/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 30 Replace each \(\circ\) with \(&l... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 30

Replace each \(\circ\) with \(<,>,\) or \(=\) to make a true sentence. $$\frac{5}{8} \circ 0.65$$

Short Answer

Expert verified
The correct symbol is <.

Step by step solution

01

Convert the Decimal to a Fraction

To compare a fraction and a decimal, it's often easier to convert the decimal to a fraction. The decimal 0.65 can be written as a fraction. Since 0.65 means 65 per 100, it can be expressed as \( \frac{65}{100} \).
02

Simplify the Fraction

Simplify the fraction \( \frac{65}{100} \) by finding the greatest common divisor of 65 and 100. These two numbers share a common divisor of 5. Dividing the numerator and denominator by 5 gives: \( \frac{65 \div 5}{100 \div 5} = \frac{13}{20} \).
03

Find a Common Denominator

Now, compare \( \frac{5}{8} \) and \( \frac{13}{20} \). Find a common denominator for these fractions. The least common multiple of 8 and 20 is 40. Convert each fraction to have this common denominator: \( \frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40} \), and \( \frac{13}{20} = \frac{13 \times 2}{20 \times 2} = \frac{26}{40} \).
04

Compare the Numerators

With both fractions having the same denominator of 40, compare their numerators: 25 and 26. Since 25 is less than 26, \( \frac{25}{40} < \frac{26}{40} \).
05

Write the Conclusion

Therefore, \( \frac{5}{8} < 0.65 \) when comparing these values.

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Key Concepts

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

Decimal to Fraction Conversion
Converting decimals to fractions is a handy skill that can make it easier to work with different forms of numbers. Let's break it down using the example of converting 0.65 into a fraction. The decimal 0.65 tells us that we have 65 parts out of 100. Thus, it can be represented as the fraction \( \frac{65}{100} \).

Converting from decimal to fraction consists of the following steps:
  • Write the decimal number over its place value. For 0.65, the last digit (5) is in the hundredths place, so it's \( \frac{65}{100} \).
  • Simplify, if needed. Find the greatest common divisor (GCD) for both numerator and denominator. Here, 65 and 100 are both divisible by 5. Dividing them by 5 simplifies the fraction to \( \frac{13}{20} \).
Decimal to fraction conversion is invaluable, especially when comparing numbers or doing further mathematical operations. It helps to simplify calculations or gain a clearer analytical perspective.
Comparing Fractions
To determine which of two fractions is larger, we often need to make their denominators the same. This process is called finding a common denominator, which allows us to compare fractions directly.

Follow these steps when comparing fractions:
  • Identify the least common multiple (LCM) of the denominators. For \( \frac{5}{8} \) and \( \frac{13}{20} \), the denominators are 8 and 20. The LCM is 40.
  • Adjust each fraction to have this common denominator. Multiply the numerator and the denominator by the same number to change each fraction. Thus, \( \frac{5}{8} = \frac{25}{40} \) and \( \frac{13}{20} = \frac{26}{40} \).
  • Compare the numerators. Here, 25 is less than 26, so \( \frac{25}{40} < \frac{26}{40} \).
Comparing fractions becomes straightforward with a universal denominator. This method is effective as it avoids confusion and provides a clear comparative view.
Least Common Multiple
The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. It's particularly useful for finding common denominators when comparing or adding fractions.

Here's how to find the LCM:
  • List the multiples of each number. For example, when working with 8 and 20, list out the multiples:
    • Multiples of 8: 8, 16, 24, 32, 40...
    • Multiples of 20: 20, 40, 60...
  • ID the smallest number common to both lists. Here, 40 appears in both sets of multiples.
When you establish the LCM, you can seamlessly rewrite fractions with identical denominators. This technique ensures fair comparisons and supports computations with ease.

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Most popular questions from this chapter

Estimate each product. $$1 \frac{2}{3} \cdot 4 \frac{1}{8}$$

Solve each equation. Check your solution. $$2 x-1=9$$

Write each improper fraction as a mixed number in simplest form. $$\frac{8}{7}$$

Evaluate \(a-b\) if \(a=9 \frac{5}{6}\) and \(b=1 \frac{1}{6}\)

Solve each equation. Check your solution. $$-\frac{7}{9} t=-\frac{28}{36}$$
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