Use ordinary language (and algebraic symbolism if necessary) to explain the proc

Question

Use ordinary language (and algebraic symbolism if necessary) to explain the process of solution in Diophantus Book I, problem 15. Try to follow the same process with your choice of numbers that satisfy the initial conditions.

To find two numbers such that each, after receiving from the other a given number, is in a given ratio with the remaining number.

Let it be required that the first number, after receiving 30 units from the second number, becomes twice as big as the second number, and the second number, after receiving 50 units from the first, becomes three times the first number.

Let the second number be 1 arithmos plus 30 units, which that number will give. From then on, the first number will be 2 arithmoi minus 30 units, and if it receives 30 units from the second number it becomes twice as big as the second number after giving its 30 units. Also, it is necessary that the second number after receiving 50 units from the first becomes three times the first after giving its 50 units. But if the first number gives 50 units, the remainder will be 2 arithmos minus 80 units, while the second number after receiving 50 units becomes 1 arithmos plus 80 units. Finally, it is necessary that 1 arithmos plus 80 units be 3 times 2 arithmoi minus 80 units; so the least of these numbers, taken 3 times will be equal to the greater number and the arithmos becomes 64 units.

Accordingly, the first number will be 98 units, the second 94 units and these numbers solve the problem.

Explanation / Answer

# To find two numbers such that each, after receiving from the other a given number, is in a given ratio with the remaining number.

Let p and q be the 2 numbers which are in relation with one another . Now let n be the number that p recieves from q so that p and q comes in a given ratio. The same must be true when q recieves a number from p.

the representation would be : p/q = n or p = nq or q = p/n

# Let it be required that the first number, after receiving 30 units from the second number, becomes twice as big as the second number, and the second number, after receiving 50 units from the first, becomes three times the first number.

let the first number be p and the second number be q

=> p + 30 = 2q

q + 50 = 3p

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