Question

The product of two numbers is $4\dfrac{4}{9}$. If one of the numbers is $6\dfrac{2}{9}$, find the other.

Answer 1

Hint: If the product of two numbers and a number is given, then the second number can be calculated by dividing the product by the first number. In order to make the calculation simpler convert the mixed form of the numbers into a normal fraction. And then use the concept of product of two numbers.

Complete step by step solution: The given product of two numbers is $4\dfrac{4}{9}$and one of the numbers is $6\dfrac{2}{9}$.
The above numbers are in mixed fractions, to do operations like multiplication and division, we have to convert them into improper fractions.
$4\dfrac{4}{9} = \dfrac{{9 \times 4 + 4}}{9} = \dfrac{{40}}{9}$
$6\dfrac{2}{9} = \dfrac{{6 \times 9 + 2}}{9} = \dfrac{{56}}{9}$
So, now the given product of two numbers is $\dfrac{{40}}{9}$ and one of the numbers is $\dfrac{{56}}{9}$
Now if two numbers are x and y and their product is c.
$ \Rightarrow xy = c$
$ \Rightarrow y = \dfrac{c}{x}$
Thus, the second number can be calculated by dividing the given product by the first number.
So, second number is: $\dfrac{{\dfrac{{40}}{9}}}{{\dfrac{{56}}{9}}}$
On simplification we get,
second number is: $\dfrac{{40 \times 9}}{{9 \times 56}}$
On cancelling common factors we get,
second number is: $\dfrac{{40}}{{56}}$
Further simplification we get,
second number is: $\dfrac{5}{7}$

Hence, the required number is $\dfrac{5}{7}$

Additional Information:
The above question is a basic problem of the product of improper fractions.

Note: Note:
Students must convert mixed fractions into improper fractions carefully without any mistake and then proceed further as per the question. An improper fraction is the one whose numerator is greater than the denominator, whereas a mixed fraction is the one which contains a whole number part and other part having a proper fraction. Following is the explanation of converting improper fractions into mixed fractions:
For $\dfrac{a}{b}$where $a > b$, and a is not a multiple of b, we have $\dfrac{a}{b}$ as an improper fraction.
Let, $a = nb + r$ where b and r the quotient and the remainder when a is divided by b.
$\dfrac{a}{b} = \dfrac{{nb + r}}{b}$
$\dfrac{a}{b} = \dfrac{{nb}}{b} + \dfrac{r}{b}$
$\dfrac{a}{b} = n + \dfrac{r}{b}$
Now, n is a whole number, and $\dfrac{r}{b}$ is a proper fraction as r is less than b.
$\dfrac{a}{b} = n\dfrac{r}{b}$
Now, to convert mixed fractions into improper fractions:
Consider \[n\dfrac{a}{b}\]to be a mixed fraction, where n is a whole number and $a < b$ so that \[\dfrac{a}{b}\]is a proper fraction.
Now, $n\dfrac{a}{b} = n + \dfrac{a}{b}$
$n\dfrac{a}{b} = \dfrac{{nb}}{b} + \dfrac{a}{b}$
$n\dfrac{a}{b} = \dfrac{{a + nb}}{b}$
Let $a + nb = p$
$n\dfrac{a}{b} = \dfrac{p}{b}$where $p = a + nb > b$
So here we have learnt converting one type of fraction into another.