Problem 25 - Prealgebra

Question

Find the following quotients. $$\left(\frac{3}{4} \div 2 \frac{1}{2}\right) \div 3$$

Step-by-Step Solution

Answer

The quotient is \(\frac{1}{10}\).

Step 1: Convert Mixed Number to Improper Fraction

Change the mixed number, \(2 \frac{1}{2}\), into an improper fraction. Multiply the whole number by the denominator (2*2) and add the numerator. This equals 4 + 1 = 5, so the improper fraction is \(\frac{5}{2}\).

Step 2: Perform the First Division

Divide \(\frac{3}{4}\) by \(\frac{5}{2}\) by multiplying \(\frac{3}{4}\) by the reciprocal of \(\frac{5}{2}\). Therefore, multiply \(\frac{3}{4}\) by \(\frac{2}{5}\), which equals \(\frac{3 \times 2}{4 \times 5} = \frac{6}{20}\).

Step 3: Simplify the Resulting Fraction

Simplify \(\frac{6}{20}\) by dividing the numerator and the denominator by their greatest common divisor (2). This gives \(\frac{3}{10}\).

Step 4: Perform the Second Division

Divide \(\frac{3}{10}\) by 3. To do this, multiply \(\frac{3}{10}\) by the reciprocal of 3, which is \(\frac{1}{3}\). This results in \[\frac{3}{10} \times \frac{1}{3} = \frac{3 \times 1}{10 \times 3} = \frac{3}{30}\].

Step 5: Simplify the Final Result

Simplify \(\frac{3}{30}\) by dividing the numerator and the denominator by their greatest common divisor (3). This gives the final simplified fraction, \(\frac{1}{10}\).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is a way of expressing numbers that are between two whole numbers. For example, the mixed number \(2 \frac{1}{2}\) represents the quantity that is the sum of 2 and a half, or an additional \(\frac{1}{2}\).

• To convert a mixed number to an improper fraction, follow these steps:
• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator.
• Place this sum over the original denominator.

For example, \(2 \frac{1}{2}\) is converted to \(\frac{5}{2}\) by calculating \((2 \times 2) + 1 = 5\). This means 2 wholes and 1 half are now expressed as an improper fraction.

Reciprocal

Reciprocals are key in fraction division. Understanding them can make division less confusing. A reciprocal of a fraction is simply flipping its numerator and denominator.

• For instance, the reciprocal of \(\frac{5}{2}\) is \(\frac{2}{5}\).
• Likewise, the reciprocal of a whole number, like 3, is \(\frac{1}{3}\).

To divide by a fraction, multiply by its reciprocal. This is how complex division problems can be simplified into multiplication, such as turning \(\frac{3}{4} \div \frac{5}{2}\) into \(\frac{3}{4} \times \frac{2}{5}\). This approach streamlines problem-solving in fractions.

Simplifying Fractions

Simplifying, or reducing fractions, is the process of making them as simple as possible. This means shortening the fraction to its smallest equivalent expression, while keeping its value.

• To simplify, identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by this number.

For example, in the fraction \(\frac{6}{20}\), the GCD is 2. So, dividing both 6 and 20 by 2, you get \(\frac{3}{10}\). The fraction is now simplified, making it easier to work with and understand.

Improper Fractions

Improper fractions might seem unusual because the numerator is larger than the denominator. This means the fraction represents a value greater than one.

• Converting mixed numbers to improper fractions is useful, especially in mathematical operations.
• For instance, \(2 \frac{1}{2}\) gets converted into \(\frac{5}{2}\), clarifying the calculation process.

Improper fractions help simplify multiplication and division because they remove the complexity of dealing with whole and fractional parts separately. Mastering this conversion sets a strong foundation for understanding more advanced math topics.