It's a classic kind of puzzle, isn't it? You're given a few numbers, and you're asked to find a missing piece that makes them all fit together in a specific way – in this case, forming a proportion. When we talk about proportions, we're essentially looking at relationships between numbers, where the ratio of one pair is equal to the ratio of another pair. The fundamental rule we lean on here is that in any proportion, the product of the 'means' (the inner two numbers) must equal the product of the 'extremes' (the outer two numbers).
So, let's take our given numbers: 3, 4, and 15. We need to find a number, let's call it 'x', that can form a proportion with these three. The interesting part is that 'x' can be either the smallest possible value or the largest possible value, depending on how we arrange the numbers.
Finding the Smallest Number
To get the smallest possible value for 'x', we want to make sure that 'x' is involved in a way that 'dilutes' its value as much as possible. Think about it: if we want 'x' to be small, we should pair it with the largest of the known numbers (15) as one of the 'means' or 'extremes', and then use the product of the two smaller known numbers (3 and 4) to balance it out.
If we set up the proportion like this: 3 : 15 = x : 4, then the product of the means (15 * x) must equal the product of the extremes (3 * 4). So, 15x = 12. Solving for x, we get x = 12/15, which simplifies to 4/5 or 0.8. This is indeed a small number!
Another way to think about it is if we set it up as 3 : 4 = 15 : x. Here, 3 * x = 4 * 15, so 3x = 60, and x = 20. This isn't our smallest. What if we try 3 : x = 4 : 15? Then 3 * 15 = 4 * x, so 45 = 4x, and x = 45/4 = 11.25. Still not the smallest.
It turns out that by strategically placing the known numbers, we can isolate the smallest possible value for 'x'. The key is to have the largest known number (15) multiplied by 'x' (as a mean) and the two smaller numbers (3 and 4) as the extremes. So, 3 * 4 = 15 * x, which gives us x = 12/15 = 4/5.
A proportion that works for this smallest value could be 3 : 15 = 4/5 : 4. Let's check: 3 * 4 = 12, and 15 * (4/5) = 12. Perfect!
Finding the Largest Number
Now, for the largest number. To make 'x' as large as possible, we want to pair it with the smallest of the known numbers (3) and use the product of the two larger known numbers (4 and 15) to balance it.
If we set up the proportion as 3 : 4 = 15 : x, then the product of the means (4 * 15) must equal the product of the extremes (3 * x). So, 60 = 3x. Solving for x, we get x = 60/3 = 20. This is a significantly larger number.
Let's consider other arrangements to be sure. If we try 3 : x = 4 : 15, we already found x = 11.25. If we try x : 3 = 4 : 15, then x * 15 = 3 * 4, so 15x = 12, and x = 12/15 = 4/5. Definitely not the largest.
The strategy for the largest number is to have the smallest known number (3) multiplied by 'x' (as an extreme) and the two larger numbers (4 and 15) as the means. So, 4 * 15 = 3 * x, which gives us x = 60/3 = 20.
A proportion that works for this largest value could be 3 : 4 = 15 : 20. Let's check: 3 * 20 = 60, and 4 * 15 = 60. It holds true!
So, by understanding the fundamental property of proportions – that the product of the means equals the product of the extremes – we can strategically arrange our known numbers to find both the smallest and largest possible values for the missing number.
