Question

Find equivalent fractions of \[\dfrac{4}{9}\] having
i) Numerator $16$
ii) Denominator $81$
iii) Numerator $24$
iv) Denominator $45$

Answer 1

Hint: If two fractions are equivalent we can reduce one to another by cancelling a common factor from the numerator and denominator. Here one of the numerator and denominator are given. We can find the other by finding which number is multiplied by the numerator or denominator of the given fraction to get the new numerator or denominator.

Useful formula:
A fraction $\dfrac{x}{y}$ is equal to another fraction $\dfrac{a}{b}$ if and only if there exist a $k$ such that $a = kx$ and $b = ky$.

Complete step-by-step answer:
We are given the fraction \[\dfrac{4}{9}\].
We have to find its equivalent fractions with given numerators and denominators.
A fraction $\dfrac{x}{y}$ is equal to another fraction $\dfrac{a}{b}$ if and only if there exist a $k$ such that $a = kx$ and $b = ky$.
i) First we have to find the equivalent fraction of \[\dfrac{4}{9}\] with the numerator $16$.
That is, we have to find $b$ such that $\dfrac{4}{9} = \dfrac{{16}}{b}$.
We have, $16 = 4 \times 4$.
This gives, $b = 9 \times 4 = 36$.
Therefore the equivalent fraction is $\dfrac{{16}}{{36}}$.
ii) Now we have to find the equivalent fraction of \[\dfrac{4}{9}\] with denominator $81$.
That is, we have to find $a$ such that $\dfrac{4}{9} = \dfrac{a}{{81}}$.
We have $81 = 9 \times 9$.
This gives, $a = 4 \times 9 = 36$.
Therefore, the equivalent fraction is $\dfrac{{36}}{{81}}$.
iii) Here we have to find the equivalent fraction of \[\dfrac{4}{9}\] with the numerator $24$.
That is, we have to find $b$ such that $\dfrac{4}{9} = \dfrac{{24}}{b}$.
We have $24 = 4 \times 6$.
This gives, $b = 9 \times 6 = 54$.
Therefore the equivalent fraction is $\dfrac{{24}}{{54}}$.
iv) Now we have to find the equivalent fraction of \[\dfrac{4}{9}\] with denominator $45$.
That is, we have to find $a$ such that $\dfrac{4}{9} = \dfrac{a}{{45}}$.
We have, $45 = 9 \times 5$.
This gives, $a = 4 \times 5 = 20$.
Therefore, the equivalent fraction is $\dfrac{{20}}{{45}}$.


Note: In this question, all the numerators and denominators given were multiples of the numerator or denominator of the fraction. But this may not be the case always. The number multiplied to get the new numerator or denominator need not be an integer. For example, $\dfrac{3}{6} = \dfrac{4}{8}$.