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When dealing with equations involving fractions, understanding fraction multiplication is essential. Multiplying fractions means you multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For instance, if you have to multiply \(\frac{1}{4} n\) by 4 to isolate \(n\), you multiply the entire side of the equation by 4: \(4 \times \frac{1}{4} n\).

This turns into: \(\left(4 \times \frac{1}{4}\right)n = \frac{7}{10} \times 4\).
Since \(4 \times \frac{1}{4} = 1\), it simplifies to: \(n = 4 \times \frac{7}{10}\).

So, \( \frac{7}{10} \times 4 = \frac{7 \times 4}{10} = \frac{28}{10}\). Now, we have that \(n = \frac{28}{10}\). Great! Next, let's understand 'isolating the variable'.

Isolating the variable means getting the variable, in this case, \(n\), on one side of the equation by itself. To do this, you'll often need to perform operations inversely.

In our example: \(\frac{1}{4} n = \frac{7}{10}\), the variable \(n\) is multiplied by \(\frac{1}{4}\). To isolate \(n\), you need to do the opposite operation, which is multiplying by the reciprocal (which is 4 in this case).

By multiplying both sides by 4, we essentially 'cancel out' the \(\frac{1}{4}\) on the left side:
\(4 \times \frac{1}{4} n = 4 \times \frac{7}{10}\).

This simplifies to: \(n = \frac{28}{10}\). Finally, make sure the fraction is in the simplest form.

Once you've isolated the variable and have an answer, it typically needs to be simplified. Simplifying a fraction means reducing it to its smallest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).

In our example, we have \(n = \frac{28}{10}\). To simplify this:

• Find the GCD of 28 and 10, which is 2.
• Divide both the numerator and the denominator by 2.

Here’s the math:
\( \frac{28}{10} \div \frac{2}{2} = \frac{28 \div 2}{10 \div 2} = \frac{14}{5}\). So, \(n = \frac{14}{5}\) is the simplified form.

Understanding these core concepts will help you in solving fraction equations more confidently.