Proportion is a fundamental concept in mathematics that deals with the equality of ratios. When four quantities are in proportion, it implies that the ratio of the first two quantities is equal to the ratio of the last two quantities. This can be represented as:...
a : b :: c : d
where a, b, c, and d represent the four quantities. This notation reads as "a is to b as c is to d."
A key property of proportions is that the product of the means (b and c) is equal to the product of the extremes (a and d):
a * d = b * c
Applying Proportion to the Problem
In this problem, we are given four numbers: 2, 7, 17, and 37. Our goal is to find a constant number that, when added to each of these numbers, will make them proportional. Let's represent this constant number as 'x'.
Therefore, after adding 'x' to each number, we get the following sequence: 2 + x, 7 + x, 17 + x, and 37 + x. For these numbers to be in proportion, the following equation must hold:
(2 + x) * (37 + x) = (7 + x) * (17 + x)
Solving the Equation
Expanding the equation, we get:
74 + 2x + 37x + x² = 119 + 7x + 17x + x²
Simplifying the equation, we get:
39x - 45 = 24x
Solving for x, we get:
15x = 45
x = 3
Verification
To verify that adding 3 to each of the original numbers creates a proportion, let's check the ratios:
(2 + 3) : (7 + 3) :: (17 + 3) : (37 + 3)
5 : 10 :: 20 : 40
The ratios are indeed equal, confirming that adding 3 to each of the original numbers results in a proportional sequence.
Conclusion
Therefore, the constant number that must be added to each of 2, 7, 17, and 37 to make them in proportion is 3. By adding 3 to each number, we obtain a proportional sequence: 5, 10, 20, and 40, where the ratio of the first two numbers is equal to the ratio of the last two numbers.