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Chapter 0: Problem 25

Simplify the given expression as much as possible. $$ \frac{\frac{5}{7}}{\frac{2}{3}} $$

Short Answer

Expert verified
The short answer is: \(\frac{15}{14}\).

Step by step solution

01

Write down the expression

Rewrite the given expression to make it clear what should be done: \[ \frac{\frac{5}{7}}{\frac{2}{3}} \]
02

Use the reciprocal

Replace the division with multiplication and the second fraction with its reciprocal, which is swapping the numerator and the denominator of the second fraction: \[ \frac{5}{7} \cdot \frac{3}{2} \]
03

Multiply the fractions

Multiply the numerators and the denominators: \[ \frac{5 \cdot 3}{7 \cdot 2} \]
04

Perform calculations

Perform the multiplication: \[ \frac{15}{14} \]
05

Write the final simplified expression

Write down the simplified form of the expression: \[ \frac{15}{14} \] Our final simplified expression is \(\frac{15}{14}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Fractions
Simplifying fractions is an essential skill in mathematics, helping to make fractions easier to understand and compare. Simplification involves reducing a fraction to its simplest form, which means the numerator and the denominator cannot both be divided equally by any other number than 1.

To simplify a fraction, follow these steps:
  • Find the greatest common divisor (GCD) of the numerator and denominator.
  • Divide both the numerator and the denominator by their GCD.
A fraction like \(\frac{8}{12}\) can be simplified by dividing both 8 and 12 by their GCD, which is 4, leaving us with \(\frac{2}{3}\). Notice that the values are smaller, yet the fraction equals the same value as the non-simplified version. This process makes calculations and comparisons more manageable.
Reciprocal
The concept of a reciprocal plays a significant role, especially when dividing fractions. The reciprocal of a fraction is simply flipping it; swap the numerator with the denominator. For example, the reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\).

This concept becomes helpful when we need to divide one fraction by another, such as in our given exercise \(\frac{\frac{5}{7}}{\frac{2}{3}}\). Instead of dividing directly, we multiply by the reciprocal. This means \(\frac{5}{7}\) is multiplied by \(\frac{3}{2}\) so that division transforms into multiplication, which is easier to perform. Always remember that every fraction, and every whole number too, has a reciprocal.
Multiplication of Fractions
Multiplying fractions might seem daunting at first, but it's simpler than it looks. To multiply two fractions, you multiply the numerators together to get the new numerator, and the denominators together to get the new denominator.

Let's break it down:
  • Consider two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\).
  • To multiply them, compute \(\frac{a \cdot c}{b \cdot d}\).
For example, in our solution, \(\frac{5}{7} \times \frac{3}{2}\) turns into \(\frac{5 \cdot 3}{7 \cdot 2} = \frac{15}{14}\).

Multiplying fractions is all about straightforward multiplication without any carrying over or regrouping, making it a clean and straightforward process.

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Most popular questions from this chapter

Explain why the product of a nonzero rational number and an irrational number is an irrational number.

Simplify the given expression as much as possible. $$ 3(2 m+4(n+5 p))+6 n $$

In Exercises \(19-30,\) write each set as an interval or as a union of two intervals. $$ \left\\{x:|3 x-2|<\frac{1}{4}\right\\} $$

In Exercises \(7-16,\) write each union as a single interval. $$ (-3, \infty) \cup[-5, \infty) $$

For Exercises 1-4, determine how many different values can arise by inserting one pair of parentheses into the given expression. $$ 19-12-8-2 $$
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