Question

How do you add $\dfrac{3}{7}+\dfrac{7}{9}$?

Answer 1

Hint: We will look at the necessary conditions for adding two fractions. We will have to make the denominators of both the fractions the same. To do this, we will find the least common multiple of both the denominators. We will multiply the numerators to get the equivalent fractions. After that we will add the two fractions and then simplify the resulting fraction if needed.

Complete step-by-step solution:
We can add two fractions when the denominators of the two fractions are the same. So, we first need to make the denominators same of both the fractions. For this, we will find the least common multiple of the denominators. The two denominators are 7 and 9. The prime factor of 7 is 7. The prime factors of 9 are 3 and 3. So, the least common multiple is the product of the prime factors of both the numbers. Therefore, we have
$\begin{align}
  & LCM=3\times 3\times 7 \\
 & \therefore LCM=63 \\
\end{align}$
Now, to make the denominator of the first term as 63 and obtain the equivalent fraction, we will multiply the numerator and denominator of the first fraction by 9. Also, to make the denominator of the second term as 63 and obtain the equivalent fraction, we will multiply the numerator and denominator of the second fraction by 7.
Therefore, we have the following expression,
$\begin{align}
  & \dfrac{3}{7}+\dfrac{7}{9}=\dfrac{3}{7}\times \dfrac{9}{9}+\dfrac{7}{9}\times \dfrac{7}{7} \\
 & \therefore \dfrac{3}{7}+\dfrac{7}{9}=\dfrac{27}{63}+\dfrac{49}{63} \\
\end{align}$
Now, we have the same denominators in both the fractions. So, the addition of the fractions is now the addition of numerators. Hence, we have
$\begin{align}
  & \dfrac{3}{7}+\dfrac{7}{9}=\dfrac{27+49}{63} \\
 & \therefore \dfrac{3}{7}+\dfrac{7}{9}=\dfrac{76}{63} \\
\end{align}$

Note: The prime factors of 76 are 2, 2 and 19. The prime factors of 63 are 3, 3 and 7. As there are no common factors in the numerator and denominator of the resulting fraction, we cannot reduce it any further. The least common multiple of two numbers is the smallest number that is divisible by both the numbers.