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Dividend, Divisor and Quotient:
In any division problem, we have three parts:
- Dividend – the number being divided (\(15\))
- Divisor – the number we divide by (\(3\))
- Quotient – the result of the division (\(5\))
Key idea: Every division problem can be rewritten as a multiplication problem.
\(15 \div 3 = 5\) is the same as \(3 \times\) ? \(= 15\)
This idea of converting division into multiplication is the foundation for dividing fractions.
Reciprocal:
To divide by a fraction, we first need the idea of a reciprocal. The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). We obtain it by interchanging the numerator and denominator.
Property: When a fraction is multiplied by its reciprocal, the product is always \(1\).
Example: \(\frac{6}{7} \times \frac{7}{6} = 1\)
Brahmagupta's formula for division of fractions
To divide two fractions
Step 1:
Find the reciprocal of the divisor.
Step 2:
Multiply this by the dividend to get the quotient.
Thus, we get
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}\)
This method and formula for division of fractions, in this general form, was first explicitly stated by Brahmagupta in his Brahmasphutasiddhanta (628 CE).
Example:
\(\frac{2}{3} \div \frac{1}{5}\)
Solution:
From the above steps and the formula for division, we get,
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)
\(\frac{2}{3} \div \frac{1}{5}\)
\(=\frac{2}{3} \times \frac{5}{1}\)
\(=\frac{2 \times 5}{3 \times 1}\)
\(=\frac{10}{3}\)
Relationship between Dividend, Divisor and Quotient:
Situation \(1\): Dividing by a whole number
If dividing by a whole number greater than \(1\).
Example: \(4 \div 2 = 2\)
Here the dividend is \(4\) and the quotient is \(2\), here \(2 < 4\).
Dividing by a whole number greater than \(1\) gives a quotient smaller than the dividend because the dividend is being divided into more than one equal group.
Situation \(2\): Whole number divided by a fraction
If dividing by a fraction less than \(1\).
Example: \(5 \div \frac{1}{5} =\) \(5 \times \frac{5}{1} = 25\).
Here the dividend is \(5\) and the quotient is \(25\), here \(25 > 5\).
Because we are finding how many one-fifths are in \(5\). Since each part is very small, many such parts fit into \(5\).
Situation \(3\): Dividing by a fraction less than \(1\)
If dividing a fraction by a larger fraction.
Example: \(\frac{1}{12} \div \frac{1}{2} =\) \(\frac{1}{12} \times \frac{2}{1} = \frac{1}{6}\).
Here the dividend is \(\frac{1}{12}\) and the quotient is \(\frac{1}{6}\), here \(\frac{1}{6} > \frac{1}{12}\).
Because even though both are fractions, dividing by a fraction less than \(1\) increases the value.
Key points to remember:
| If the divisor is | And the quotient becomes |
| Greater than \(1\) | Smaller than the dividend |
| Equal to \(1\) | Same as dividend |
| Between \(0\) and \(1\) | Greater than the dividend |
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