Problem 34 - Prealgebra

Question

What is the quotient of \(1 \frac{1}{5}\) and \(2 \frac{2}{5} ?\)

Step-by-Step Solution

Answer

The quotient is \( \frac{1}{2} \).

Step 1: Convert Mixed Numbers to Improper Fractions

The first step in this process is to convert the mixed numbers to improper fractions. For the first number, \(1 \frac{1}{5}\), multiply the whole number \(1\) by the denominator \(5\) and add the numerator \(1\), resulting in \( \frac{6}{5} \). For the second number, \(2 \frac{2}{5}\), multiply the whole number \(2\) by the denominator \(5\) and add the numerator \(2\), resulting in \( \frac{12}{5} \).

Step 2: Set Up the Division Problem

Now, using the improper fractions you found in Step 1, set up the division problem. You need to divide \( \frac{6}{5} \) by \( \frac{12}{5} \). In fraction division, this means you will multiply by the reciprocal of the second fraction.

Step 3: Multiply by the Reciprocal

To divide by a fraction, multiply by its reciprocal. Thus, the reciprocal of \( \frac{12}{5} \) is \( \frac{5}{12} \). Therefore, the operation becomes \( \frac{6}{5} \times \frac{5}{12} \).

Step 4: Perform the Multiplication

Multiply the numerators: \(6 \times 5 = 30\). Then multiply the denominators: \(5 \times 12 = 60\). This gives the fraction \( \frac{30}{60} \).

Step 5: Simplify the Fraction

Finally, simplify the fraction \( \frac{30}{60} \) by dividing the numerator and the denominator by their greatest common denominator, which is \(30\). This results in \( \frac{1}{2} \).

Step 6: Write the Conclusion

We have simplified the fraction \( \frac{30}{60} \) to \( \frac{1}{2} \), so the quotient of the original mixed numbers is \( \frac{1}{2} \).

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator, the top number, is larger than or equal to the denominator, the bottom number. These fractions might look a bit intimidating at first, but they are actually very useful, especially when performing operations like multiplication or division. When you have a mixed number, converting it to an improper fraction can make calculations easier and more straightforward.
Let's consider an example of converting a mixed number to an improper fraction. Suppose we have the mixed number 3 2/3. To convert it:

• Multiply the whole number by the fraction's denominator: 3 × 3 = 9.
• Add this number to the original numerator: 9 + 2 = 11.
• The resulting improper fraction is 11/3.

Improper fractions are not "wrong" or "incorrect" - their name simply comes from the fact that their top number is not less than the bottom number. Once you get comfortable with them, they can be quite handy.

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often easy to understand because they tell us exactly how many whole units there are plus an extra part. For instance, 2 3/4 tells us we have 2 whole units and three-quarters of another unit.
However, when it comes to performing arithmetic operations like division and multiplication, mixed numbers can make the process complex. That's why they're often converted to improper fractions before those operations.
When dealing with real-life scenarios or interpretations, mixed numbers offer an intuitive understanding of quantity, making them valuable in practical mathematics. Yet, always remember that for calculations like the one we discussed, switching to improper fractions helps ensure efficiency and accuracy.

Reciprocal

The reciprocal of a fraction is simply what you get when you swap its numerator and denominator. The concept of reciprocal is vital, especially for division in fractions, because instead of directly dividing by a fraction, we multiply by its reciprocal.
For example, if you have a fraction like 7/9, its reciprocal would be 9/7. When you multiply a fraction by its reciprocal, the result is always 1. This is because:

• 7/9 × 9/7 = (7×9)/(9×7) = 63/63 = 1.

In math problems involving division of fractions, using reciprocals simplifies the process. If you need to divide by a fraction, merely multiply by the reciprocal, making your work both easier and less error-prone.

Simplification of Fractions

Simplifying fractions is crucial for ensuring answers are accurate and presented in their most understandable form. Simplification involves reducing the fraction to its lowest terms, which means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number.
For instance, consider the fraction 28/42.

• Both 28 and 42 can be divided by their common divisor, which is 14.
• Dividing both the numerator and denominator by 14 gives us: 28 ÷ 14 = 2 and 42 ÷ 14 = 3.
• The simplified fraction is 2/3.

A well-simplified fraction provides clarity and ensures that the value is really understood, especially in both real-world applications and math problems. Simplification also helps in comparing fractions or performing further mathematical operations efficiently.