ABACUS: MYSTERY OF THE BEAD
The Bead Unbaffled - An Abacus Manual

Linear Equations - Fernando Tejón

In order to solve the following examples we use the Fernando Tejón's multifactorial multiplication method. As an alternative, Kojima's standard method could be used if preferred.

SYSTEM OF LINEAR EQUATIONS

Using the soroban, it's also possible to solve systems of linear equations. Here's a classic problem solved in two different ways:


PROBLEM 1:

In a farmyard there are both chickens and rabbits. Altogether there are 35 animals with a total of 94 legs. How many chickens and rabbits are there in the farmyard?

SOLUTION A.

One can surmise; if all the animals were rabbits the total number of legs would be 140 (4 * 35 = 140). On the soroban first place 35 and multiply by 4 using the Fernando Tejón's Multifactorial Multiplication method:

B+3 A+5
. FEDCBA
. 000035

3 * 3 = 09 - > C+1 B-1
3 * 5 = 15 - > B+2 A-5
. FEDCBA
. 000140

The actual number of legs totals 94, which is obviously smaller than 140. Therefore we know that not all the animals are rabbits. In other words some of the animals are chickens, but how many?

Supposing initially that the 35 animals are rabbits, each rabbit replaced by a hen loses a total of 2 legs (-4 + 2 = -2). The difference between 140 and 94 divided by 2 will give us the number of chickens:

First subtraction 140 - 94 = 46:
C-1 A+5+1
. FEDCBA
. 000046

Next 46 is divided by 2:
B-2 A-5+2
. FEDCBA
. 000023

The soroban shows that the number of chickens is 23. With this we can determine the number of rabbits. (35 - 23 = 12), which can be done mentally or easily with the soroban.

***An alternative using negative numbers***

The last part of the solution can also be solved by using Kojima's methods for working with negative numbers.

Since 23 is already displayed on the soroban, the subtraction could become:
23 - 35 = -12 instead of 35 - 23 = 12:
[C+1 C-1] B+5+2 B-1 A+5 (or also: [C+1 C-1] B+5+1 A+5)
. FEDCBA
. 000088
As Kojima shows us, the final result is obtained by adding 1 to the complementary number. The complementary number for 88 is 11; this becomes 11 + 1 = 12.

In the farmyard there are 23 chickens and 12 rabbits.


SOLUTION B.

Algebra can also easily solve the problem.

If "x" is the number of chickens and "y" is the number of rabbits, the solution to the problem is obtained by using the system of linear equations:
x + y = 35
2*x + 4*y = 94.

There are many methods to solve systems of linear equations, among them is the Gaussian-Jordan method of elimination which works well on the soroban.

Express the equations as a vector:
x + y = 35 -> [001001035]
2*x + 4*y = 94 -> [002004094]

Place each vector on the soroban, the first vector (V1) on the rods of R through J and the second (V2) on the rods of I through A (the soroban is used to display each vector in this way):
. RQPONMLKJIHGFEDCBA
. 001001035002004094

V1 is multiplied by 2 and the result is then subtracted from V2:
P+1 M+1 K+5-1 J-5
G-2 D-5+3 B-5-2
. RQPONMLKJIHGFEDCBA
. 002002070000002024

Now subtract V2 from V1:
M-2 K-5+2 J+5+1
. RQPONMLKJIHGFEDCBA
. 002000046000002024

Finally V1 and V2 are each divided by 2:
P-1 K-2 J-5+2 D-1 B-1 A-2
. RQPONMLKJIHGFEDCBA
. 001000023000001012

The solution is on the soroban: there are 23 chickens (x = 23) and 12
rabbits (y = 12).

Warm greetings

Fernando Tejón.

Another example from Gary Flom
 

▪ Abacus: Mystery of the Bead
▪ Advanced Abacus Techniques

September 2008
Fernando Tejón
Totton Heffelfinger