Solving the Ratio Problem: Finding the Numbers with a Given Gap

In this article, we will explore a common category of ratio problems in mathematics. Specifically, we will tackle the problem of finding two numbers in a given ratio that also differ by a certain amount. This type of problem is a fundamental concept in algebra and helps build a strong foundation in mathematical reasoning.

The Verbal Statement of the Problem

The problem at hand is to find two numbers that are in a ratio of 4:7 and have a difference of 69. This means that if we denote the numbers as 4x and 7x, the difference between them should equal 69.

Solution 1: Using Simple Algebraic Manipulation

Let's use the method of simple algebraic manipulation to solve this problem. We start by setting up the equation based on the given information:

7x - 4x 69

Hence, 3x 69

Hence, x 23

Now that we have solved for x, we can determine the two numbers:

4x 4 * 23 92

7x 7 * 23 161

Therefore, the two numbers are 92 and 161.

Solution 2: Using the Larger and Smaller Number Method

Another approach is to define the larger number as x and the smaller number as (x - 69). We then solve the equation based on the given ratio:

x / (x - 69) 4 / 7

By cross-multiplying, we get:

7x 4(x - 69)

7x 4x - 276

3x 276

x 92

The smaller number is then:

92 - 69 23

So, the two numbers are 92 and 23. This solution aligns with the previous one.

Solution 3: Using a Simple Hit-and-Trial Method

A more intuitive approach is to use a hit-and-trial method, given the gap between the numbers (69) and the difference in the ratio (3). We can divide 69 by 3 to get 23. Then, we can assume the numbers to be 23 * 4 and 23 * 7. This gives us:

23 * 4 92

23 * 7 161

This again confirms that the two numbers are 92 and 161.

Solution 4: Detailed Calculation with Fractions

An alternative method involves setting up the equation with fractions:

x - y 68

x 4y

4y - y 68

3y 68

y 68 / 3 ≈ 22.67

x - y 68

x 68 22.67 ≈ 90.67

The numbers are approximately 22.67 and 90.67.

Solution 5: Complex Fractional Method

Another complex method involves a more complicated set of equations and fractions:

x - y 68

x 4y

4y - y 68

3y 68

y 68 / 3 22 2/3

x - 68 / 3 68

x 68 68 / 3 90 2/3

Therefore, the numbers are 22 2/3 and 90 2/3.

In conclusion, solving ratio problems by finding numbers with a given gap involves a variety of methods. Simple algebraic manipulation or intuitive methods can provide clear solutions. Both methods confirm that the two numbers in the ratio 4:7 with a difference of 69 are 92 and 161, or approximately 22.67 and 90.67.

Conclusion

The solutions to the ratio problem of finding two numbers in a given ratio with a specific gap are consistent across various methods. By understanding and practicing these methods, students can develop strong problem-solving skills in mathematics.

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