Smallest Integer Calculation Sum And Ratio Of 3 Integers

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The sum of 3 positive integers is 180, and the ratio of the integers is 5:3:2. What is the value of the smallest of the integers? F. 18 G. 36 H. 54 T. 90

Solution

To solve this problem, we need to find three positive integers that add up to 180 and have a ratio of 5:3:2. Let's break down the solution step by step.

1. Understanding Ratios

The ratio 5:3:2 means that the three integers can be represented as 5x, 3x, and 2x, where x is a common factor. This implies that the integers are proportional to each other, maintaining the given ratio. Understanding ratios is crucial in solving many mathematical problems, especially those involving proportions and scaling. The ratio provides a clear relationship between the quantities, making it easier to determine their individual values when the total sum or another related parameter is known.

2. Setting Up the Equation

Since the sum of the three integers is 180, we can write the equation as:

5x + 3x + 2x = 180

This equation represents the sum of the three integers in terms of the common factor 'x'. By setting up the equation, we can now solve for 'x' and subsequently find the values of the individual integers. The equation accurately translates the problem's condition into a mathematical statement, allowing us to use algebraic techniques to find the unknown value of 'x'. The ability to set up equations from word problems is a fundamental skill in algebra, enabling us to model and solve a wide range of real-world scenarios.

3. Solving for x

Combining the terms on the left side, we get:

10x = 180

Dividing both sides by 10, we find:

x = 18

Solving for 'x' involves simplifying the equation and isolating the variable. This step is crucial because the value of 'x' will be used to determine the values of the three integers. The arithmetic operations performed here (combining like terms and dividing both sides by a constant) are fundamental algebraic techniques used to solve linear equations. Once 'x' is found, it acts as a key to unlocking the individual values of the integers in the ratio.

4. Finding the Integers

Now that we know x = 18, we can find the three integers:

  • First integer: 5x = 5 * 18 = 90
  • Second integer: 3x = 3 * 18 = 54
  • Third integer: 2x = 2 * 18 = 36

Calculating the integers involves substituting the value of 'x' back into the expressions 5x, 3x, and 2x. This step directly applies the ratio concept to find the individual values of the integers. Each integer is a multiple of 'x', determined by its corresponding ratio number. The arithmetic here is straightforward multiplication, but it's a critical step in translating the value of 'x' into the actual integers that satisfy the problem's conditions.

5. Identifying the Smallest Integer

The three integers are 90, 54, and 36. The smallest of these is 36.

Identifying the smallest integer involves comparing the three calculated values. This is a simple but crucial step to answer the specific question asked in the problem. The comparison provides the final answer, which is the value of the smallest integer in the set. It ensures that we address the original question directly and provides a clear, concise solution.

Answer

The value of the smallest of the integers is 36.

Thus, the correct answer is G. 36.

Additional Notes

Alternative Approach: Proportional Parts

Another way to think about this problem is in terms of proportional parts. The ratio 5:3:2 means that the whole (180) is divided into 5 + 3 + 2 = 10 parts. Each part is worth 180 / 10 = 18. The integers are then 5 parts (5 * 18 = 90), 3 parts (3 * 18 = 54), and 2 parts (2 * 18 = 36). This method provides an alternative perspective on solving ratio problems, emphasizing the concept of dividing a whole into proportional segments. It can be particularly useful for visualizing the problem and understanding the distribution of the total value among the different parts.

Importance of Ratios in Problem Solving

Ratios are a fundamental concept in mathematics and are used extensively in various fields, including finance, science, and engineering. Understanding how to work with ratios is crucial for solving a wide range of problems. Ratios help us compare quantities and understand their relative sizes. They are also essential for scaling quantities up or down while maintaining proportions. Mastering ratio problems enhances problem-solving skills and provides a valuable tool for mathematical reasoning.

Common Mistakes to Avoid

When solving ratio problems, common mistakes include: Misinterpreting the ratio: Ensure you understand what each part of the ratio represents. Incorrectly setting up the equation: Double-check that your equation accurately represents the problem's conditions. Arithmetic errors: Be careful with your calculations to avoid mistakes in finding the value of x or the integers. Failing to answer the question asked: Always make sure you are providing the specific value requested in the problem (in this case, the smallest integer). Avoiding these common mistakes can significantly improve accuracy and efficiency in solving ratio-based problems.

Practice Problems

To reinforce your understanding of ratios, try solving similar problems. For example:

  1. The ratio of apples to oranges in a basket is 4:3. If there are 28 apples, how many oranges are there?
  2. Divide 360 in the ratio 2:3:5. What is the largest part?
  3. The angles of a triangle are in the ratio 1:2:3. Find the measure of each angle.

Solving these practice problems will solidify your understanding of ratios and improve your problem-solving skills. Each problem offers a slightly different context for applying the concept of ratios, helping you develop a versatile approach to solving such problems.

Conclusion

Solving the problem involves setting up an equation based on the given ratio and the sum of the integers. By finding the common factor and then calculating the individual integers, we can easily identify the smallest one. Remember, ratios are a powerful tool in mathematics, and mastering them can help solve a variety of problems efficiently. The key to solving ratio problems lies in understanding the proportional relationships and setting up accurate equations. Consistent practice and attention to detail will enhance your proficiency in this area of mathematics.