Solving the Equation: x/3 27/45

In this article, we will explore the steps needed to solve the equation x/3 27/45. We will break down the problem into several methods, including simplifying fractions, cross multiplication, and converting improper fractions to mixed numbers. Let's dive in!

Simplifying Fractions

To simplify the fraction on the right side of the equation, we first need to find the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 27 and 45 is 9.

Divide both the numerator and the denominator by 9: [frac{27 div 9}{45 div 9} frac{3}{5}]

Now that we have simplified the fraction, the equation becomes:

[frac{x}{3} frac{3}{5}]

Solving for x

Next, we solve for x. To do this, we can multiply both sides of the equation by 3:

[x 3 cdot frac{3}{5} frac{9}{5}]

Thus, the value of x is 9/5 or 1.8.

Cross Multiplication

Another approach to solving the equation is using cross multiplication. The principle of cross multiplication is:

[frac{a}{b} frac{c}{d} quad Rightarrow quad a cdot d b cdot c]

Applying this method to our problem:

[frac{x}{3} frac{27}{45}] [x cdot 45 3 cdot 27] [45x 81] [frac{45x}{45} frac{81}{45}] [x frac{81}{45}] [x frac{9}{5}] [x 1.8]

Converting to Mixed Numbers and Improper Fractions

The value of x can also be expressed as a mixed number or an improper fraction. Here, we have:

[x frac{9}{5}] as an improper fraction [x 1 frac{4}{5}] as a mixed-fractional number [x 1.8] as a decimal

This gives us a complete solution to the equation x/3 27/45 with multiple representations of the value of x.

Verification

To verify the solution, we can plug the value of x back into the original equation:

[frac{x}{3} frac{27}{45}] [frac{frac{9}{5}}{3} frac{27}{45}] [frac{9}{5 cdot 3} frac{27}{45}] [frac{9}{15} frac{27}{45}] [frac{3}{5} frac{3}{5}]

This confirms that the equation holds true, establishing the root zero x 9/5 of the equation x/3 27/45.

Conclusion

In this article, we explored different methods for solving the equation x/3 27/45. Whether you choose to simplify fractions, use cross multiplication, or convert to mixed numbers and improper fractions, the solution is consistent. Understanding these methods can be invaluable for solving a wide range of mathematical problems.