Question

What fraction of an hour is 40 minutes?

Hint: Convert hour into minutes and find the value of fraction by taking 40 minutes in the numerator and one hour in the denominator (converted into minutes). Reduce the fraction into lowest terms by taking out common terms from numerator and denominator.

We have to find the value of a fraction of 40 minutes of one hour.
We will write one hour in terms of minutes and then calculate the value of fraction.
We know that $1hr=60\min$.
To find the value of fraction, we will write 40 minutes as numerator and one hour as denominator.
Thus, the value of fraction is $\dfrac{40}{60}$ as $1hr=60\min$.
We will now reduce this fraction into lowest terms.
To do so, we will find the HCF of numerator and denominator.
To find the HCF, we will write the prime factorization of each of the numbers. We will choose the common prime factors of all the three numbers. The number obtained by choosing the common prime factors of all the numbers will be the HCF of three numbers.
The prime factorization of 40 is $40={{2}^{3}}\times 5$. The prime factorization of 60 is $60={{2}^{2}}\times 3\times 5$. The terms common to prime factorization of both the numbers is ${{2}^{2}}\times 5=20$.
Thus, the prime factorization of 40 and 60 is 20.
We will now divide the numerator and denominator by their HCF, which is 20.
Thus, we have $\dfrac{40}{60}=\dfrac{\dfrac{40}{20}}{\dfrac{60}{20}}=\dfrac{2}{3}$.
Hence, 40 minutes is $\dfrac{2}{3}$ the value of one hour, i.e., 60 minutes.

Note: Fraction represents a part of a whole. We should be careful while writing the fractions as the units of numerator and denominator should always be the same. Otherwise, we will get an incorrect answer. To write the prime factorization of any number, start by dividing the number by the first prime number, which is 2 and then continue to divide by 2 until you get a number which is not divisible by 2 (which means that you get a decimal or remainder on dividing the number by 2). Then start dividing the number by the next prime number which is 3. Continue dividing the number by 3 until you get a number which is not divisible by 3. Thus, continuing the process, keep dividing the numbers by series of prime numbers $5,7,...$ until the only numbers left are prime numbers. Write the given number as a product of all the prime numbers (considering the fact to count each prime number as many times as it divides the given number) to get the prime factorization of the given number.