Solving for the Smaller Number Using Given Ratio and Larger Number

When faced with a problem involving the ratio of two numbers and the value of one of the numbers, it is often straightforward to determine the other number. In this article, we will explore the steps to find the smaller number when the ratio and the larger number are provided.

Step-by-Step Solution

Given problem: The ratio of two numbers is 5:6 and the larger number is 72.

To solve this, let's break it down:

1. Define the Variables:

Let the smaller number be x. Let the larger number be given as 72. The ratio is expressed as: x : 72 5 : 6.

2. Set Up the Equation:

Using the ratio, we can write the following equation:

x72frac{x}{72}72x? 65?

65?

3. Cross Multiply to Solve for x:

6x 5 × 72

4. Calculate the Product:

5 × 72 360

5. Form and Solve the Equation:

6x 360

6. Divide Both Sides by 6:

x 360 / 6 60

Therefore, the smaller number is 60.

Alternative Approaches

There are other ways to solve for the smaller number based on the same given information. Here are a few additional methods:

Using the Unit Number Approach: p > Let k be the common unit number, then both numbers are 5k and 7k. Since their difference is 60, we can set up the equation: 7k - 5k 60 2k 60 k 30 The greater number 7k 7 × 30 210 The smaller number 5k 5 × 30 150

Another Method:

Let the two numbers be x and y, with y being the greater number. The ratio is x / y 5 / 7. Rearranging the equation: 7x 5y Substitute y - x 60: y - 5/7y 60, simplifying to 2/7y 60 y 210, and x 150

Conclusion

Understanding ratios and how to apply them to find the values of numbers is a fundamental skill in mathematics. By following the steps outlined above, you can easily determine the smaller number when given the ratio and the larger number. Whether using cross-multiplication, the unit number approach, or other methods, the core principles remain the same.