Problem 89 - Algebra A Combined Function

Question

If Tucson's average rainfall is \(11 \frac{1}{4}\) inches and Yuma's is \(3 \frac{3}{5}\) inches, how much more rain, on the average, does Tucson get than Yuma?

Step-by-Step Solution

Answer

Tucson gets \(7 \frac{13}{20}\) inches more rain than Yuma on average.

Step 1: Convert Mixed Numbers to Fractions

First, convert the mixed numbers to improper fractions to make subtraction easier. \(11 \frac{1}{4} = \frac{45}{4}\) and \(3 \frac{3}{5} = \frac{18}{5}\).

Step 2: Find Common Denominator

The least common denominator of 4 and 5 is 20. Convert the fractions to have this common denominator: \(\frac{45}{4} = \frac{225}{20}\) and \(\frac{18}{5} = \frac{72}{20}\).

Step 3: Subtract Fractions

Subtract the second fraction from the first: \(\frac{225}{20} - \frac{72}{20} = \frac{153}{20}\).

Step 4: Convert to Mixed Number

The result \(\frac{153}{20}\) is converted back to a mixed number: \(7 \frac{13}{20}\).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. In our exercise, Tucson's rainfall is given as the mixed number \(11 \frac{1}{4}\), while Yuma's is \(3 \frac{3}{5}\). These represent more than just single fractions and are often used in daily life when numbers need to express something beyond a whole unit.
To handle operations such as addition or subtraction effectively with mixed numbers, it is often necessary to convert them into improper fractions, as we will see.

Improper Fractions

Improper fractions have numerators greater than (or equal to) their denominators, which means they represent quantities greater than or equal to one. In the subtraction problem you have on average rainfall, we transformed the mixed numbers into improper fractions:

• Tucson's rainfall \(11 \frac{1}{4} = \frac{45}{4}\)
• Yuma's rainfall \(3 \frac{3}{5} = \frac{18}{5}\)

This transformation simplifies calculations as it allows us to work with numerators and denominators directly, which aligns well with the process of finding common denominators.

Common Denominator

A common denominator is crucial when adding or subtracting fractions, as it ensures the fractions are comparable. In our example, the least common denominator for 4 (from \(\frac{45}{4}\)) and 5 (from \(\frac{18}{5}\)) is 20.
We converted the fractions to have a common denominator:

• Converting \(\frac{45}{4}\) to the equivalent fraction with denominator 20: \(\frac{225}{20}\).
• Converting \(\frac{18}{5}\) to the equivalent fraction with denominator 20: \(\frac{72}{20}\).

Having both fractions with the same denominator allows for straightforward subtraction or addition.

Converting Fractions to Mixed Numbers

After executing the subtraction of fractions, resulting in \(\frac{153}{20}\), it is often necessary to convert back these improper fractions into mixed numbers for clarity and understanding. This process can be achieved by dividing the numerator by the denominator:

• \(\frac{153}{20}\) converts to the mixed number: \(7 \frac{13}{20}\)

Here, 7 is the whole number obtained from the division, and \(\frac{13}{20}\) is the fractional part leftover. This conversion helps in presenting the difference in a more interpretable form.