Give four rational numbers equivalent to: $\dfrac{4}{9}$
Answer
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Hint: Rational numbers can always be multiplied by a common factor in the numerator and denominator to give another rational number that, on reducing, gives back the original rational number.
Complete step-by-step answer:
You’re all familiar with the way we reduce fractions that have a common factor in the numerator and denominator, to its simplest form, to get the simplest rational number that we can obtain from that original fraction, one where the HCF of the numerator and denominator is essentially equal to $1$. Such a fraction can’t be reduced to a simpler form.
Now, let’s analyse the fraction given to us. It’s equal to $\dfrac{4}{9}$. Let’s find the HCF of the numerator and denominator and know if the fraction provided can be reduced further, to a simpler form. The numerator is equal to $4$, and the denominator is equal to $9$, and the HCF of these two numbers, will obviously be equal to $1$, since they don’t have any common factors. To elaborate, the factors of $4$ are $1,2,4$ and that of $9$ are $1,3,9$. As you can see, the only factor common to the numerator and the denominator is $1$. Hence, this is the simplest form.
But, we still have to find four rational numbers that are equal to $\dfrac{4}{9}$. To find them, let’s back calculate and multiply the numerator and denominator with the same factors, to essentially find a non-reduced form of the same fraction.
Thus, let’s multiply the numerator and denominator with $2$ first. Doing so, we get $\dfrac{4.2}{9.2}=\dfrac{8}{18}$. $\dfrac{8}{18}$ is considered a new rational number, but one whose value is essentially equal to $\dfrac{4}{9}$ only.
Next, let’s multiply the numerator and denominator with $3$. Doing so, the new rational number we get will be equal to $\dfrac{4.3}{9.3}=\dfrac{12}{27}$. Thus, we have the second rational number equal to $\dfrac{4}{9}$.
Next, let’s multiply the numerator and denominator with $4$. Doing so, the new rational number we’ll get is $\dfrac{4.4}{9.4}=\dfrac{16}{36}$. Thus, we have the third rational number equal to $\dfrac{4}{9}$.
Lastly, let’s the numerator and denominator with $5$. Doing so, we’ll finally get the fourth rational number equal to $\dfrac{4}{9}$, which is equal to $\dfrac{4.5}{9.5}=\dfrac{20}{45}$.
Thus, we have four rational numbers that are equal to $\dfrac{4}{9}$, and they are : $\dfrac{8}{18},\dfrac{12}{27},\dfrac{16}{36},\dfrac{20}{45}$.
Note: You can multiply the numerator and denominator with numbers apart from these as well. Even then, you’ll get four rational numbers that are equal to $\dfrac{4}{9}$, just, they’ll not be the same as the ones we found out. Rational numbers are any number that can be written in the form of a fraction, or in the form $\dfrac{p}{q}$.
Complete step-by-step answer:
You’re all familiar with the way we reduce fractions that have a common factor in the numerator and denominator, to its simplest form, to get the simplest rational number that we can obtain from that original fraction, one where the HCF of the numerator and denominator is essentially equal to $1$. Such a fraction can’t be reduced to a simpler form.
Now, let’s analyse the fraction given to us. It’s equal to $\dfrac{4}{9}$. Let’s find the HCF of the numerator and denominator and know if the fraction provided can be reduced further, to a simpler form. The numerator is equal to $4$, and the denominator is equal to $9$, and the HCF of these two numbers, will obviously be equal to $1$, since they don’t have any common factors. To elaborate, the factors of $4$ are $1,2,4$ and that of $9$ are $1,3,9$. As you can see, the only factor common to the numerator and the denominator is $1$. Hence, this is the simplest form.
But, we still have to find four rational numbers that are equal to $\dfrac{4}{9}$. To find them, let’s back calculate and multiply the numerator and denominator with the same factors, to essentially find a non-reduced form of the same fraction.
Thus, let’s multiply the numerator and denominator with $2$ first. Doing so, we get $\dfrac{4.2}{9.2}=\dfrac{8}{18}$. $\dfrac{8}{18}$ is considered a new rational number, but one whose value is essentially equal to $\dfrac{4}{9}$ only.
Next, let’s multiply the numerator and denominator with $3$. Doing so, the new rational number we get will be equal to $\dfrac{4.3}{9.3}=\dfrac{12}{27}$. Thus, we have the second rational number equal to $\dfrac{4}{9}$.
Next, let’s multiply the numerator and denominator with $4$. Doing so, the new rational number we’ll get is $\dfrac{4.4}{9.4}=\dfrac{16}{36}$. Thus, we have the third rational number equal to $\dfrac{4}{9}$.
Lastly, let’s the numerator and denominator with $5$. Doing so, we’ll finally get the fourth rational number equal to $\dfrac{4}{9}$, which is equal to $\dfrac{4.5}{9.5}=\dfrac{20}{45}$.
Thus, we have four rational numbers that are equal to $\dfrac{4}{9}$, and they are : $\dfrac{8}{18},\dfrac{12}{27},\dfrac{16}{36},\dfrac{20}{45}$.
Note: You can multiply the numerator and denominator with numbers apart from these as well. Even then, you’ll get four rational numbers that are equal to $\dfrac{4}{9}$, just, they’ll not be the same as the ones we found out. Rational numbers are any number that can be written in the form of a fraction, or in the form $\dfrac{p}{q}$.
Last updated date: 21st Sep 2023
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