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There's more to building an LCD than simply creating a sheet of liquid crystals. The combination of four facts makes LCDs possible:
Light can be polarized. (See How Sunglasses Work for some fascinating information on polarization!)
Liquid crystals can transmit and change polarized light.
The structure of liquid crystals can be changed by electric current.
There are transparent substances that can conduct electricity.
An LCD is a device that uses these four facts in a surprising way.
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To create an LCD, you take two pieces of polarized glass. A special polymer that creates microscopic grooves in the surface is rubbed on the side of the glass that does not have the polarizing film on it. The grooves must be in the same direction as the polarizing film. You then add a coating of nematic liquid crystals to one of the filters. The grooves will cause the first layer of molecules to align with the filter's orientation. Then add the second piece of glass with the polarizing film at a right angle to the first piece. Each successive layer of TN molecules will gradually twist until the uppermost layer is at a 90-degree angle to the bottom, matching the polarized glass filters.
As light strikes the first filter, it is polarized. The molecules in each layer then guide the light they receive to the next layer. As the light passes through the liquid crystal layers, the molecules also change the light's plane of vibration to match their own angle. When the light reaches the far side of the liquid crystal substance, it vibrates at the same angle as the final layer of molecules. If the final layer is matched up with the second polarized glass filter, then the light will pass through.
If we apply an electric charge to liquid crystal molecules, they untwist. When they straighten out, they change the angle of the light passing through them so that it no longer matches the angle of the top polarizing filter. Consequently, no light can pass through that area of the LCD, which makes that area darker than the surrounding areas.
Building a simple LCD is easier than you think. Your start with the sandwich of glass and liquid crystals described above and add two transparent electrodes to it. For example, imagine that you want to create the simplest possible LCD with just a single rectangular electrode on it. The layers would look like this:
The LCD needed to do this job is very basic. It has a mirror (A) in back, which makes it reflective. Then, we add a piece of glass (B) with a polarizing film on the bottom side, and a common electrode plane (C) made of indium-tin oxide on top. A common electrode plane covers the entire area of the LCD. Above that is the layer of liquid crystal substance (D). Next comes another piece of glass (E) with an electrode in the shape of the rectangle on the bottom and, on top, another polarizing film (F), at a right angle to the first one.
The electrode is hooked up to a power source like a battery. When there is no current, light entering through the front of the LCD will simply hit the mirror and bounce right back out. But when the battery supplies current to the electrodes, the liquid crystals between the common-plane electrode and the electrode shaped like a rectangle untwist and block the light in that region from passing through. That makes the LCD show the rectangle as a black area.
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The least common denominator (LCD) is the smallest number divisible by all denominators of the given set of fractions. It is the smallest number among the common multiples of the denominators.
In simple words, LCD is the LCM of the denominators of the given fractions.
The concept of LCD in math is really useful when it comes to comparing, adding or subtracting unlike fractions.
Example: Add the fractions $\frac{1}{9}$ and $\frac{3}{5}$.
To add any two fractions, firstly we check if the denominators are the same.
Here, the denominators are 9 and 5.
Find the least common denominator.
Multiples of $9 = 9,\; 18,\; 27,\; 36,\; 45$, …
Multiples of $5 = 5,\; 10,\; 15,\; 20,\; 25,\; 30,\; 35,\; 40,\; 45$, …
Common multiples of 9 and $5 = 45,\; 50,\; 95$, …
LCM (9, 5) $=$ LCD $(\frac{1}{9}$ and $\frac{3}{5})= 45$
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The least common denominator of a set of fractions is the smallest number of all the common multiples of denominators. It is also known as the Lowest Common Denominator (abbreviated as LCD).
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To find the least common denominator, we can use either of the ways as given below:
Listing Method
One way is to list the multiples of both the denominators. This method is convenient to use when the denominators are small numbers.
Example: Find the least common denominator of $\frac{5}{8}$ and $\frac{11}{12}$.
Multiples of $8 = 8,\; 16,\; 24,\; 32,\; 40,\; 48$,…
Multiples of $12 = 12,\; 24,\; 36,\; 48$,…
Common Multiples of 8 and $12 = 24,\; 48$,…
LCD $(\frac{5}{8} ,\; \frac{11}{12}) =$ LCM (8,12) $= 24$
We can make the denominators of $\frac{5}{8}$ and $\frac{11}{12}$ same by finding the LCD. Multiply both numerator and denominator of $\frac{5}{8}$ with 3. Multiply both numerator and denominator of $\frac{11}{12}$ with 2.
$\frac{5}{8} \times \frac{3}{3}= \frac{15}{24}$
$\frac{11}{12}\times \frac{2}{2} = \frac{22}{24}$
Prime Factorization Method
Find the prime factorization of the denominators. Identify the common (matching) factors. Note down the remaining factors. Multiply them together.
Example: $\frac{5}{21},\; \frac{3}{30}$
Prime factorization of $21 = 3\times7$
Prime factorization of $30 = 3\times2\times5$
Common factors $= 3$
Uncommon factors $= 2,\; 7,\; 5$
LCD $= 2 \times7 \times5 \times3 = 210$
NOTE: If the two or more denominators have HCF $= 1$, simply multiply the denominators to find the LCD.
For example, $\frac{1}{9}$ and $\frac{4}{7}$.
Since the HCF of 9 and 7 is 1, the Least Common Denominator is the product of two denominators. On multiplying the denominators, we get $9 \times 7 = 63$.
The concept of LCD in math is really helpful when working with fractions. Let’s see how to simplify operations on fractions using the least common denominator.
We will discuss two points.
We can easily compare and order unlike fractions by finding LCD.
Example: Find the LCD of the fractions: $\frac{3}{5},\;\frac{4}{6},\;\frac{9}{20}$
5 5 10 15 20 25 30 35 40 45 50 55 60 6 6 12 18 24 30 36 42 48 54 60 66 72 20 20 40 60 80 100 120 140 160 180 200 220 240Using the table of multiples above, we can observe that
LCM of 5, 20 and $6 = 60$.
Thus, LCD of the given fractions is 60
The fractions can be rewritten as: $\frac{36}{60},\;\frac{40}{60},\;\frac{27}{60}$
Ascending order: $\frac{27}{60}\lt\frac{36}{60}\lt\frac{40}{60} \Rightarrow \frac{9}{20}\lt\frac{3}{5}\lt\frac{4}{6}$
Descending order: $\frac{40}{60}\gt\frac{36}{60}\gt\frac{27}{60} \Rightarrow \frac{4}{6}\gt\frac{3}{5}\gt\frac{9}{20}$
Using the least common denominator, fractions can be added and subtracted.
Example 1: Find: $\frac{5}{6}\;-\;\frac{9}{20}$.
$6 = 2 \times 3$
$20 = 2 \times 2 \times 5$
LCM (6, 20) $= 2 \times2 \times3 \times5 = 60$
LCD $(\frac{5}{6},\;\frac{9}{20}) = 60$
$\frac{5\times10}{6\times10} = \frac{50}{60}$
$\frac{9\times3}{20\times3} = \frac{27}{60}$
We get
$\frac{5}{6}\;-\;\frac{9}{20} = \frac{50}{60}\;-\;\frac{27}{60} = \frac{13}{60}$
Example 2: Find $\frac{3}{4} + \frac{1}{5}$.
Since GCD$(4,\; 5) = 1$, LCM $(4,\; 5) = 4\times 5 = 20$
LCD$(\frac{3}{4},\;\frac{1}{5}) = 20$
The fractions can be rewritten as $\frac{15}{20}$ and $\frac{4}{20}$.
Sum $= \frac{15}{20} + \frac{4}{20} = \frac{19}{20}$
In this article, we learned about Least Common Denominator, its definition, applications along with examples on how to find LCD. Let’s solve a few more examples and practice problems for better understanding.
1. Find the LCD for $\frac{2}{5},\;\frac{1}{7}$ and $\frac{4}{9}$.
Solution:
The denominators 5, 7, and 9 have no common factors other than 1.
HCF (5, 7 and 9) $= 1$
Thus, LCM (5, 7 and 9) $= 5 \times 7 \times 9 = 315$
LCD$(\frac{2}{5},\;\frac{1}{7},\;\frac{4}{9}) = 315$.
2. Simplify: $\frac{21}{4}\;-\;\frac{7}{3}$
Solution:
We will first find the LCD of the denominators.
LCM (3, 4) $= 12$
LCD $(\frac{21}{4},\;\frac{7}{3}) = 12$
$\frac{21\times3}{4\times3} = \frac{63}{12}$ and $\frac{7\times4}{3\times4} = \frac{28}{12}$
$\frac{21}{4}\;-\;\frac{7}{3} = \frac{63}{12}\;-\;\frac{28}{12} = \frac{35}{12}$
3. Find the LCD of $\frac{7}{8}$ and $\frac{1}{6}$ by listing multiples.
Solution:
Multiples of $8 = 8,\; 16,\; 24,\; 32,\; 40,\; 48$, …
Multiples of $6 = 6,\; 12,\; 18,\; 24,\; 30,\; 36$, …
LCM(8, 6) $= 24$
Thus, LCD$(\frac{7}{8},\; \frac{1}{6}) = 24$
4. Compare the fractions $\frac{2}{9},\;\frac{3}{4}$.
Solution:
9 and 4 have no common factor other than 1.
Thus, LCM(4, 9) $= 9\times4 = 36$
Thus, LCD$(\frac{7}{8}$ and $\frac{1}{6}) = 36$
Let’s rewrite the fractions using the common denominator.
$\frac{2}{9} = \frac{8}{36}$ and $\frac{3}{4} = \frac{27}{36}$
Here, $\frac{8}{36} \lt \frac{27}{36}$
Thus, $\frac{2}{9} \lt \frac{3}{4}$
Attend this Quiz & Test your knowledge.
1
$\frac{1}{6}\lt\frac{5}{8}$
$\frac{2}{7}\lt\frac{1}{11}$
$\frac{2}{3}\lt\frac{8}{13}$
$\frac{1}{10}\gt\frac{7}{8}$
Correct
Incorrect
Correct answer is: $\frac{1}{6}\lt\frac{5}{8}$
LCD $=$ LCM $(6,\; 8) = 24$
$\frac{1\times4}{6\times4} = \frac{4}{24}$ and $\frac{5\times3}{8\times3} = \frac{15}{24}$
$\frac{4}{24}\lt\frac{15}{24}$
2
GCF
HCF
GCD
LCM
Correct
Incorrect
Correct answer is: LCM
The LCD of fractions is calculated by finding the LCM of the denominators.
3
$\frac{1}{12}$
$\frac{7}{12}$
$\frac{5}{12}$
$\frac{11}{12}$
Correct
Incorrect
Correct answer is: $\frac{5}{12}$
3 and 4 are coprimes. So, HCF$(3,\; 4) = 1$
LCM $(3,\; 4) = 12$
Thus, LCD of $\frac{1}{3}$ and $\frac{1}{4}$ is $3\times4 = 12$.
4
divisible by
a factor of
a divisor of
None of the above
Correct
Incorrect
Correct answer is: divisible by
Since the LCD is a LCM of denominators. Thus, it is basically a multiple of denominators. Thus, it is divisible by all denominators.
What is the difference between LCM and LCD? Are LCM and LCD the same or different?
LCD of fractions is the LCM of the denominators of the fractions. LCM of two or more numbers is the smallest number of common multiples of given numbers.
How is LCD different from the common denominator?
Least Common Denominator is the smallest common multiple of the common multiples of the denominators of a set of fractions. On the other hand, the common denominator is the common multiple of the denominators. For example: For the fractions $\frac{3}{5}$ and $\frac{2}{7}$, the least common denominator is 35. The common denominator can be 35, 70, 105, etc.
How are the LCD and GCF different?
LCD stands for Least Common Denominator and GCF stands for Greatest Common Factor. They are just about opposites. LCD is the least multiple that is the same for two or more denominators whereas, the GCF of two or more numbers is the greatest factor that these numbers share.
Can you find the LCD by simply multiplying the denominators?
Multiplying all of the denominators results in a common denominator between the fractions, it does not always give you the LCD. If the GCF of denominators is 1, then the LCD of fractions can be calculated by simply multiplying the denominators.