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How do you use the LCD of $10$ and $15$ to solve $\dfrac{3}{10}+\dfrac{5}{15}?$

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Last updated date: 20th Jun 2024
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Answer
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Hint: The lowest common denominator or least common denominator (LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting and comparing fractions.
For example:
$\dfrac{1}{2}+\dfrac{2}{3}=\dfrac{3}{6}+\dfrac{4}{6}=\dfrac{7}{6}$
Here $6$ is the least common multiple of $2$ and $3.$ Their product is $36$ is also a common denominator but calculating with that denominator involves larger numbers.
$\dfrac{1}{2}+\dfrac{2}{3}=\dfrac{18}{36}+\dfrac{24}{36}=\dfrac{42}{36}$

Complete step by step solution:
To add or subtract fractions they must be over a common denominator.
Give that $\dfrac{3}{10}+\dfrac{5}{15}$
To find least or lowest common denominator of $10$ and $15.$
So first we will multiply it by $1$
$10\times 1=10$
$\Rightarrow 15\times 1=15$
So, it is not possible
Now, we will multiply $10$ by $3$ and $15$ by $2$
$10\times 3=30$
$\Rightarrow 15\times 2=30$
So, the lowest common denominator of $10$ and $15$ is $30.$
Now, to make the denominator of both the numbers same.
We have to have the same numbers.
We have to multiply the fractions $\dfrac{3}{10}$ with $\dfrac{8}{3}$ and we have to multiply the fraction.
$\dfrac{5}{15}$ with $\dfrac{2}{2}$
$\Rightarrow \left( \dfrac{3}{3}\times \dfrac{3}{10} \right)+\left( \dfrac{2}{2}\times \dfrac{5}{15} \right)$
$\Rightarrow$ \[\dfrac{3\times 3}{3\times 10}+\dfrac{2\times 5}{2\times 15}\]
$\Rightarrow \dfrac{9}{30}+\dfrac{10}{30}$
Now, we can add a numerator because the denominator is the same.
$\dfrac{9+10}{30}=\dfrac{19}{30}$
By using LCD of $10$ and $15$ the value of
$\dfrac{3}{10}+\dfrac{5}{15}=\dfrac{19}{30}$

Additional Information:
The LCD lowest common denominator has many practice uses, such as determining the number of objects of two different lengths necessary to align them in allow which starts and ends at the same place such as in brickwork filling etc. It is also useful in planning work schedules with employees with $y$ days off every $x$ days.

Note: LCD is only calculated. When the denominators of two fractions are different, if the denominators are the same then that number becomes the LCD.
LCD is finded when we need to add or subtract the two fractions with different denominators.
Do not get confused between LCD and LCM they both are different from each other.
LCD is used when denominators are different and LCM is the lowest common multiple of two numbers.