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Chapter 0: Problem 72

Evaluate each expression. $$ \frac{21}{25} \cdot \frac{100}{3} $$

Short Answer

Expert verified
28

Step by step solution

01

Multiply the Numerators

To begin, multiply the numerators of the given fractions: \[ 21 \times 100 = 2100 \]
02

Multiply the Denominators

Next, multiply the denominators of the given fractions: \[ 25 \times 3 = 75 \]
03

Form the New Fraction

Combine the results from Step 1 and Step 2 to form a new fraction: \[ \frac{2100}{75} \]
04

Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). Here, the GCD of 2100 and 75 is 75: \[ \frac{2100 \, \text{(divided by 75)}}{75 \, \text{(divided by 75)}} = \frac{28}{1} = 28 \]

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Key Concepts

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

Understanding Fractions
Fractions represent a part of a whole. A fraction consists of two numbers: the numerator and the denominator. The numerator is the top number, indicating how many parts we have. The denominator is the bottom number, showing into how many parts the whole is divided.
  • For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and means 3 parts of the whole.
  • 4 is the denominator and indicates the whole is divided into 4 parts.
Fractions can be proper (numerator smaller than the denominator) or improper (numerator equal to or larger than the denominator).
This basic understanding of fractions is essential when you begin to multiply and simplify fractions.
Multiplying Fractions
Multiplying fractions involves straightforward steps. Unlike adding or subtracting, there is no need for a common denominator. Here's a step-by-step guide to multiply fractions:
1. **Multiply the Numerators**: Multiply the top numbers (numerators) of the fractions. Example: For \(\frac{21}{25} \cdot \frac{100}{3}\), multiply 21 by 100 to get 2100.
2. **Multiply the Denominators**: Multiply the bottom numbers (denominators) of the fractions. Using our example, multiply 25 by 3 to get 75.
3. **Form the New Fraction**: Combine the results from the numerators and denominators as a new fraction. In our case, \(\frac{2100}{75}\).
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. To do this, divide the numerator and the denominator by their greatest common divisor (GCD).
For instance, to simplify \( \frac{2100}{75} \), we find the GCD of 2100 and 75, which is 75.
  • Divide the numerator (2100) by the GCD (75) to get 28.
  • Divide the denominator (75) by the GCD (75) to get 1.
The simplified fraction is \(\frac{28}{1}\), which equals 28.
Always check to make sure the fraction cannot be simplified further. Simplifying fractions helps to easily interpret the result and is a key part of working with fractions.

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Most popular questions from this chapter

The period \(T\), in seconds, of a pendulum of length \(l,\) in feet, may be approximated using the formula $$T=2 \pi \sqrt{\frac{l}{32}}$$ Express your answer both as a square root and as a decimal approximation. Find the period \(T\) of a pendulum whose length is 64 feet.

Simplify each expression. Assume that all variables are positive when they appear. $$(\sqrt[3]{3} \sqrt{10})^{4}$$

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$2 x(3 x+4)^{4 / 3}+x^{2} \cdot 4(3 x+4)^{1 / 3}$$

Simplify each expression. $$\left(\frac{27}{8}\right)^{2 / 3}$$

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{\sqrt[3]{8 x+1}}{3 \sqrt[3]{(x-2)^{2}}}+\frac{\sqrt[3]{x-2}}{24 \sqrt[3]{(8 x+1)^{2}}} \quad x \neq 2, x \neq-\frac{1}{8}$$
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