Problem 59 - Prealgebra

Question

Solve each equation. Check your solution. $$15=60 h$$

Step-by-Step Solution

Answer

The solution is \( h = \frac{1}{4} \).

Step 1: Isolate the Variable

We begin by isolating the variable \( h \). This means we need to solve for \( h \) in the equation. The equation given is \( 15 = 60h \). To isolate \( h \), we need to divide both sides of the equation by 60. This gives us:\[ h = \frac{15}{60} \]

Step 2: Simplify the Fraction

Next, we simplify the fraction \( \frac{15}{60} \). We look for the greatest common divisor (GCD) of 15 and 60, which is 15. Dividing both the numerator and the denominator by 15, we get:\[ h = \frac{15 \div 15}{60 \div 15} = \frac{1}{4} \]

Step 3: Check the Solution

Now, we check our solution to ensure it's correct. Substitute \( h = \frac{1}{4} \) back into the original equation to see if both sides are equal:\[ 15 = 60 \times \frac{1}{4} \]Calculate the right side:\[ 60 \times \frac{1}{4} = 60 \div 4 = 15 \]Since both sides of the equation are equal, the solution is verified.

Key Concepts

Isolate the Variable

When solving linear equations, the first step is often to isolate the variable. In this context, the variable is the letter that represents an unknown number, which in our example is \( h \). To isolate \( h \), our goal is to get \( h \) on one side of the equation by itself. The equation given is \( 15 = 60h \). Here, \( h \) is being multiplied by 60. To reverse this operation and solve for \( h \), we need to divide both sides of the equation by 60. This operation does not change the equality, but it simplifies the equation to:

• \( h = \frac{15}{60} \)

This process ensures that \( h \) stands alone on one side of the equation, allowing us to further evaluate and find its value.

Simplify the Fraction

Once the variable \( h \) is isolated, the next step is to simplify the fraction \( \frac{15}{60} \). Simplifying fractions involves finding the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 15 and 60 is 15. Simplifying means we divide both 15 and 60 by their GCD:

• The numerator becomes \( 15 \div 15 = 1 \)
• The denominator becomes \( 60 \div 15 = 4 \)

Thus, simplifying \( \frac{15}{60} \) results in \( \frac{1}{4} \). Recognizing this step is crucial because it reduces the fraction to its simplest form, making further computation and interpretation much more straightforward.

Check the Solution

Checking your solution is an essential step to ensure accuracy. In this case, we need to verify that \( h = \frac{1}{4} \) satisfies the original equation \( 15 = 60h \). To do this, substitute \( h = \frac{1}{4} \) back into the equation:

• Calculate \( 60 \times \frac{1}{4} \), which equals \( 60 \div 4 = 15 \)

Since the left side of the equation \( 15 \) matches the right side after substitution, \( 15 \), the solution \( h = \frac{1}{4} \) is verified as correct. This step ensures that no errors were made during the process, and confirms the accuracy of your work.