Problem 49 - Prealgebra

Question

Solve each equation. $$\frac{x}{10}=\frac{270}{6}$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 450 \).

Step 1: Simplify the Right Side

Simplify the fraction on the right side of the equation: \( \frac{270}{6} \). This means dividing 270 by 6.

Step 2: Perform Division

Calculate \( 270 \div 6 \) to simplify the fraction. This results in 45 since 270 divided by 6 is 45.

Step 3: Update the Equation

Rewrite the equation with the simplified fraction: \( \frac{x}{10} = 45 \).

Step 4: Eliminate the Fraction

To solve for \( x \), multiply both sides of the equation by 10 to eliminate the fraction. This results in \( x = 45 \, \times \, 10 \).

Step 5: Perform the Multiplication

Calculate \( 45 \, \times \, 10 \) to find \( x \). This results in \( x = 450 \).

Key Concepts

Simplifying Fractions

In mathematics, simplifying fractions means reducing them to their simplest form. This process involves dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). This makes working with the fraction easier and the results more understandable.
For example, consider the fraction \( \frac{270}{6} \). To simplify this, you need to divide both 270 and 6 by their GCD, which is 6. Here's how you do it:

• Divide the numerator: \( 270 \div 6 = 45 \)
• Divide the denominator: \( 6 \div 6 = 1 \)

After simplification, \( \frac{270}{6} \) becomes 45. Whenever you simplify, always check that the fraction is as reduced as possible by ensuring the numerator and denominator have no common divisors other than 1.

Multiplication in Equations

When solving equations, multiplication is a key operation often used to isolate the variable. The objective is to 'cancel out' denominators or to make the variable stand alone on one side of the equation.
In our solved example, once we simplified \( \frac{270}{6} = 45 \), the equation became \( \frac{x}{10} = 45 \). To solve for \( x \), you need to eliminate the fraction by multiplying both sides of the equation by 10, which is the denominator of the fraction.
Here's the reasoning:

• By multiplying both sides by the denominator, you effectively 'clear' the fraction.
• In our example, we multiplied both sides by 10: \[ \frac{x}{10} \times 10 = 45 \times 10 \]
• This simplifies the equation to: \( x = 450 \)

Remember, whatever operation you apply to one side of the equation, you must also apply to the other side to maintain equality.

Division in Equations

Division is another essential mathematical operation often utilized in equations to simplify parts of the equation or to solve for the variable. When you have division in an equation, reversing it typically involves multiplication.
In our exercise, the fraction \( \frac{x}{10} \) indicated division. To solve the equation, we had to reverse this division by multiplying. When you want to solve for a variable that's part of a division, remember:

• Identify what is being divided, like \( \frac{x}{10} \), which means \( x \div 10 \).
• Multiply both sides by that divisor (10 in this case) to cancel it out.
• This operation effectively helps you determine the value of \( x \) independently.

The key to using division in equations is understanding that you can always perform the opposite operation (multiplication) to help isolate and solve for the unknown variable.