Which expression is equivalent to $\frac{e}{f}+\frac{g}{h}$?
To add two fractions with different denominators, we need to find a common denominator. The expression given is \(\frac{e}{f} + \frac{g}{h}\). The denominators are \(f\) and \(h\). The smallest common denominator is the product of these two, which is \(f \times h\) (assuming \(f, h eq 0\)).
To convert \(\frac{e}{f}\) to a fraction with the common denominator \(fh\), we multiply both the numerator and the denominator by \(h\):
\[\frac{e}{f} = \frac{e \times h}{f \times h} = \frac{eh}{fh}\]
Similarly, to convert \(\frac{g}{h}\) to a fraction with the common denominator \(fh\), we multiply both the numerator and the denominator by \(f\):
\[\frac{g}{h} = \frac{g \times f}{h \times f} = \frac{gf}{fh}\]
Now that both fractions have the same denominator, we can add them by summing their numerators:
\[\frac{eh}{fh} + \frac{gf}{fh} = \frac{eh + gf}{fh}\]
Comparing this result with the provided options, we see that it matches the third option, \(\frac{eh + gf}{fh}\).
Imagine you are splitting a pizza with friends. You have \(\frac{e}{f}\) of one pizza and \(\frac{g}{h}\) of another. To find out exactly how much pizza you have in total, you need to add these two fractions. By finding a common denominator, you can combine the portions accurately. This basic skill is essential for everything from cooking with recipes to managing shared resources.