laya and layakari
Laya
Laya means tempo. It is the distance between two maatras. As the distance decreases, laya increases and vice versa. Laya also means destruction  when you play the second matra the time interval of the first maatra is destroyed. Every aspect of music is dependent on the laya, and laya is a universal theme (ex. Planets revolve at a specific speed).
There are three main types of laya:

Madhya laya  Is the medium tempo, approximately equal to heart rate, 80 bpm.

Vilambit laya Is the slow tempo played at approximately ½ of the speed of the madhya laya, 40 bpm (bada khayal is performed in vilambit). Ektaal, jhoomra, teentaal are taals than can be appropriately played in vilambit.

Drut laya  Is the fast tempo played at approximately 2 times the speed of the madhya laya, 160 bpm.
There aren’t precise numbers for each laya, it is relative to the composition.
Normally performances begin with vilambit laya, then go to madhya laya, and end in drut or atidrut laya. (Ex. peshkaar, kayda, rela, tukdas)
Baraabar laya/Thaa(ठाह), dugun (2 maatras in 1), tigun (3 in 1), chaugun (4 in 1)
Layakari
The skill of controlling the laya and changing how the maatras are divided to create rhythmic variation. In other words, it is the practical and artistic implementation of various layas.
Relation between Laya and Layakari
Laya and layakari are closely related ideas. Whenever we create and perform a specific layakari, we require a specific laya. Music can’t simply jump from 102 bpm to 22 bpm to 15 bpm during a theka or kayda, for example. When performing, the tabla composition and lehra will be of a particular laya, and the laya will generally increase throughout the performance. The word “layakari” is “laya” + “kari” meaning “to do” laya. Whole number layakaris: Dugun, Tigun, Chougun. Fractional layakaris: Kuwaad laya (5 maatras in the span of 4 maatras  1.25), Aad laya (3 maatras in the span of two maatras  1.5), Biad laya (7 maatras in the span of 4 maatras  1.75), Navamgun laya (9 maatras in the span of 4 maatras  2.25), Ekadashguna laya (11 maatras in 4 maatras  2.75).
How to write fractional layakari of any taal
Laya is the distance between two maatras and the repetition of the interval over multiple maatras in a sequence. During accompaniment or in a solo, the original laya is kept the same, and the number of beats in the theka can be increased or decreased, and this process describes performing layakari during a musical performance. (Ex. going from 4 beats/maatra to 6/8/12 beats/maatra to complement the main artist’s focus. However we should not do layakari or play too much where it’s not appropriate).
When performing layakari, complex fractional forms can be used.

Double 2/1 (Duppat)

Triple 3/1 (Tippat)

Quadruple 4/1 (Chaupat)

3/4 (Poundpat)

4/3 (Inverse of Poundpat)

5/4 (Kuwaad)

4/5 (Inverse of Kuwaad)

3/2 (Aad)

2/3 (Inverse of Aad)

7/4 (Biaad)

4/7 (Inverse of Biaad)
*The above method of referring to layas with fractions is adopted because it is easier to understand the speed in relation to the value of the fraction (as taught in school). For example, if the fraction is 7/4, it is faster than baraabar laya because 7/4 is greater than 1. If the fraction is 3/4 (Poundpat), then it is slower than baraabar laya because 3/4 is less than 1. The above method of referring to layas in terms of fractions smaller than or larger than 1 is found to be efficient because it directly indicates if the speed is slower or faster than baraabar. Think of the respective fractions as "multipliers".

if the fraction is less than 1, it goes slower than baraabar

Ex. (2/3)


if the fraction is more than 1, it goes faster than baraabar

Ex. (3/2)

In order to master layakari in any taal, understanding the layakari with mathematical formulas is essential. For example, let’s try writing jhaptaal in kuwaad (5/4) laya.
Step 1: The numerator of the fraction (5) represents the number of beats in each maatra while writing kuwaad.
Step 2: The denominator minus 1 (3) represents the number of pauses to leave after each bol while writing kuwaad.
Step 3: Write the kuwaad of jhaptaal with 5 beats per maatra and 3 beats of pauses after each bol.
→ DhinSSSNa SSSDhiS SSDhiSS SNaSSS TinSSSNa SSSDhiS SSDhiSS SNaSSS
Because writing the kuwaad of jhaptaal took 8 maatras, we can fill the first two maatras with pauses to complete one awartan of jhaptaal (10 maatras).
→ SSSSS SSSSS DhinSSSNa SSSDhiS SSDhiSS SNaSSS TinSSSNa SSSDhiS SSDhiSS SNaSSS
We can use those steps to write the layakari of any taal in any laya. The above method can be used for writing layakari so that it fits within one awartan (sumtosum). However, we can also write the kuwaad of jhaptaal 5 times without the two maatras of pause in the beginning so that it fits in 4 awartans of jhaptaal (40 maatras).
We were able to find out that doing the kuwaad of jhaptaal takes 8 maatras by practically writing and finding out. But there is also another way of finding out how many maatras the layakari of a particular taal will be. Simply flip the numerator and denominator and multiply that by the number of maatras in the taal.
Ex. The juwaad of jhaptaal will be 8 maatras long
The inverse of kuwaad fraction (5/4) is 4/5
4/5 times 10 maatras in jhaptaal is 8
Ex. The aad of teentaal will be 10 and 2/3 maatras long
The inverse of aad fraction (3/2) is 2/3
2/3 times 16 maatras in teentaal is 10 and 2/3
Jati
The concept of layakari can be demonstrated in jati as well. Jati indicates creating a distribution of syllables in a particular maatra in various ways. In general, a maatra is divided in 4 syllables, but we can divide it in other whole numbers as well.

Tisra jati: 3 beats in 1

Chatushra jati: 4 beats in 1

Khand jati: 5 beats in 1

Mishra jati: 7 beats in 1

Sankirna jati: 9 beats in 1

Divyasankirna jati: 11 beats in 1
*Note: multiples are also allowed. Example: tisra jati can be 6 beats in 1, mishra jati can be 14 beats in 1