UPSKILL MATH PLUS
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Learn moreMultiplication in real - life context:
\(\text{Multiplication}\) is an arithmetic operation that is known as a \(\text{repeated addition}\).
Example:
It takes \(30\) minutes for \(\text{Meena}\) to walk \(1 km\). How many minutes it will take for \(\text{Meena}\) to walk for \(3\) kms?
Solution:
It is given that
\(\text{Meena}\) walk \(1 \text{km}\) within \(30\) \(\text{minutes}\).
\(\implies\) \(\text{Meena}\) walk \(3\) \(\text{km}\) within
\(= 3 \times 30\) \(\text{minutes}\)
\(=30+30+30\)
\(=90\) \(\text{minutes}\)
Answer:
It takes \(90\) \(\text{minutes}\) for \(\text{Meena}\) to walk \(3\) \(\text{km}\)
From the above example, it is clear that \(\text{multiplication}\) is an arithmetic operation that is known as (\text{repeated addition}\).
Multiplying a whole number by a fraction:
Multiplying a \(\text{whole number}\) by a \(\text{fraction}\) is nothing but a \(\text{repeated addition}\) of that particular fraction.
Example:
Charu take her pet dog to walk \(\frac{1}{2}\) \(\text{km}\) in \(1\) \(\text{hour}\). How far can it walk in \(5\) \(\text{hours}\)?
Solution:
It is given that Charu takes her pet dog for a walk \(\frac{1}{2}\) \(\text{km}\) in \(1\) \(\text{hour}\)
Here, the distance covered in an hour is a fraction.
The total distance covered is the product.
Distance covered in \(1\) \(\text{hour}\) \(=\) \(\frac{1}{2}\) \(\text{km}\)
Therefore, the distance covered in \(5\) \(\text{hours}\)
\(= 5 \times \frac{1}{2}\)
\(=\) \(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\)
\(=\) \(\frac{5}{2}\)
The dog can walk \(\frac{5}{2}\) \(\text{km}\) in \(5\) \(\text{hours}\).
Multiplying two fractions:
To multiply two fractions, we have two methods,
(i) Unit square method
(ii) Formula method
Unit square method
Steps to find the product of fractions using the unit square.
Step 1:
Consider the \(\text{first}\) fraction,
Divide the unit square into many vertical strips as indicated by the denominator of the given fraction.
Step 2:
Consider the \(\text{second}\) fraction,
Divide the unit square into many horizontal strips as indicated in the denominator of the given fraction.
Step 3:
To multiply both fractions,
We need to \(\text{overlap}\) both of the above unit squares together in step 1 and step 2.
Then the solution would be,
The product of the given fractions is found by the number of overlapping rectangles divided by the total number of rectangles.
Example:
Solve \(\frac{1}{3} \times \frac{1}{5}\), using a unit square as a whole for the representing fraction.
Solution:
Fraction \(\frac{1}{3}\) represented by the unit square is,
Fraction \(\frac{1}{5}\) represented by the unit square is,
Thus, \(\frac{1}{3} \times \frac{1}{5}\) represented by the unit square is,
The product of the given fractions is found by the number of overlapping rectangles divided by the total number of rectangles.
Hence, \(\frac{1}{3} \times \frac{1}{5} = \frac{1}{15}\)
Formula method:
When two fractional units are multiplied, their product is \(\frac{1}{\text{product of denominators}}\). (This formula applies only to fractions where the numerator is \(1\).)
\(\frac{1}{b} \times \frac{1}{d} = \frac{1}{b \times d}\)
Example:
Solve \(\frac{1}{3} \times \frac{1}{5}\)
Solution:
To solve the above fraction \(\frac{1}{3} \times \frac{1}{5}\) we can use the formula,
\(\frac{1}{b} \times \frac{1}{d} = \frac{1}{b \times d}\)
Thus, we get,
\(\frac{1}{3} \times \frac{1}{5}\)
\(=\frac{1}{3 \times 5}\)
\(= \frac{1}{15}\)
Important!
In both the unit square method and the formula method, we get the same answer.
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