Smallest Integer: Sum 180, Ratio 5:3:2

Welcome to a fun mathematical puzzle where we need to find the smallest of three positive integers. We're given two key pieces of information: the sum of these three integers is 180, and their ratio is 5:3:2. This means that for every 5 parts of the first integer, there are 3 parts of the second, and 2 parts of the third. Our goal is to pinpoint the exact value of the smallest integer among them. Let's dive into how we can solve this intriguing problem step-by-step.

Understanding Ratios and Sums in Mathematics

In mathematics, understanding ratios and sums is fundamental to solving a wide array of problems, especially those involving proportions and distributions. A ratio, like the 5:3:2 we're dealing with, expresses the relative sizes of different quantities. It tells us how much of one thing there is compared to another. When we talk about the sum of integers, we're simply adding them up to find a total. In this specific problem, the ratio 5:3:2 implies that the three integers can be represented as 5x, 3x, and 2x, where 'x' is a common multiplier. This 'x' is crucial because it maintains the given proportion between the numbers. The sum of these integers is then 5x + 3x + 2x. We are told that this sum equals 180. Therefore, we have an equation: 5x + 3x + 2x = 180. This equation allows us to solve for 'x'. Once we find the value of 'x', we can easily determine the value of each of the three integers. The smallest integer will correspond to the smallest part of the ratio, which is 2. So, the smallest integer will be 2x. The process of solving for 'x' involves combining the terms on the left side of the equation: (5 + 3 + 2)x = 180, which simplifies to 10x = 180. To isolate 'x', we divide both sides of the equation by 10. This gives us x = 180 / 10, resulting in x = 18. Now that we have the value of 'x', we can find the individual integers. The first integer is 5x = 5 * 18 = 90. The second integer is 3x = 3 * 18 = 54. And the third, which is the smallest, is 2x = 2 * 18 = 36. To verify our answer, we can sum these integers: 90 + 54 + 36 = 180. This matches the given sum, confirming our calculations are correct. The question asks for the value of the smallest of the integers, which we found to be 36.

Decoding the Ratio: Breaking Down the Proportions

Let's delve deeper into decoding the ratio to fully grasp how it guides us to the solution. The ratio 5:3:2 means that the whole sum (180) is divided into a total of 5 + 3 + 2 = 10 parts. Imagine you have 10 equal slices of a pie, and these slices are distributed among three people according to the ratio. One person gets 5 slices, another gets 3 slices, and the third gets 2 slices. The total number of slices is 10, which corresponds to our total sum of 180. To find the value of one 'part' or one 'slice' of this pie, we divide the total sum by the total number of parts: 180 / 10 = 18. This value, 18, is our common multiplier, the 'x' we used earlier. Now, to find the actual value of each integer, we multiply its corresponding ratio part by this value of one part (18). The first integer, represented by the ratio 5, is 5 * 18 = 90. The second integer, with a ratio of 3, is 3 * 18 = 54. The third integer, with the smallest ratio of 2, is 2 * 18 = 36. This method clearly illustrates how the ratio dictates the proportion of the total sum each integer receives. The smallest integer is naturally associated with the smallest number in the ratio, which is 2. Therefore, the smallest integer's value is 2 times the value of one part, giving us 36. This systematic breakdown ensures that we don't miss any details and accurately identify the smallest integer. It's like distributing a treasure chest of 180 gold coins according to a specific distribution plan (the ratio 5:3:2) – we first figure out how much each 'share' is worth and then distribute accordingly.

The Algebraic Approach: Solving for the Unknown Multiplier

To formally solve this problem, we employ an algebraic approach, which is a powerful tool in mathematics for handling unknown quantities. We represent the three positive integers based on their given ratio. Let the three integers be $5x$, $3x$, and $2x$. Here, $x$ is a positive constant, our unknown multiplier, that ensures the integers maintain the ratio of 5:3:2. The problem states that the sum of these three integers is 180. Therefore, we can write the equation: $5x + 3x + 2x = 180$ To solve for $x$, we first combine the like terms on the left side of the equation: $(5 + 3 + 2)x = 180$ $10x = 180$ Now, to find the value of $x$, we divide both sides of the equation by 10: $x = \frac{180}{10}$ $x = 18$ With the value of $x$ determined, we can now find the value of each of the three integers. The first integer is $5x = 5 \times 18 = 90$. The second integer is $3x = 3 \times 18 = 54$. The third integer, which is the smallest since it corresponds to the smallest ratio part (2), is $2x = 2 \times 18 = 36$. The question specifically asks for the value of the smallest of the integers. Comparing the values 90, 54, and 36, the smallest integer is indeed 36. This algebraic method provides a clear and structured way to solve ratio problems, ensuring accuracy and efficiency in finding the desired values. It's a testament to how algebra can simplify complex relationships into manageable equations.

Identifying the Smallest Integer: The Power of Proportionality

Ultimately, the core of this problem lies in identifying the smallest integer, which is directly tied to the concept of proportionality. The ratio 5:3:2 inherently tells us the relative sizes of the integers. The number '2' in the ratio is the smallest value, indicating that the integer represented by '2x' will be the smallest among the three. Once we establish the value of our common multiplier, $x$, finding the smallest integer becomes a straightforward multiplication. As we calculated, $x = 18$. Therefore, the smallest integer is $2x = 2 \times 18 = 36$. It's a direct consequence of the proportional relationship defined by the ratio. If the ratio were, for instance, 1:2:7, the smallest integer would correspond to the '1' part. The larger the number in the ratio, the larger the actual integer value, assuming a positive multiplier. This principle of proportionality is a cornerstone of mathematics and applies to countless real-world scenarios, from scaling recipes to understanding economic models. In this particular case, recognizing that the smallest ratio component corresponds to the smallest integer is a key step in quickly arriving at the correct answer, especially in timed tests or when seeking an efficient solution.

Verification: Checking Our Sum and Ratio

Before we conclude, it's always a good practice to perform a verification to ensure our answer is correct. We found the three integers to be 90, 54, and 36. Let's check if they satisfy the two conditions given in the problem: their sum is 180, and their ratio is 5:3:2.

First, let's check the sum: $90 + 54 + 36 = 144 + 36 = 180$ The sum is indeed 180, which matches the information provided in the problem. This is a great first step in confirming our results.

Next, let's check the ratio. We need to see if the ratio 90:54:36 simplifies to 5:3:2. We can do this by dividing each number by their greatest common divisor (GCD). We already know from our calculation that the common multiplier $x$ was 18. So, if we divide each number by 18:

$$\frac{90}{18} = 5$$

$$\frac{54}{18} = 3$$

$$\frac{36}{18} = 2$$

The ratio 90:54:36 simplifies to 5:3:2. This matches the given ratio perfectly.

Since both the sum and the ratio conditions are met, we can be highly confident that our calculated integers are correct, and therefore, the smallest integer, 36, is the correct answer. This verification step is crucial for building confidence in mathematical solutions.

Conclusion: The Smallest Integer Revealed

We embarked on a mathematical journey to find the smallest of three positive integers whose sum is 180 and whose ratio is 5:3:2. Through a combination of understanding ratios, applying algebraic principles, and careful calculation, we determined that the three integers are 90, 54, and 36. The smallest among these is 36. This problem beautifully illustrates how ratios dictate the proportional distribution of a total sum, and how simple algebraic equations can unlock the values of unknown quantities. The steps involved converting the ratio into algebraic expressions ($5x, 3x, 2x$), setting up an equation based on the given sum ($5x + 3x + 2x = 180$), solving for the common multiplier ($x = 18$), and finally identifying the smallest integer ($2x = 36$). Remember, in ratio problems, the smallest part of the ratio always corresponds to the smallest quantity when the multiplier is positive.

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