PDF chapter test TRY NOW
Multiplication of Fractions - Simplifying to Lowest Form
Multiply the following fractions and express the product in its lowest form:
\(\frac{3}{4} \times \frac{2}{9}\)
Instead of multiplying the numerators \(3\) and \(2\) and the denominators \(4\) and \(9\) first and then simplifying, we could do the following:
\(\frac{3}{4} \times \frac{2}{9}\)
Similarly, the denominator of the first fraction and the numerator of the second fraction have a common factor of \(2\).
We know that a fraction remains the same when the numerator and denominator are divided by the common factor.
In this case, we can divide them by their common factors.
\(= \frac{3}{2 \times 2} \times \frac{2}{3 \times 3}\)
\(=\frac{\not3}{\not2 \times 2} \times \frac{\not2}{\not3 \times 3}\)
\(=\frac{1}{2} \times \frac{1}{3}\)
Using the formula, we get,
\(\frac{1}{b} \times \frac{1}{d} = \frac{1}{b \times d}\)
\(=\frac{1}{2 \times 3}\)
\(=\frac{1}{6}\)
When multiplying fractions, we can first divide the numerator and denominator by their common factors before multiplying the numerators and denominators.
This is called cancelling the common factors.
Is the product always greater than the numbers multiplied?
We know that, when a number is multiplied by \(1\), the product remains unchanged.
Similarly, we have some situations as follows,
- When we multiply two numbers greater than \(1\), the product is greater than both numbers.
- Consider a number between \(0\) and \(1\) and one number is greater than \(1\), then their product is greater than the number between \(0\) and \(1\), and lesser than the number greater than \(1\).
- Consider two numbers between \(0\) and \(1\); their product is less than those two numbers.
| Situation | Example | Relationship |
| Two numbers greater than \(1\) |
\(6\) and \(4\)
\(6 \times 4 = 24\)
|
Here \(24 > 6\) and \(4\)
The product is greater than both the numbers being multiplied.
|
|
One number is between \(0\) and \(1\) and
another number is greater \(1\)
|
\(\frac{1}{3}\) and \(6\)
\(\frac{1}{3} \times 6 = \frac{6}{3} = 2\)
|
Here \(2>\frac{1}{3}\) and \(2<6\)
The product is greater than the number taken between \(0\) and \(1\), and
the product is less than the number greater than \(1\).
|
| Two numbers between \(0\) and \(1\) |
\(\frac{4}{5}\) and \(\frac{3}{7}\)
\(\frac{4}{5} \times \frac{3}{7}\)
\(= \frac{4 \times 3}{5 \times 7}\)
\(= \frac{12}{35}\)
|
Let us compare the taken numbers and the product
\(\frac{4}{5} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}\)
\(\frac{3}{7} = \frac{3 \times 5}{7 \times 5} = \frac{15}{35}\)
\(\frac{21}{35} > \frac{12}{35}\)
\(\frac{15}{35}> \frac{12}{35}\)
Thus, their product is less than those two numbers.
|
Order of Multiplication:
Example:
Example 1:
We know that
\(\frac{1}{2} \times \frac{1}{5} = \frac{1}{10}\)
Using unit square, we get,
Similarly,
\(\frac{1}{5} \times \frac{1}{2} = \frac{1}{10}\)
Using the unit square we get,
Example 2:
In general, consider a rectangle.
The area of a rectangle remains the same even if the length and breadth are interchanged.
From the above two examples, we conclude that the order of multiplication does not matter.
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
This can also be seen from Brahmagupta's formula for multiplying fractions.
The Pinch of History:
Long ago in India, reducing a fraction to its simplest form was known as apavartana. This method was so popular that it appeared not only in mathematics but also in other types of books. Around 150 CE, the Jain scholar Umasvati used this idea as an example in one of his philosophical works.
Register for free to see more content