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To create an equivalent fraction, we often need to transform the existing fraction to match a new denominator while retaining its value. This is done by using the mathematical concept known as the "multiplication factor." A multiplication factor is a number used to multiply both the numerator and the denominator in a fraction to produce an equivalent fraction.

Consider why the multiplication factor is crucial: It allows us to scale the original fraction to a new one with a specified denominator while keeping the same proportion. In our example, we need to convert the denominator of \(\frac{7}{8}\) into 40.

To find the multiplication factor, we simply divide the desired denominator by the original denominator. In the formula:

• Original denominator = 8
• Desired denominator = 40

Thus, the multiplication factor is \(\frac{40}{8} = 5.\) This factor tells us how much we need to scale up both the numerator and the denominator to maintain the fraction's value.

The numerator might seem trivial, but it plays a vital role in forming equivalent fractions as it determines how many parts of the whole we're discussing. In the original fraction, \(\frac{7}{8}\), 7 is our numerator.

To convert this into an equivalent fraction with a new denominator, we need to adjust our numerator using the multiplication factor found previously. By multiplying the original numerator by the multiplication factor, we obtain the numerator of our equivalent fraction.

Here's how you can calculate it:

• Original numerator = 7
• Multiplication factor = 5

The calculation becomes \(7 \times 5 = 35.\)

Thus, 35 stands as the new numerator of the equivalent fraction, ensuring that the value of the fraction remains identical despite its altered form.

The denominator is the bottom part of a fraction which represents how many equal parts the whole is divided into. It is vital in understanding the fraction itself. In our task, we were instructed to change the denominator from 8 to 40.

Understanding the importance of the denominator helps us grasp how fractions are adjusted while maintaining their original value. The original denominator (8) needs to be multiplied by our determined multiplication factor (5) to get the new denominator, 40.

Although we don't physically change the original denominator in calculations, we use this multiplication factor to affect both numerator and denominator uniformly to find the equivalent fraction:

• Original denominator = 8
• Desired denominator = 40
• The multiplication factor needed = \(\frac{40}{8} = 5\)

The new fraction \(\frac{35}{40}\) mirrors the original fraction's value, thanks to this calculated transformation of both numerator and denominator.