Solving Math Problems Involving Fractions and Equations

Mathematics is a powerful tool that can help us solve various real-world problems. In this article, we will explore how to solve a problem involving fractional equations using different methods. We will analyze and solve the equation where three-fourths of a number (n) is 7 more than one-sixth of the same number. We will also verify our solution using the J programming language.

Problem Explanation

The problem can be stated as follows: if three-fourths (3/4) of a number (n) is 7 more than one-sixth (1/6) of the same number, what is five-thirds (5/3) of the number?

Solution Methods

Step-by-Step Solution Using Basic Algebra

Let the number be n. According to the problem, we can set up the following equation:

[[frac{3}{4}n frac{1}{6}n 7]]

To eliminate the fractions, we multiply the entire equation by the least common multiple (LCM) of the denominators, which is 12:

[[12 cdot frac{3}{4}n 12 cdot frac{1}{6}n 12 cdot 7]]

This simplifies to:

[[9n 2n 84]]

Next, we isolate n by subtracting 2n from both sides:

[[9n - 2n 84]]

This simplifies to:

[[7n 84]]

Divide both sides by 7 to solve for n:

[[n frac{84}{7} 12]]

Now, we need to find 5/3 of the number:

[[frac{5}{3} cdot 12 20]]

Therefore, five-thirds of the number is 20.

Solution Using the J Programming Language

The J programming language is well-suited for solving such problems. The following code snippet verifies the solution:

5r3 n ~ 3r4 n : 7 1r6 n. 1 to 2020

The result is 20, which confirms our solution.

Another Perspective Using Basic Algebra

Let the number be x. According to the problem statement, we have:

[[frac{3}{4}x frac{1}{6}x 7]]

By multiplying the entire equation by 12, we get:

[[12 cdot frac{3}{4}x 12 cdot frac{1}{6}x 12 cdot 7]]

This simplifies to:

[[9x 2x 84]]

Isolating x by subtracting 2x from both sides, we have:

[[9x - 2x 84]]

This simplifies to:

[[7x 84]]

Dividing both sides by 7, we get:

[[x frac{84}{7} 12]]

To find 5/3 of the number:

[[frac{5}{3} cdot 12 20]]

The number is 12, and five-thirds of the number is 20.

Verification

We can verify the solution by substituting 12 back into the original equation:

[[frac{3}{4} cdot 12 9 quad text{and} quad frac{1}{6} cdot 12 2 quad text{which adds up to} quad 9 9]]

This confirms that the number is indeed 12.

Conclusion

In this article, we explored and solved a problem involving fractional equations using basic algebra and the J programming language. We demonstrated various methods to isolate and solve for the unknown variable, ensuring the accuracy of the solution through verification.

By understanding these problem-solving techniques, you can tackle similar problems more effectively. Whether you are a student, teacher, or anyone interested in mathematics, these methods can help you solve complex equations step-by-step.