Consider this scenario:
Marissa has a job that pays her m dollars an hour. Her manager delivers some good news: Her wages are being raised by 5%. As shown in previous examples, Marissa’s new earnings per hour can be expressed as m + 0.05m because 5% is equal to 0.05. That amount can also be written more compactly as 1.05m .
Marissa’s new earnings can also be written using fractions, however. Because 5% is equal to 120 , think of Marissa’s new earnings per hour as m + 120m , which can be written as m + m20 . The fact that this expression is equivalent to m + 0.05m demonstrates that increasing a number by 5% is the same as multiplying it by 1120 .
To recap:
Marissa's new earnings per hour can be expressed as m + 0.05m or 1.05m.
Marissa's new earnings per hour can also be expressed as m + 120m or m + m/20.
Both expressions are equivalent and demonstrate that increasing a number by 5% is the same as multiplying it by 1.05 or 1120.
To summarize, the scenario presented involves Marissa receiving a wage raise of 5%. Her new earnings per hour can be expressed in three different ways:
1. In decimal form: Marissa's new earnings per hour can be represented as m + 0.05m or 1.05m.
2. In fractional form: The 5% raise can be written as m + (1/20)m or m + m/20.
3. Using the conversion factor: Increasing a number by 5% is equivalent to multiplying it by 1.05 or dividing it by (1+0.05) which simplifies to 1.05 or 1120.
In this scenario, Marissa's job pays her "m" dollars per hour. Her manager has announced a raise of 5% in her wages.
To calculate Marissa's new earnings per hour, you can use the formula:
New Earnings = m + (0.05m)
This is because a 5% raise can be expressed as 0.05. You add this amount to her initial earnings "m" to get the new amount.
Simplifying this expression further, you can factor out the common term "m":
New Earnings = (1 + 0.05)m = 1.05m
So Marissa's new earnings per hour can also be written as 1.05m.
However, there's another way to express this using fractions. Since 5% is equivalent to 5/100, we can rewrite 5% as 5/100 or 1/20 (because 5 and 100's common factor is 20).
Now, instead of adding m + (0.05m), we can represent Marissa's new earnings per hour as m + (1/20)m.
Simplifying this expression, we have:
New Earnings = m + (1/20)m = (20/20)m + (1/20)m = (21/20)m = (1 + 1/20)m
Therefore, Marissa's new earnings per hour can also be written as m + (1/20)m or (1 + 1/20)m.
To further simplify this expression, we can convert the fraction 1/20 to the decimal form, which is 0.05.
So, (1 + 1/20)m can also be written as (1 + 0.05)m.
We observe that (1 + 0.05)m is equivalent to (1.05)m.
Hence, increasing a number by 5% is the same as multiplying it by 1.05 or (21/20).