Problem 50 - Prealgebra

Question

Find the quotient of \(\frac{8}{15}\) and \(\frac{8}{35}\).

Step-by-Step Solution

Answer

The quotient of \(\frac{8}{15}\) and \(\frac{8}{35}\) is \(\frac{7}{3}\).

Step 1: Understand the Division of Fractions

To find the quotient of two fractions, you need to divide the first fraction by the second. In general, dividing by a fraction is the same as multiplying by its reciprocal.

Step 2: Reciprocal of the Second Fraction

The reciprocal of a fraction is obtained by swapping its numerator and denominator. For the fraction \(\frac{8}{35}\), the reciprocal is \(\frac{35}{8}\).

Step 3: Multiply the First Fraction by the Reciprocal

Now, multiply the first fraction \(\frac{8}{15}\) by the reciprocal of the second fraction, which is \(\frac{35}{8}\). This gives: \[\frac{8}{15} \times \frac{35}{8} = \frac{8 \times 35}{15 \times 8}.\]

Step 4: Simplify the Multiplication

In the expression \(\frac{8 \times 35}{15 \times 8}\), the 8 in the numerator and denominator cancel out, simplifying the expression to:\[\frac{35}{15}.\]

Step 5: Simplify the Resulting Fraction

Now, simplify \(\frac{35}{15}\) by dividing both the numerator and denominator by their greatest common divisor, which is 5. Thus:\[\frac{35 \div 5}{15 \div 5} = \frac{7}{3}.\]

Step 6: Final Step: Conclusion

The simplified quotient of \(\frac{8}{15}\) and \(\frac{8}{35}\) is \(\frac{7}{3}\).

Key Concepts

Reciprocal of a Fraction

Understanding the concept of the reciprocal of a fraction is crucial for dividing fractions. The reciprocal is essentially what you flip in a fraction: you swap the positions of its numerator and denominator. So, if you have a fraction like \(\frac{8}{35}\), its reciprocal becomes \(\frac{35}{8}\). This action of flipping creates the multiplicative inverse, which, when multiplied by the original fraction, results in 1.

• This swapping is a key step when dividing fractions since dividing by a fraction equates to multiplying by its reciprocal.
• The notion of reciprocals extends beyond fractions and can apply to numbers as well; for instance, the reciprocal of 5 is \(\frac{1}{5}\).

Reciprocals are a handy tool because they turn a potentially complex division problem into a simpler multiplication problem.

Simplifying Fractions

When dealing with fractions, simplifying them to their lowest terms makes them easier to handle. To simplify a fraction, you need to ensure that the numerator and denominator are as small as possible, while still retaining the same value of the fraction.

• The process involves finding a common factor between the numerator and the denominator.
• Once identified, both parts of the fraction are divided by this common factor.

For example, consider the fraction \(\frac{35}{15}\). To simplify, you divide the numerator and the denominator by their greatest common divisor (GCD), which is the highest number that evenly divides both numbers. Simplifying \(\frac{35}{15}\) by dividing by 5, the GCD, results in \(\frac{7}{3}\), delivering an equivalent yet simpler form of the fraction.
Regular simplification helps in comparing, adding, subtracting, multiplying, or dividing fractions by reducing computation complexity.

Greatest Common Divisor

The greatest common divisor (GCD) is a fundamental concept in simplifying fractions. The GCD is the largest positive integer that divides two or more integers without a remainder. By connecting this with fractions, it empowers you to simplify them effectively by reducing both the numerator and the denominator by this shared divisor.

• Identifying the GCD involves listing out the divisors of both numbers and finding the highest one common to both.
• For example, the divisors of 35 are 1, 5, 7, and 35, while those for 15 are 1, 3, 5, 15. Therefore, the shared factor is 5, making it the GCD.

Understanding the GCD enables you to simplify fractions systematically, ensuring calculations are clear and efficient. This concept is not only central to simplifying fractions but also plays a significant role in many areas of mathematics, such as solving equations and finding rational roots.