Problem 60 - Prealgebra

Question

Solve each equation. Check your solution. $$28 g=1.4$$

Step-by-Step Solution

Answer

The solution is \(g = \frac{1}{20}\).

Step 1: Isolate the Variable

We are given the equation \(28g = 1.4\). To solve for \(g\), we need to isolate it. To do this, divide both sides of the equation by 28. This gives \(g = \frac{1.4}{28}\).

Step 2: Perform the Division

Now, calculate \(\frac{1.4}{28}\). First, convert this division into a form that's easier to handle without a calculator: \(1.4\) can be rewritten as \(\frac{14}{10}\). Now, the equation becomes \(g = \frac{14}{10} \times \frac{1}{28}\). Simplify this expression: \(g = \frac{1}{20}\).

Step 3: Check the Solution

To ensure our solution is correct, substitute \(g = \frac{1}{20}\) back into the original equation \(28g = 1.4\). Calculate \(28 \times \frac{1}{20} = \frac{28}{20} = 1.4\). Since this matches the right side of the original equation, \(g = \frac{1}{20}\) is indeed correct.

Key Concepts

Isolating the Variable

When solving equations, one of the first steps is to isolate the variable. In our given problem, the equation is in the form of \(28g = 1.4\). Here, "g" is the variable we want to find. Isolating the variable means rearranging the equation such that "g" appears alone on one side, typically the left. By dividing both sides by 28, which is the coefficient of the variable "g", we eliminate its association with "g". Thus, the structure is adjusted to \(g = \frac{1.4}{28}\). This step is essential as it simplifies the equation into a more straightforward form where the value of "g" can be easily calculated. The goal, every time, is to make "g" stand alone so that what remains is an equation in its simplest form which allows us to determine the value of "g" efficiently.

Division in Equations

Once we have isolated the variable, the next step often involves division. In the example, we arrive at \(g = \frac{1.4}{28}\). Performing division here means essentially finding out how many times 28 fits into 1.4, when expressed as a fraction. To make calculations easier, converting numbers into more manageable fractions can help. For instance, 1.4 can be expressed as \(\frac{14}{10}\), enabling a clearer approach to division. By rewriting the division expression as \(g = \frac{14}{10} \times \frac{1}{28}\), we simplify the division process by mean of multiplication of fractions. Multiplication of fractions requires multiplying nominator with nominator and denominator with denominator, simplifying it to \(g = \frac{1}{20}\).

• Remember that division by a number is the same as multiplication by its reciprocal.
• Converting decimals to fractions can simplify calculations, especially when working without a calculator.
• Simplifying fractions earlier in the problem makes it easier to find the final answer.

Check Solutions to Equations

After solving an equation, verifying your solution ensures accuracy. This is crucial because simple mistakes can occur during calculations. To check a solution, substitute the value you found back into the original equation. For example, with \(g = \frac{1}{20}\), substitute it into the original equation \(28g = 1.4\). Calculate \(28 \times \frac{1}{20}\). This should equal 1.4. Breaking this down, \(28 \times \frac{1}{20} = \frac{28}{20}\), which simplifies to 1.4 upon division. Since 1.4 equals the original right side of the equation, this confirms that our solution is correct. Verification is like a tool to cross-check your understanding and calculations. It gives confidence that what you've done is right, especially in more complex problems.

• Replacing the variable back into the original equation is a reliable method to check solutions.
• If the original equation equals after substitution, the solution is correct.
• Taking the time to check solutions can help avoid errors and deepen your comprehension of the problem-solving process.