Problem 75 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \frac{7}{18} \div \frac{14}{36} $$

Step-by-Step Solution

Answer

1

Step 1: Rewrite the Division as Multiplication

To divide fractions, you multiply the first fraction by the reciprocal of the second fraction. The reciprocal of \( \frac{14}{36} \) is \( \frac{36}{14} \). Therefore, the expression \( \frac{7}{18} \div \frac{14}{36} \) becomes \( \frac{7}{18} \times \frac{36}{14} \).

Step 2: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together:\[\frac{7}{18} \times \frac{36}{14} = \frac{7 \times 36}{18 \times 14}.\] This gives us \( \frac{252}{252} \).

Step 3: Simplify the Fraction

The fraction \( \frac{252}{252} \) simplifies to \( 1 \) because any number divided by itself is 1.

Key Concepts

Division of Fractions

When you hear about dividing fractions, it might sound tricky at first. But it's simpler than it seems. Here's the golden rule: To divide one fraction by another, you need to turn division into multiplication. How do you do that? By using the reciprocal!

When you're asked to divide, say \( \frac{7}{18} \) by \( \frac{14}{36} \), you actually multiply \( \frac{7}{18} \) by the reciprocal of \( \frac{14}{36} \). Once you've changed the division to multiplication, just multiply the fractions as usual—numerators together and denominators together. This clever trick makes dividing fractions as easy as pie!

• Change division to multiplication.
• Use the reciprocal of the divisor.
• Multiply across numerators and denominators.

Understanding this process helps eliminate confusion and simplifies your fractional calculations.

Reciprocal

The term 'reciprocal' might sound like something from an advanced math class, but it really just means flipping a fraction upside down.

For example, the reciprocal of \( \frac{14}{36} \) is \( \frac{36}{14} \). It's as easy as switching places of the numerator and the denominator!

• Reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).
• Always used in fraction division.

Why is finding a reciprocal important? In the world of math, especially when dividing fractions, it allows us to turn division into multiplication. This means once you find the reciprocal of the divisor, you just multiply it by the dividend, making the calculation straightforward.
Whenever you're unsure, just flip the fraction—you'll find the reciprocal easily fitting into your problem-solving toolkit.

Simplifying Fractions

Simplifying fractions is like tidying up a math problem. It means reducing a fraction to its smallest form, where the numerator and denominator have no common factors other than 1. Why do we simplify fractions? To make them easier to work with and interpret.

Consider this: After multiplying \( \frac{7}{18} \) and its reciprocal of \( \frac{14}{36} \), you land on \( \frac{252}{252} \). This simplifies directly to 1 because any fraction divided by itself is 1. Simplification doesn't always stop with easy numbers—sometimes you need to find the greatest common divisor (GCD) to reduce fractions to their simplest form.

• Check for common factors in numerator and denominator.
• Divide both by greatest common divisor if necessary.
• Fractions express essences when in simplest terms.

Keeping fractions simplified not only makes math problems look cleaner but also strengthens your math communication clarity while solving problems.