Question

Someone is thinking of two numbers that are natural, and one is bigger than the other one by 10, if you multiply those two numbers you get 375 what are those two numbers.

Answer 1

So the two numbers are {15, 25}.

Explanation:

Simultaneous Equations:

xy = 375

x = y + 10

Subsitute the Second One into the First:

(y + 10)y = 375

y^2 + 10y - 375 = 0

y = 15 [and not -25 (natural numbers are not negative)]

Solve for x by Substituing y in the Second Equation:

x = 15 + 10

x = 25

Solution:

So the two numbers are {15, 25}.

Answer 2

15 and 25

Explanation:

The given relations between the two numbers allow you to write a system of equations. Algebraic solution of the system involves the solution of a quadratic equation. The numbers are "natural numbers", so are both positive integers.

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setup

Let x and y represent the smaller and larger of the two numbers, respectively. The given relations are ...

y = x +10 . . . . . . the larger is 10 more than the smaller

xy = 375 . . . . . . their product is 375

solution

Substituting for y in the second equation, we have ...

x(x +10) = 375

Adding (10/2)² will complete the square.

x² +10x +25 = 375 +25 . . . . add 25

(x +5)² = 400 . . . . . . . . . . . write as a square

x +5 = √400 = 20 . . . . . . positive square root

x = 20 -5 = 15

y = x +10 = 25

Those two numbers are 15 and 25.
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The attachment shows a graphical solution of the equations.

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Additional comment

Your number sense tells you that both numbers end in 5. Their geometric mean is √375 ≈ 19.4, so the first numbers that deserve consideration are numbers ending in 5 that are either side of this value: 15 and 25.

You can also consider their arithmetic mean. If that is represented by z, then we have (z-5)(z+5)=375 ⇒ z²-25=375 ⇒ z²=400 ⇒ z=20, and the two numbers are 20±5.