Problem 78 - Prealgebra

Question

Simplify. $$\frac{\frac{5}{7}}{\frac{6}{7}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5}{6} \).

Step 1: Identify the Outer Division

We are given an expression with a fraction in the numerator and a fraction in the denominator. The expression is \( \frac{\frac{5}{7}}{\frac{6}{7}} \). We need to think of this as dividing one fraction by another fraction.

Step 2: Apply the Division Rule

When dividing fractions, we multiply by the reciprocal of the fraction in the denominator. Therefore, \( \frac{\frac{5}{7}}{\frac{6}{7}} = \frac{5}{7} \times \frac{7}{6} \).

Step 3: Multiply the Numerators

Multiply the numerators of the two fractions: \( 5 \times 7 = 35 \).

Step 4: Multiply the Denominators

Multiply the denominators of the two fractions: \( 7 \times 6 = 42 \).

Step 5: Simplify the Resulting Fraction

Now, we have \( \frac{35}{42} \). Find the greatest common divisor of 35 and 42, which is 7. Divide both the numerator and the denominator by 7: \( \frac{35 \div 7}{42 \div 7} = \frac{5}{6} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is like tidying up fractions, making them as neat and small as possible. Think of simplifying as reducing a fraction to its simplest form, where the numerator and the denominator are as small as they can be while still keeping the same value. To do this, you need to find a number that goes neatly into both the top (numerator) and the bottom (denominator).

• Find a common factor between the numerator and the denominator.
• Divide both parts of the fraction by that common factor.

Let's take the fraction \( \frac{35}{42} \). We need to find the greatest number that divides evenly into both 35 and 42. That number is 7. We then divide both the top and bottom by 7 to simplify the fraction to \( \frac{5}{6} \). That’s simplified, as you can't make it any smaller without losing its value. Simplifying helps to easily compare fractions or perform arithmetical operations more efficiently.

Reciprocal

The reciprocal is like turning a fraction on its head. It's a key part in dividing fractions and it means simply flipping the fraction upside down.For a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). When you have to divide by a fraction, you multiply by its reciprocal instead. For example, when looking at the division \( \frac{\frac{5}{7}}{\frac{6}{7}} \), you're really doing multiplication by \( \frac{7}{6} \), the reciprocal of \( \frac{6}{7} \).

• Take the fraction you want the reciprocal of.
• Flip the numerator and the denominator.

This flip is the magic trick that makes fraction division so much easier! By multiplying by a reciprocal, you neatly sidestep the tricky division and turn a dividing problem into a multiplying one.

Greatest Common Divisor

The greatest common divisor (GCD), or greatest common factor, is the largest number that can perfectly divide two or more numbers without leaving a remainder. Knowing how to find the GCD is crucial for simplifying fractions.To determine the GCD of two numbers, you can:

• List the factors of each number, then choose the biggest factor they share.
• Use the Euclidean algorithm, which involves a series of divisions.

For example, when simplifying \( \frac{35}{42} \), we find that the numbers share the factor 7. Dividing both numbers by 7 gives us the simplified fraction \( \frac{5}{6} \). Finding the GCD keeps fractions clean and manageable, reducing them to their simplest form while maintaining the same value. It’s a handy tool for not only simplifying but also for comparing fractions and ensuring efficiency in calculations!