Explore Fraction Concepts with Examples and Questions

When is the lcm of a fraction sum the actual denominator.
Consider a sum
${\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{x}{y}}$
where each fraction is reduced. Alternatively using the familiar process of lowest common denominators, we have
${\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{a\cdot \frac{d}{(b,d)}+c\cdot \frac{b}{(b,d)}}{\begin{array}{cc}b& d\end{array}}}$
where ${\displaystyle (b,d)}$ denotes the gcd and ${\displaystyle [b,d]}$ denotes the lcm. My question is, when is it true that ${\displaystyle y=[b,d]}$? For example, this does not hold for
${\displaystyle \frac{5}{6}+\frac{1}{14}=\frac{38}{42}=\frac{19}{21}}$
where ${\displaystyle 21\ne [14,6]}$
Are there any simple necessary and sufficient conditions for ${\displaystyle y=[b,d]}$?
Edit It has been suggested that
${\displaystyle y=[b,d]\iff (b,d)=1}$
is a necessary and sufficient condition. I'm interested in either a proof or a counterexample if possible.
Edit 2 After a bit of searching, I've found
${\displaystyle \frac{1}{24}+\frac{1}{16}=\frac{5}{48}}$
as a counterexample.
I am still looking for nice conditions for this to be satisfied and I feel that I should give an explanation of exactly what type of condition I am seeking. Angela has provided a necessary and sufficient condition, but it does not seem to be "simpler" than simply adding the fraction and seeing if it reduces. This is perhaps ambitious, but I am looking for a condition which is simple enough to use by inspection for simple fractions.

Writing 1 in form of ${\displaystyle \frac{1}{t_1}+\cdots +\frac{1}{t_n}}$

what is equivilet to 1/3

find the rational number halfway between the numbers 7/8 and 2/3

What is the value of ${\displaystyle 5\frac{1}{4}\times \frac{4}{9}}$?

What is the value of ${\displaystyle (n!)}$ by formula?

How do you write 0.24 in percent form?

Jeanie has a ${\displaystyle \frac{3}{4}}$ yard piece of ribbon. She needs one ${\displaystyle \frac{3}{8}}$ yard piece and one ${\displaystyle \frac{1}{2}}$ yard piece. Can she cut the piece of ribbon into the two smaller pieces? Why?

What is the value of ${\displaystyle 5.79\cdot 3.6}$?

How do you write ${\displaystyle 6\frac{7}{8}}$ as an improper fraction?

What type of number is ${\displaystyle -\frac{15}{3}}$?

How do you write ${\displaystyle \frac{99}{50}}$ as a percentage?

Consider a sector with angle ${\displaystyle \frac{\pi }{3}}$ radians, and radius 3.5 m. What is the arc length and area?

How do you simplify ${\displaystyle \frac{5}{12}+\frac{2}{3}}$?

How do you convert ${\displaystyle 0.004\mathrm{\%}}$ into a fraction?

Express ${\displaystyle 183\frac{1}{3}\mathrm{\%}}$ as fraction?

In Mr. Prime's class, ${\displaystyle \frac{6}{10}}$ of the students had done their homework. Of these ${\displaystyle \frac{2}{3}}$ had all correct answers. What fraction of the whole class had all correct answers?

How do you write the following as a mixed number: ${\displaystyle \frac{43}{8}}$?

How do you simplify ${\displaystyle \frac{1}{18}+\frac{4}{18}}$?

What is 115% of 600?