Question

Evaluate each expression. $$ \frac{\frac{5}{21}}{\frac{2}{35}} $$

Short answer

\(\frac{25}{6}\)

Step-by-step solution

Write Down the Initial Expression

Begin by writing down the given expression: \(\frac{\frac{5}{21}}{\frac{2}{35}}\).

Invert the Divisor

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{2}{35}\) is \(\frac{35}{2}\). Thus, our expression becomes: \(\frac{5}{21} \times \frac{35}{2}\).

Multiply the Numerators

Multiply the numerators of the fractions: \(5 \times 35 = 175\).

Multiply the Denominators

Multiply the denominators of the fractions: \(21 \times 2 = 42\).

Simplify the Fraction

Simplify the resulting fraction \(\frac{175}{42}\). Both the numerator and the denominator can be divided by their greatest common divisor, which is 7: \(\frac{175 \div 7}{42 \div 7} = \frac{25}{6}\).

Key concepts

Reciprocal

The reciprocal of a number is basically its flipped version, where the numerator and the denominator are swapped. For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). For example, if we have the fraction \(\frac{2}{35}\), its reciprocal would be \(\frac{35}{2}\). Understanding reciprocals is crucial when dealing with division of fractions, as it transforms the division problem into a multiplication problem. A simple way to remember it is, whenever you see a division of fractions, just flip the second fraction and change the division sign to multiplication.

Simplifying Fractions

Simplifying fractions means making them as simple as possible, often by dividing both the numerator and the denominator by their common factors. Sometimes, fractions can look messy and large until we reduce them to their simplest form. For example, if you have a fraction like \(\frac{42}{56}\), you look for common factors of 42 and 56. You can divide both numbers by 14 (the greatest common divisor, which we will detail in the next section). So, \(\frac{42 \div 14}{56 \div 14} = \frac{3}{4}\). This makes calculations easier and your final answer neat. It’s a great skill to master because it will always help in making numbers more manageable.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can evenly divide two or more numbers. For example, the GCD of 56 and 42 is 14 because 14 is the highest number that both 42 and 56 can be divided by without leaving a remainder. Finding the GCD is useful when simplifying fractions because it helps reduce the fraction to its simplest form. To find the GCD:

• List down the factors of each number.
• Identify the common factors.
• Choose the largest one among those common factors.

For example, the factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 and the factors of 56 are 1, 2, 4, 7, 8, 14, 28, 56. The common factors are 1, 2, 7, 14. So, the GCD is 14. Mastering this will help simplify even the most complex fractions easily and accurately!