Question

Solve the equation. $$\frac{c}{4}=\frac{6}{8}$$

Short answer

c = 3. The solution to the given equation is c = 3.

Step-by-step solution

Identify the problem

We identify the equation that needs to be solved which is \( \frac{c}{4} = \frac{6}{8} \). We need to find the value of 'c'.

Cross-multiply

In order to solve for 'c', we cross multiply the 4 with the 6 and the 8 with 'c'. This results in \(8c = 24\).

Solve for 'c'

To isolate 'c' and find its value, we divide both sides of the equation by 8. Thus, \(c = \frac{24}{8}\).

Key concepts

Cross Multiplication

Cross multiplication is a powerful technique used to solve equations that involve two fractions set equal to each other. When you encounter an equation like \( \frac{c}{4} = \frac{6}{8} \), cross multiplication helps you eliminate the fractions and find a simpler equation that's easier to solve. To cross multiply, you take the numerator of the first fraction and multiply it by the denominator of the second fraction, and vice versa, setting the two products equal to each other. In our example:
\[ 4 \times 6 = c \times 8 \]
After performing the multiplication, you'll end up with an equation without fractions: \( 24 = 8c \). This simplifies the process of finding the unknown variable, leading us to the next step of isolating the variable to solve for its value.

Isolating Variables

Isolating the variable, in essence, means to get the unknown variable on one side of the equation and everything else on the other side. This is a fundamental step in solving simple equations, and the goal is to have the variable with a coefficient of one. After cross multiplying our equation \( \frac{c}{4} = \frac{6}{8} \), we got to \( 8c = 24 \).
To isolate the variable 'c', we need to undo the multiplication by 8. We achieve this by dividing both sides of the equation by 8, effectively reversing the previous operation. So, our equation becomes:
\[ \frac{8c}{8} = \frac{24}{8} \]
Simplifying both sides gives us \( c = 3 \), which is our final answer. The process of isolating the variable is a key skill to master for algebra and beyond, as it applies to a wide range of mathematical problems.

Equivalent Fractions

When working with fractions, the concept of equivalent fractions is crucial. Two fractions are considered equivalent if they have the same value when simplified. For instance, \( \frac{6}{8} \) can be simplified to \( \frac{3}{4} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2 in this case.
Understanding equivalent fractions is important for simplifying equations and comparing fractions. It's also a handy way to check your work when solving equations. In our original equation \( \frac{c}{4} = \frac{6}{8} \), recognizing that \( \frac{6}{8} \) is equivalent to \( \frac{3}{4} \) could help us see that the value of 'c' must be 3, because \( \frac{c}{4} \) would need to be equivalent to reduce to the same value of the fraction. Having a strong grasp of equivalent fractions is beneficial not only for solving equations, but also for other applications in mathematics, such as finding common denominators or simplifying complex expressions.