Problem 41 - Fundamentals of Mathematics

Question

What part of \(\frac{14}{15}\) is \(\frac{7}{30} ?\)

Step-by-Step Solution

Answer

\(\frac{7}{30}\) is \(\frac{1}{4}\) of \(\frac{14}{15}\).

Step 1: Understand the question

We need to find what fraction of \(\frac{14}{15}\) is \(\frac{7}{30}\). This can be expressed as a division of fractions: \(\frac{7}{30} \div \frac{14}{15}\).

Step 2: Division of Fractions

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. Thus, \(\frac{7}{30} \div \frac{14}{15}\) becomes \(\frac{7}{30} \times \frac{15}{14}\).

Step 3: Multiply the Fractions

Multiply the numerators and the denominators: \(\frac{7 \times 15}{30 \times 14}\). This simplifies to \(\frac{105}{420}\).

Step 4: Simplify the Fraction

To simplify \(\frac{105}{420}\), find the greatest common divisor (GCD) of 105 and 420, which is 105. Divide both numerator and denominator by 105: \(\frac{105 \div 105}{420 \div 105} = \frac{1}{4}\).

Step 5: Conclusion

Thus, \(\frac{7}{30}\) is \(\frac{1}{4}\) of \(\frac{14}{15}\).

Key Concepts

Greatest Common Divisor

To simplify fractions, the greatest common divisor (GCD) is your best friend. It allows you to reduce fractions to their simplest form. In our exercise, we needed to find the GCD of 105 and 420. But what exactly is GCD? It's the largest number that can evenly divide both numbers.

• For example, the divisors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105.
• The divisors of 420 are 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, and 420.

The greatest number common to both lists is 105. By dividing the numerator and denominator of the fraction by their GCD, you effectively simplify the fraction. For example, \[\frac{105}{420} \rightarrow \frac{105 \div 105}{420 \div 105} = \frac{1}{4}.\]
This process helps ensure that fractions are reduced to their simplest form, making them easier to interpret and compare.

Reciprocal

The concept of reciprocal is central to dividing fractions. When we hear 'reciprocal,' we simply think of flipping a fraction. For example, the reciprocal of \(\frac{14}{15}\) is \(\frac{15}{14}\). We use reciprocals when dividing fractions because it turns the division into a multiplication problem:

• To find the reciprocal, swap the numerator (top number) with the denominator (bottom number).
• This transformation is simple yet powerful when paired with multiplication.

In the exercise, to divide \(\frac{7}{30}\) by \(\frac{14}{15}\), we changed it into \(\frac{7}{30} \times \frac{15}{14}\) by using the reciprocal of \(\frac{14}{15}\). This approach simplifies the calculation and is a fundamental concept in fraction division.

Fraction Multiplication

Understanding how to multiply fractions is crucial for fraction division, especially after applying the reciprocal. To multiply two fractions, follow these straightforward steps:

• Multiply the numerators (top numbers) together to get the new numerator.
• Multiply the denominators (bottom numbers) together to get the new denominator.

For the exercise, after finding the reciprocal, we have:\[\frac{7}{30} \times \frac{15}{14} = \frac{7 \times 15}{30 \times 14}.\]This results in \(\frac{105}{420}\). By multiplying across the top and bottom, you combine the fractions into a single fraction. Then, you can further simplify it by finding the greatest common divisor, as we discussed earlier.

• Fraction multiplication requires attention to detail but is easier with practice.
• Always look for opportunities to simplify fractions before or after multiplication.