Problem 11 - Fundamentals of Mathematics

Question

Perform the following divisions. $$ 8 \div \frac{8}{15} $$

Step-by-Step Solution

Answer

The result of the division is 15.

Step 1: Understand the Problem

The problem requires dividing the number 8 by the fraction \( \frac{8}{15} \). This can be recast as multiplying by the reciprocal of the fraction \( \frac{8}{15} \).

Step 2: Identify the Reciprocal

To divide by a fraction, we multiply by its reciprocal. The reciprocal of \( \frac{8}{15} \) is \( \frac{15}{8} \).

Step 3: Perform the Multiplication

Now, multiply 8 by the reciprocal of \( \frac{8}{15} \), which is \( \frac{15}{8} \). So we calculate \( 8 \times \frac{15}{8} \).

Step 4: Simplify the Multiplication

When multiplying \( 8 \times \frac{15}{8} \), the 8s cancel, leaving \( 15 \). This simplifies because \( 8 = \frac{8}{1} \), so \( \frac{8}{8} = 1 \), leading to \( 1 \times 15 = 15 \).

Key Concepts

Reciprocal of a Fraction

When dealing with fractions, understanding the concept of the reciprocal is essential. The reciprocal of a number is simply what you get when you flip it upside down. For example, the reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). This idea comes in very handy when you want to divide by a fraction. Instead of actually performing division, you can multiply by the reciprocal.
To find the reciprocal:

• Take the given fraction.
• Swap the numerator and the denominator.

Knowing this saves a lot of time and makes complex calculations simpler. It's an essential tool for understanding division involving fractions.

Simplifying Multiplication

After identifying the reciprocal of a fraction, we often need to perform multiplication. Simplifying multiplication, especially involving fractions, is an important step to avoid unnecessary calculations.
Here's how we go about it:

• Write the whole number as a fraction by placing it over 1 (e.g., 8 becomes \( \frac{8}{1} \)).
• Multiply the numerators and denominators across: \( 8 \times \frac{15}{8} \) becomes \( \frac{8 \times 15}{1 \times 8} \).
• Cancel any common factors in the numerator and the denominator before multiplying the remaining numbers.

In our example, the number 8 appears in both the numerator and denominator, so they cancel out, leaving us with 15. This simplifies the calculation and prevents unnecessary complication, leading to faster results.

Fraction Division

Division of fractions can seem tricky, but it becomes simple when applying the reciprocal rule. The basic idea is to convert the division into multiplication, which is much easier to manage. If you have a problem like \( a \div \frac{b}{c} \), you can rewrite it as \( a \times \frac{c}{b} \).
Here’s how you can tackle any fraction division problem:

• Find the reciprocal of the fraction you are dividing by.
• Change the division operation to multiplication.
• Multiply as usual, and then simplify your result.

Mastering this method allows you to understand and solve fraction division problems quickly and effectively. It changes a potentially complex operation into one that's straightforward and easy to handle.