Understanding Ratio and Proportion: Solving for the Smaller Number Given a Ratio of 4:5 and a Larger Number of 450
Understanding the relationship between two numbers given their ratio is a fundamental skill in mathematics. This article will explore how to determine the smaller number when the ratio is 4:5 and the larger number is 450, providing detailed explanations and examples.
Introduction to Ratios and Proportions
In mathematics, a ratio is a comparison of two numbers that indicates how many times one number contains or is contained within the other. Ratios are often expressed as two numbers separated by a colon, such as 4:5. In this ratio, the first number (4) represents a proportion of the whole, while the second number (5) represents a larger proportion.
Solving for the Smaller Number
Given that the ratio of two numbers is 4:5 and the larger number is 450, we need to find the smaller number. Here is a step-by-step breakdown:
1. Let the smaller number be x.
2. The ratio can be set up as follows:
(frac{x}{450} frac{4}{5})
3. To find x, cross-multiply:
5x 4 × 450
4. Calculate 4 × 450:
4 × 450 1800
5. Now we have:
5x 1800
6. Divide both sides by 5:
x (frac{1800}{5}) 360
Therefore, the smaller number is 360.
Alternative Methods
Several alternative methods can be used to solve this problem:
Method 1: Using the ratio directly:
The larger ratio is 5. If 5 parts equal 450, then 1 part equals 90 (450 ÷ 5). The smaller number has 4 parts, so:
4 × 90 360
The smaller number is 360.
Method 2: Using the total parts:
The total number of parts in the ratio is:
4 5 9 parts
Each part is:
450 ÷ 5 90
Therefore, the smaller number is:
4 × 90 360
Discussion and Examples
Let's consider another example to solidify the concept. If the ratio is 4:5, and the larger number is 600, we can follow a similar approach:
1. The larger number (5 parts) is 600.
2. One part is:
600 ÷ 5 120
3. The smaller number (4 parts) is:
4 × 120 480
Thus, the smaller number is 480.
Common Pitfalls and Misconceptions
A common misconception is that if the larger number is 450 and the ratio is 4:5, the smaller number is simply 120. However, this is incorrect because the ratio tells us that the larger number (5 parts) is 450, not 150. The calculation should follow the steps outlined above.
Conclusion
Understanding ratios and proportions is crucial in various fields, including mathematics, physics, and economics. By following the steps outlined in this article, you can confidently solve problems involving ratios, such as finding the smaller number given a larger number and a specific ratio. Remember to always check your calculations and ensure that each step logically follows from the previous one.
Keywords: ratio, proportion, smaller number