 \frac{3}{4} \) by \(4 \frac{1}{5} \)

To convert mixed fractions into improper fractions, we can use the formula:

\(\text{Improper fraction} = \frac{(\text{Whole number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}} \)

For \(3 \frac{3}{4} \): 

Improper fraction = \(\frac{(3 \times 4) + 3}{4} = \frac{15}{4} \)

For \(4 \frac{1}{5} \):

\(\text{Improper fraction} =\frac{(4 \times 5) + 1}{5} = \frac{21}{5} \)

So, we get \(\frac{15}{4} \times \frac{21}{5} \)

Now we can multiply the numerators:

\(15 × 21 = 315\)

Multiply the denominators: \(4 × 5 = 20\)

The obtained fraction is: 

\(\frac{315}{20}\)

To simplify the fraction, we need to find the greatest common factor (GCF) of 315 and 20. 

To find this, first, we need to identify the prime factorization of both numbers.  

\(315 = 3^2 × 5 × 7\)

\(20 = 2^2 × 5\)

5 is the only common factor of 315 and 20. 

Hence, 5 is the GCF. 

Now we can divide 315 and 20 by its GCF.

\(\frac{315 \div 5}{20 \div 5} = \frac{63}{4} \)

\(\frac{63}{4}\) is an improper fraction, we must convert it into a mixed number. 

\(63 ÷ 4\) gives 15 as the quotient and 3 as the remainder. 

So, 

\(\frac{63}{4} = 15 \frac{3}{4} \)

The product is \(15 \frac{3}{4} \).

We refer to a fraction as improper when its numerator is greater than its denominator. Multiplying two improper fractions often gives an improper fraction as a result.

For example, multiply \(\frac{5}{2}\) by \(\frac97\)

Now, we can multiply the numerators: \(5 × 9 = 45\)

Multiply the denominators: \(2 × 7 = 14\)

The fraction \(\frac{45}{14}\) cannot be simplified further. 

Hence, the product is \(\frac{45}{14}\) which can be written as \(3 \frac{3}{14} \)

Let us represent the product of \(\frac25\) and \(\frac34\) using models. 

We know that finding \(\frac25 \times \frac34\) is equivalent to finding \(\frac25\) of \(\frac34.\)

Therefore, we will show three-fourths and then chase the two-fifths of it. 

Step 1: Divide a rectangle into four equal parts, and shade three of them to represent \(\frac34.\)

Step 2: Now, divide the five sections in each box.

Step 3: Shade two parts of the five sections in a different color. Now, we can represent \(\frac25\) in this. 

Step 4: Let us now identify the represented fraction.

We can notice that 6 out of 20 parts are shaded in a different color.

As always, there are certain tips and tricks that can be used to our advantage while solving problems involving the multiplication of fractions. Some of them are mentioned below:

• Simplify fractions before multiplying them. Instead of multiplying first and then simplifying, check if fractions can be simplified beforehand. For example, when we multiply 154 × 215 before simplifying, it will become complex to solve. 
   
• If it is possible, we can simplify two fractions. If the numerator of one fraction and the denominator of another share a common factor, we can simplify before multiplying. 
   
• Teachers should start teaching with the idea of a part. Teach them that decimals are all about parts. Before any rules, we must build the conceptual understanding. Use pizza, chocolate bars, or paper strips to explain the parts concept.
   
• Parents can use area models as visuals. Draw a rectangle and split it into parts. We can use different colors or different directions of shade to differentiate the fractional parts. The overlapping area is the product of the fractions. 
   
• Teachers can teach the learners to cancel or simplify before multiplying the fractions. Name it “cross-simplify trick” so they remember it. This would make simplifying and multiplying fractions easier. 
   
• Teachers should teach the children how to convert mixed numbers into improper fractions. Many kids multiply whole numbers and fractions incorrectly. Clarify their confusion and then teach multiplication of mixed numbers. 
   
• Parents can help them identify fractions in everyday life and ask them to perform multiplication on them. Suppose they eat \(\frac34\) of a \(\frac12\) sized chocolate bar; let them determine the fraction of the chocolate they ate. 
   
• Learners can start practicing with easy, smaller fractions first, then move on to mixed, improper, and negative fractions.