Problem 50 - Prealgebra

Question

Find the quotients. (Divide.) $$\frac{59}{10} \div 2$$

Step-by-Step Solution

Answer

The quotient is \( \frac{59}{20} \).

Step 1: Understanding the Problem

We need to find the quotient of \( \frac{59}{10} \) when divided by 2. This means we need to perform the division operation on the mixed fraction \( \frac{59}{10} \) and 2.

Step 2: Rewrite the Division

Divide fractions by first expressing them using the multiplication of the reciprocal: \( \frac{59}{10} \div 2 = \frac{59}{10} \times \frac{1}{2} \).

Step 3: Multiply the Fractions

Now, multiply the numerators together and the denominators together: \( \frac{59 \times 1}{10 \times 2} = \frac{59}{20} \).

Step 4: Simplify the Fraction

Check if the fraction \( \frac{59}{20} \) can be simplified. Since 59 and 20 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Multiplication by Reciprocal

When dividing fractions, one of the foundational concepts students encounter is the idea of 'multiplying by the reciprocal'. This may sound a bit complex, but it's quite simple once you get the hang of it. Suppose you have two fractions, and you need to divide one by the other. Instead of dividing directly, you flip the second fraction. This flipped fraction is called the 'reciprocal'. For example, if you're dividing by 2, you take the reciprocal of 2, which is \( \frac{1}{2} \).

• Start by identifying the numbers involved in the division.
• The number by which you're dividing is the one you need to invert (flip).
• Multiply the first fraction by the reciprocal of the second number.

So, in our exercise, dividing \( \frac{59}{10} \) by 2 becomes multiplying \( \frac{59}{10} \) by \( \frac{1}{2} \), which is simply flipping 2 into its reciprocal form \( \frac{1}{2} \). By understanding this simple rule, you can solve any fraction division problem. It's a neat trick that always works!

Simplifying Fractions

After multiplying fractions together, the next step usually involves simplifying the result. Simplifying means making the fraction as simple as possible. Your goal is to express the fraction using the smallest possible whole numbers. To do this, check both the numerator and the denominator for any common factors.

• Look for the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the top and bottom numbers by this GCD.
• If no common factors other than 1 are found, the fraction is already in simplest form.

In our example, the final fraction after multiplying by the reciprocal is \( \frac{59}{20} \). Since 59 and 20 have no common divisors other than 1, \( \frac{59}{20} \) is already in its simplest form. Simplifying fractions not only makes them easier to understand but also more concise for further calculations.

Mixed Fractions Division

Although our initial problem involved the fraction \( \frac{59}{10} \), which can be perceived as a mixed fraction, knowing how to handle mixed numbers is a handy skill. A mixed fraction is a combination of a whole number and a proper fraction, like \( 1 \frac{1}{2} \). When dividing mixed fractions, you should convert them to improper fractions first.

• Convert any mixed fraction to an improper fraction before beginning the operation. Multiply the whole number by the denominator and add it to the numerator.
• Perform the division using the multiplication by reciprocal method.
• After computation, if needed, convert the result back to a mixed fraction.

For instance, if you had a problem involving a true mixed fraction like \( 1 \frac{3}{4} \div 2 \), you would first convert \( 1 \frac{3}{4} \) to \( \frac{7}{4} \). Then, follow the division steps by multiplying by the reciprocal. This way, fraction division becomes clearer and far easier to manage, even when dealing with mixed numbers.