Multiplication of Fractions

Welcome to the exciting world of fractions! Today, we'll go on a journey to understand how to multiply fractions. Multiplying fractions can often seem tricky at first, but with simple steps and practice, it can be very easy. Let's start exploring this topic with some simple definitions and examples!

Understanding fractions

Before we move on to multiplication, let's quickly review what fractions are. A fraction represents a part of a whole. It consists of two numbers - a numerator and a denominator. The number on top is called the numerator, and it tells us how many parts we have. The number on the bottom is the denominator, which tells us how many parts there are in total.

For example, in the fraction 3/4, the numerator is 3, and the denominator is 4. This means that we have 3 equal parts out of 4 of the whole.

Basics of multiplying fractions

The process of multiplying fractions is straightforward. To multiply fractions, you multiply the numerators together and then multiply the denominators together. That's it! Here's the basic formula:

(a/b) × (c/d) = (a×c) / (b×d)

Here is a step-by-step explanation to make it more clear:

Step 1: Multiply the numerators (the numbers on top).
Step 2: Multiply the denominators (the bottom numbers).
Step 3: Simplify the fraction, if possible.

Example

Let's look at an example:

Multiply 2/3 by 3/5.

(2/3) × (3/5) = (2×3) / (3×5) = 6 / 15

And this gives us the fraction 6/15. Now, let's simplify:

The greatest common factor of 6 and 15 is 3. If we divide both the top and bottom by 3, we get:

6 ÷ 3 / 15 ÷ 3 = 2 / 5

So after simplification 2/3 × 3/5 = 2/5.

Visual example

Sometimes, looking at a picture can help to understand what is happening when multiplying fractions. Let's illustrate our previous example using a visual aid.

In this visual representation, the first rectangle divided into three parts represents the fraction 2/3. The second rectangle divided into five parts represents the fraction 3/5. The purple overlapping area represents the intersection, which shows how the parts are multiplied.

Additional examples

Let's go ahead and solve more examples so that we can become masters of multiplication of fractions.

Multiply 1/4 by 2/3.

(1/4) × (2/3) = (1×2) / (4×3) = 2 / 12

Simplify the result:

2 ÷ 2 / 12 ÷ 2 = 1 / 6

Thus, 1/4 × 2/3 = 1/6.

Multiply 5/8 by 3/7.

(5/8) × (3/7) = (5×3) / (8×7) = 15 / 56

Fortunately, the fraction 15/56 is already simplified. Therefore, the solution is:

5/8 × 3/7 = 15/56

Working with mixed numbers

Sometimes, you may encounter mixed numbers during fraction multiplication. Mixed numbers have a whole part and a fractional part, such as 1 2/3. To multiply mixed numbers, you must first turn them into improper fractions.

Steps to convert mixed numbers to improper fractions

Here's how to convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fraction.
• Add this result to the numerator of the fractional part.
• The total becomes the new numerator with the original denominator.

Convert 2 1/4 to an improper fraction.

(2 × 4) + 1 = 8 + 1 = 9

So, 2 1/4 becomes 9/4.

Convert 3 2/5 to an improper fraction.

(3 × 5) + 2 = 15 + 2 = 17

So, 3 2/5 becomes 17/5.

Multiplication of mixed numbers example

Multiply 2 1/4 by 3 1/3.

Step 1: Convert the mixed numbers

Convert each mixed number to an improper fraction:

2 1/4 = 9/4
3 1/3 = 10/3

Step 2: Multiply the improper fractions

(9/4) × (10/3) = (9 × 10) / (4 × 3) = 90 / 12

Step 3: Simplify the fraction

Simplify 90/12:

90 ÷ 6 / 12 ÷ 6 = 15 / 2

This is an improper fraction, so we convert it to a mixed number:

15 ÷ 2 = 7 with a remainder of 1

So, the answer is 7 1/2.

Guidelines for simplifying fractions

Simplifying fractions is an essential skill in working with fractions. Always follow these guidelines:

1. Find the Greatest Common Factor (GCF) of the numerator and denominator.
2. Divide both the numerator and denominator by their GCF.
3. The resulting fraction is your simplified fraction.

Simplify 16/20.

The GCF of 16 and 20 is 4.
16 ÷ 4 / 20 ÷ 4 = 4 / 5

Therefore, the simplified fraction is 4/5.

Why multiply fractions?

Understanding multiplication of fractions can be very useful. Here are some scenarios when you might need to multiply fractions:

• When cooking, modify recipes to change serving sizes.
• Determining the share of a part in real-world problems, such as discounting or probability.
• Find the area of rectangles where the sides are different.

For example, if a recipe calls for 3/4 cup sugar and you are making 1/2, you would multiply:

(3/4) × (1/2) = 3/8

So you would use 3/8 cup sugar.

Practice problems

To fully understand multiplication of fractions, try solving these practice problems. Remember to simplify!

1. 2/5 × 3/4
2. 7/8 × 2/3
3. Multiply 3 1/5 by 1 2/7
4. Simplify the product of 4/9 × 3/2
5. Calculate 5/6 of 2/5

Conclusion

Multiplying fractions isn't too difficult. By following a few simple steps - multiplying the numerators, multiplying the denominators, and simplifying the result - you can effectively solve any fraction multiplication problem. Practice regularly and use visual aids to deepen your understanding. Have fun learning!

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