CA1206455A - Manual mathematical calculator - Google Patents
Manual mathematical calculatorInfo
- Publication number
- CA1206455A CA1206455A CA000450275A CA450275A CA1206455A CA 1206455 A CA1206455 A CA 1206455A CA 000450275 A CA000450275 A CA 000450275A CA 450275 A CA450275 A CA 450275A CA 1206455 A CA1206455 A CA 1206455A
- Authority
- CA
- Canada
- Prior art keywords
- tokens
- abacus
- tracks
- numeral
- token
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Expired
Links
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06G—ANALOGUE COMPUTERS
- G06G1/00—Hand manipulated computing devices
- G06G1/02—Devices in which computing is effected by adding, subtracting, or comparing lengths of parallel or concentric graduated scales
- G06G1/04—Devices in which computing is effected by adding, subtracting, or comparing lengths of parallel or concentric graduated scales characterised by construction
Abstract
TITLE
MANUAL MATHEMATICAL CALCULATOR
INVENTOR
Werner REXER
ABSTRACT
A modified abacus which aids children in counting, ad-ding, substracting, multiplying and dividing including squares and square roots of numerals, discloses the utilization of a square matrix of slidable tokens, each token slidable on a track or rod, and preferably with 10 tokens on each track and 10 parallel tracks arranged whereby the square matrix is esta-blished. On one side each of the tokens has indicia or nume-rals, N, where N is defined by the equation:
N(a,b) = (10a-10) + b, and also indicia on the other side;
N(a,b) = a x b where;
a and b are the coordinate positions on the tokens in each matrix. Each token has a lateral width and the cumulative width of all tokens on a common track is less than the length of the track thereby allowing the tokens to move to and fro on command.
MANUAL MATHEMATICAL CALCULATOR
INVENTOR
Werner REXER
ABSTRACT
A modified abacus which aids children in counting, ad-ding, substracting, multiplying and dividing including squares and square roots of numerals, discloses the utilization of a square matrix of slidable tokens, each token slidable on a track or rod, and preferably with 10 tokens on each track and 10 parallel tracks arranged whereby the square matrix is esta-blished. On one side each of the tokens has indicia or nume-rals, N, where N is defined by the equation:
N(a,b) = (10a-10) + b, and also indicia on the other side;
N(a,b) = a x b where;
a and b are the coordinate positions on the tokens in each matrix. Each token has a lateral width and the cumulative width of all tokens on a common track is less than the length of the track thereby allowing the tokens to move to and fro on command.
Description
6i45S
This invention relates to a manual mathematical calcu-lator and specifically to a modified abacus.
In the prior art there are a large number of mechanical calculators derived from the ancient abacus which itself was of various forms. The abacus was a mechanical calculator;
some of the more recent prior art devices employ those struc-tures with the intent to aid in mathematical calculation or in the learning of mathematical calculation.
I have desiyned a modified abacus which aids children in counting, adding, substracting, multiplying and dividing including squares and square roots. I have found that with little practice, children as young as 3 can count effectively to 100, and children 4 or 5 years old can master addition, subtraction, multiplication and division. From there the learning of square roots and squaring i9 apparent. In this respect, my modified abacus facilitates counting, adding and subtracting, with the use of one matrix of slidable tokens, and multiplication, division, squares and square roots with another matrix of slidable toXens, Each matrix is preferably a two dimensional X by X where X = 10.
In the preferxed embodiment numeral indicia of one matrix are embossed on the front face of each token, and the numerals of the o~her matrix on the obverse face of the same tokens.
In that respect, the value of ~he numeral on each token of one matrix i3 deined by the equation ~(a b) = (lOa-10) + b, and for the other matrix W(a b) = a x b where a and b are the coordinate positions of the tokens.
The invention therefore contemplates an abacus for aiding in the understanding of counting, adding and substrac-ting, comprising:
(a~ an open frame carrying a plurality of parallel tracks;
(b3 a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged so that the value of each numeral i~ defined by:
...
~Z~.3~S~i (i) N(a b) = (lOa-10) ~ b where;
a,b are the coordinate positions of the tokens and is the numeral.
The invention further contemplates an abacus for aiding in the understanding of multiplying and dividing comprising:
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged so that the value of each numeral is defined by:
(i) N(a b) = a x b where;
a,b are the coordinate positions o~ the tokens and N is the numeral.
Preferably therefore the invention therefore contem-plates an abacus for aiding in the understanding of counting, adding, substracting, multiplying and dividing comprising:
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged o that the value of each numeral is defined by:
(i) N(a b) = (lOa-10) ~ b where;
a,b are the coordinate positions of the tokens and ~1 is the numeral.
(d) numerals on the surface of each token arranged so that ~he value of each numeral is deined by:
(i) ~(a b) = a x b where;
a,b are the coordinate positions of the tokens and ~ is the numeral.
In each of the foregoing, the length of each track shall be longer than the total lateral dimension o all the tokens on the track and thereb~ the tokens may be laterally moved to and fro.
The invention will now be described by way of example and reference to the accompanying drawings in which;
-Figure 1 is a perspective view of the front face of t'nenovel abacus.
Figure 2 is a similar perspective view, but of the reverse face of my abacus.
Figure 3 is an exploded, partially in section, view of the structural components of the abacus.
Figures 4, 5 and 6 respectively illustrate the operation of the front face matrix of the a-Dacus in counting, figure 5;
adding or subtracting, figure 6; and, for games and creative designs, figure 4.
Figures 7, 8 and 9, illustrate the obverse side of my novel abacus; figure 7, multiplication; figure 8, counting by multiples that employs short form multiplication; and, figure 9 division.
Referring to figures 1, 2 and 3, the abacus 10 includes a circumferential frame 11 employing two parallel pairs, of frame members, 12 and 13. One pair 13 define~ a plurality, ten in number, of sockets 14, into which respective ends of glide rods or tracks 15 fit. Preerably each member pair 13 20 i5 molded from plastic or other material and formed as two segments 13a and 13b of identical profile and in that respect define adjacently disposed fingers 17 which register when the components 13a and 13b are cemented or welded together as shown in figure 3 to orm the socket 14 into which ends of the tracks 15 are ~ecured. Similarly, paired members 12 are compo~ed of identically extruded components 12a and 12b. In that respect it is clearly seen that the components 12a and 12b, being identical, may be extruded from a single piece of material; similarly, for the componen~s 13a and 13b. Each of the components, a and b, form a sea~ 25 where th~y join and it is along the seam that the members 13 are eit~er cemented or welded as may be appropriate. If the tracks 15 and members 12 and 13 are plastic, carbon ~e~rachloride or other suitable solvent i9 a ~atisfactory cement. Each of the ten tracks 15 accommodate ten tokens 2000 Preferably the tokens dispose a rectangular upper face 205, and a similar rectangular reverse face 210. On each of these faces an appropriate indicia or 9c55 numeral 220 is affixed as by gluing. It is preferred, that the colour coordination of the indicia of the front face of the numerals as in figures 1, 4, 5 and 6, be of one colour, for example red, and those of the indicia and numerals on the reverse face, figures 2, 7, 8 and 9, be of another colour, blacX. This aids in indentifying the appropriate matrix.
Counting, Addition, Subtra~tion Referring to figures 1, 4, 5 and 6, a 10 x 10 two dimen-sional matrix of movable tokens 200 is shown organized so that the indicia 220 on each token face is defined by the equation;
a b) ~ (lOa-10) ~ b; where a and b are the coordinate positions of each token.
Hence the upper row 300 has indicia numerals 1 through 10; the second row 310, 11 through 20; and, each subsequent row numerals of the same decade so that the bottom row 390 has numerals 91 through 100, all as more clearly seen in figure 1.
Referring to figure 5, counting is achieved for example by moving the first 5 tokens on the first row 310 to the left and counting themO Thus, by moving the tokens let to right consecutively along each of ~he ~racks starting with the upper tract 310, small children are aided in counting.
Referring ~o subtraction and addition, and to figure 6 in tract 310 the first 7 tokens are placed to the left hand margin while the nex~ two tokens ~8,9~ along that track indi-cate, that if the next 2 tokens "2" are being moved rom right to left in response to arrow 400 perform the computation 'add"
to the existing "7" hence 7 + 2 = 9, the re ultant depicted on the face of the now right most token of the group. On the other hand, if 9 tokens 'nad been together and the movement of the latter 2 tokens (8 and 9) where to the right in response to arrow 450, there would be "2" taken away and hence the resultant is 7; while the computation per~ormed is subtrac-tion. With the whole matrix any ~ddition or subtraction up to 100 can be achieved by moving the tokens lef~ and right in response to the arrows. It is seen, therefore, that it is preferred that the length of each of the tracks be signifi-cantly longer than the physical dimension of 10 tokens. I
prefer that the operative length of each track 15 be at least a cumulative dimension of 15 tokens.
Multiplicatic:n, Division (Squaring and Square Roots) Referring to figures 2, 7, 8 and 9, tha upper row 600 has indicia numeraled 1 to 10 and the left hand most column i3 similar. Each subsequent row 610 through 690 has the indicia or numeral on the face 205 defined by the equation:
~(a b) ~ a x b; where a and b are the coordi-nate positions of each token.
Thus, left to right, in line 610 the token valu~s are 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20 while ~hose in row 690 are 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100.
Referring to figure 2, multiplication is achieved by moving the appropriate tokens to the left as a sub-matrix 700. In the example, it is a 3 x 4 matrix whose left hand most token has indicia "12" thereon and represents multiplication of 3 x 4 = 12.
Similarly, multiplication using sub-matrices of the balance of the tokens can be achieved. The same is illust-rated in figure 7.
Division i5 apparent by taking the highest value to token "12" and dividing it by either the hori~ontal or verti-cal number of tokens, 4 or 3 as the case might be and the opposite is the answer. Thu9 12 divided by 3 = 4 or 12 divided by 4 = 3.
Referring to figure 8, the same shows counting by multiples with base 4, because they are in the 4th row 640 binary counting is achieved on the second row or in row 620.
Each row, therefore, i~ uni~ue a~ to its base o counting.
In figure 9, fractions are illustrated where a new sub-matrix 720 is arranged with three vertical columns 721, 722, 723. Each of these columns represents the fraction of "thirds". One third of the value of the total matrix "15" is derived by reading the value of the highest valu~ token in each vertical column. Hence, 1/3 out of 15 = 5, the value at row 721; 2/3 ou~ of 15 = 10, the value of the highest tokan in row 722; and 3/3 out of 15 the ~alue of the highest token in the third column.
It will be apparent to those skilled in the art, from the foregoing, that the front and reverse face of the frame for each matrix could identify the operations which the matrix itself visibly can perform and instructions of use.
This invention relates to a manual mathematical calcu-lator and specifically to a modified abacus.
In the prior art there are a large number of mechanical calculators derived from the ancient abacus which itself was of various forms. The abacus was a mechanical calculator;
some of the more recent prior art devices employ those struc-tures with the intent to aid in mathematical calculation or in the learning of mathematical calculation.
I have desiyned a modified abacus which aids children in counting, adding, substracting, multiplying and dividing including squares and square roots. I have found that with little practice, children as young as 3 can count effectively to 100, and children 4 or 5 years old can master addition, subtraction, multiplication and division. From there the learning of square roots and squaring i9 apparent. In this respect, my modified abacus facilitates counting, adding and subtracting, with the use of one matrix of slidable tokens, and multiplication, division, squares and square roots with another matrix of slidable toXens, Each matrix is preferably a two dimensional X by X where X = 10.
In the preferxed embodiment numeral indicia of one matrix are embossed on the front face of each token, and the numerals of the o~her matrix on the obverse face of the same tokens.
In that respect, the value of ~he numeral on each token of one matrix i3 deined by the equation ~(a b) = (lOa-10) + b, and for the other matrix W(a b) = a x b where a and b are the coordinate positions of the tokens.
The invention therefore contemplates an abacus for aiding in the understanding of counting, adding and substrac-ting, comprising:
(a~ an open frame carrying a plurality of parallel tracks;
(b3 a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged so that the value of each numeral i~ defined by:
...
~Z~.3~S~i (i) N(a b) = (lOa-10) ~ b where;
a,b are the coordinate positions of the tokens and is the numeral.
The invention further contemplates an abacus for aiding in the understanding of multiplying and dividing comprising:
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged so that the value of each numeral is defined by:
(i) N(a b) = a x b where;
a,b are the coordinate positions o~ the tokens and N is the numeral.
Preferably therefore the invention therefore contem-plates an abacus for aiding in the understanding of counting, adding, substracting, multiplying and dividing comprising:
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged o that the value of each numeral is defined by:
(i) N(a b) = (lOa-10) ~ b where;
a,b are the coordinate positions of the tokens and ~1 is the numeral.
(d) numerals on the surface of each token arranged so that ~he value of each numeral is deined by:
(i) ~(a b) = a x b where;
a,b are the coordinate positions of the tokens and ~ is the numeral.
In each of the foregoing, the length of each track shall be longer than the total lateral dimension o all the tokens on the track and thereb~ the tokens may be laterally moved to and fro.
The invention will now be described by way of example and reference to the accompanying drawings in which;
-Figure 1 is a perspective view of the front face of t'nenovel abacus.
Figure 2 is a similar perspective view, but of the reverse face of my abacus.
Figure 3 is an exploded, partially in section, view of the structural components of the abacus.
Figures 4, 5 and 6 respectively illustrate the operation of the front face matrix of the a-Dacus in counting, figure 5;
adding or subtracting, figure 6; and, for games and creative designs, figure 4.
Figures 7, 8 and 9, illustrate the obverse side of my novel abacus; figure 7, multiplication; figure 8, counting by multiples that employs short form multiplication; and, figure 9 division.
Referring to figures 1, 2 and 3, the abacus 10 includes a circumferential frame 11 employing two parallel pairs, of frame members, 12 and 13. One pair 13 define~ a plurality, ten in number, of sockets 14, into which respective ends of glide rods or tracks 15 fit. Preerably each member pair 13 20 i5 molded from plastic or other material and formed as two segments 13a and 13b of identical profile and in that respect define adjacently disposed fingers 17 which register when the components 13a and 13b are cemented or welded together as shown in figure 3 to orm the socket 14 into which ends of the tracks 15 are ~ecured. Similarly, paired members 12 are compo~ed of identically extruded components 12a and 12b. In that respect it is clearly seen that the components 12a and 12b, being identical, may be extruded from a single piece of material; similarly, for the componen~s 13a and 13b. Each of the components, a and b, form a sea~ 25 where th~y join and it is along the seam that the members 13 are eit~er cemented or welded as may be appropriate. If the tracks 15 and members 12 and 13 are plastic, carbon ~e~rachloride or other suitable solvent i9 a ~atisfactory cement. Each of the ten tracks 15 accommodate ten tokens 2000 Preferably the tokens dispose a rectangular upper face 205, and a similar rectangular reverse face 210. On each of these faces an appropriate indicia or 9c55 numeral 220 is affixed as by gluing. It is preferred, that the colour coordination of the indicia of the front face of the numerals as in figures 1, 4, 5 and 6, be of one colour, for example red, and those of the indicia and numerals on the reverse face, figures 2, 7, 8 and 9, be of another colour, blacX. This aids in indentifying the appropriate matrix.
Counting, Addition, Subtra~tion Referring to figures 1, 4, 5 and 6, a 10 x 10 two dimen-sional matrix of movable tokens 200 is shown organized so that the indicia 220 on each token face is defined by the equation;
a b) ~ (lOa-10) ~ b; where a and b are the coordinate positions of each token.
Hence the upper row 300 has indicia numerals 1 through 10; the second row 310, 11 through 20; and, each subsequent row numerals of the same decade so that the bottom row 390 has numerals 91 through 100, all as more clearly seen in figure 1.
Referring to figure 5, counting is achieved for example by moving the first 5 tokens on the first row 310 to the left and counting themO Thus, by moving the tokens let to right consecutively along each of ~he ~racks starting with the upper tract 310, small children are aided in counting.
Referring ~o subtraction and addition, and to figure 6 in tract 310 the first 7 tokens are placed to the left hand margin while the nex~ two tokens ~8,9~ along that track indi-cate, that if the next 2 tokens "2" are being moved rom right to left in response to arrow 400 perform the computation 'add"
to the existing "7" hence 7 + 2 = 9, the re ultant depicted on the face of the now right most token of the group. On the other hand, if 9 tokens 'nad been together and the movement of the latter 2 tokens (8 and 9) where to the right in response to arrow 450, there would be "2" taken away and hence the resultant is 7; while the computation per~ormed is subtrac-tion. With the whole matrix any ~ddition or subtraction up to 100 can be achieved by moving the tokens lef~ and right in response to the arrows. It is seen, therefore, that it is preferred that the length of each of the tracks be signifi-cantly longer than the physical dimension of 10 tokens. I
prefer that the operative length of each track 15 be at least a cumulative dimension of 15 tokens.
Multiplicatic:n, Division (Squaring and Square Roots) Referring to figures 2, 7, 8 and 9, tha upper row 600 has indicia numeraled 1 to 10 and the left hand most column i3 similar. Each subsequent row 610 through 690 has the indicia or numeral on the face 205 defined by the equation:
~(a b) ~ a x b; where a and b are the coordi-nate positions of each token.
Thus, left to right, in line 610 the token valu~s are 2, 4, 6, 8, 10, 12, 14, 16, 18 and 20 while ~hose in row 690 are 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100.
Referring to figure 2, multiplication is achieved by moving the appropriate tokens to the left as a sub-matrix 700. In the example, it is a 3 x 4 matrix whose left hand most token has indicia "12" thereon and represents multiplication of 3 x 4 = 12.
Similarly, multiplication using sub-matrices of the balance of the tokens can be achieved. The same is illust-rated in figure 7.
Division i5 apparent by taking the highest value to token "12" and dividing it by either the hori~ontal or verti-cal number of tokens, 4 or 3 as the case might be and the opposite is the answer. Thu9 12 divided by 3 = 4 or 12 divided by 4 = 3.
Referring to figure 8, the same shows counting by multiples with base 4, because they are in the 4th row 640 binary counting is achieved on the second row or in row 620.
Each row, therefore, i~ uni~ue a~ to its base o counting.
In figure 9, fractions are illustrated where a new sub-matrix 720 is arranged with three vertical columns 721, 722, 723. Each of these columns represents the fraction of "thirds". One third of the value of the total matrix "15" is derived by reading the value of the highest valu~ token in each vertical column. Hence, 1/3 out of 15 = 5, the value at row 721; 2/3 ou~ of 15 = 10, the value of the highest tokan in row 722; and 3/3 out of 15 the ~alue of the highest token in the third column.
It will be apparent to those skilled in the art, from the foregoing, that the front and reverse face of the frame for each matrix could identify the operations which the matrix itself visibly can perform and instructions of use.
Claims (9)
1. An abacus for aiding in the understanding of counting, adding and substracting comprising:
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged so that the value of each numeral is defined by:
(i) N(a,b) = (10a-10) + b where;
a,b are the coordinate positions of the tokens and N is the numeral; and, (d) means for holding the surfaces of each of the tokens in a common coincident plane.
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged so that the value of each numeral is defined by:
(i) N(a,b) = (10a-10) + b where;
a,b are the coordinate positions of the tokens and N is the numeral; and, (d) means for holding the surfaces of each of the tokens in a common coincident plane.
2. An abacus for aiding in the understanding of multiplying and dividing comprising:
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged so that the value of each numeral is defined by:
(i) N(a,b) = a X b where;
a,b are the coordinate positions of the tokens and N is the numeral; and, (d) means for holding the surface of each of the tokens in a common coincident plane.
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the surface of each token arranged so that the value of each numeral is defined by:
(i) N(a,b) = a X b where;
a,b are the coordinate positions of the tokens and N is the numeral; and, (d) means for holding the surface of each of the tokens in a common coincident plane.
3. An abacus for aiding in the understanding of counting, adding, substracting, multiplying and dividing comprising:
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the reverse surface of each token arranged so that the value of each numeral is defined by:
(i) N(a,b) = (10a-10) + b where;
a,b are the coordinate positions of the tokens and N is the numeral; and, (e) means for holding each of the front and reverse surfaces respectively in a front and a reverse continuous coincident plane, the planes respectively parallel to each other.
(a) an open frame carrying a plurality of parallel tracks;
(b) a two dimensional matrix of movable tokens carried by the tracks;
(c) numerals on the reverse surface of each token arranged so that the value of each numeral is defined by:
(i) N(a,b) = (10a-10) + b where;
a,b are the coordinate positions of the tokens and N is the numeral; and, (e) means for holding each of the front and reverse surfaces respectively in a front and a reverse continuous coincident plane, the planes respectively parallel to each other.
4. The abacus as claimed in claim 1, 2 or 3, wherein the frame and tracks and tokens are of molded material.
5. An abacus as claimed in claim 1, 2 or 3, where a = b.
6. An abacus as claimed in claim 1, 2 or 3, where a = b =
10 .
10 .
7. An abacus as claimed in claim 1, 2 or 3, wherein the tracks have lengths longer than the cumulative latter dimension of the tokens.
8. An abacus as claimed in claim 1, 2 or 3, where a = b =
10 and the track have lengths longer than the cumulative latter dimension of the tokens.
10 and the track have lengths longer than the cumulative latter dimension of the tokens.
9. The abacus as claimed in claim 3, wherein the open frame lies in a plane parallel to the other planes.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA000450275A CA1206455A (en) | 1984-03-22 | 1984-03-22 | Manual mathematical calculator |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CA000450275A CA1206455A (en) | 1984-03-22 | 1984-03-22 | Manual mathematical calculator |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA1206455A true CA1206455A (en) | 1986-06-24 |
Family
ID=4127481
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA000450275A Expired CA1206455A (en) | 1984-03-22 | 1984-03-22 | Manual mathematical calculator |
Country Status (1)
| Country | Link |
|---|---|
| CA (1) | CA1206455A (en) |
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2017081670A3 (en) * | 2015-11-12 | 2017-10-26 | Ellis Felicity Anne | An educational apparatus |
-
1984
- 1984-03-22 CA CA000450275A patent/CA1206455A/en not_active Expired
Cited By (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2017081670A3 (en) * | 2015-11-12 | 2017-10-26 | Ellis Felicity Anne | An educational apparatus |
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| MKEX | Expiry |