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Magazine Chapter 5: Problem 145
In the following exercises, divide. $$4.75 \div 25$$
Short Answer
Expert verified
0.19
Step by step solution
01
Understand the Problem
Determine what is required: divide 4.75 by 25.
02
Convert the Division into a Fraction
Write the division expression as a fraction: \( \frac{4.75}{25} \)
03
Convert the Decimal to a Fraction
Convert 4.75 to a fraction: \( 4.75 = \frac{475}{100} \), so the expression becomes \( \frac{\frac{475}{100}}{25} \)
04
Simplify the Complex Fraction
Simplify \( \frac{\frac{475}{100}}{25} \) to \( \frac{475}{100} \times \frac{1}{25} \)
05
Perform Multiplication
Multiply the numerators and the denominators: \( \frac{475 \times 1}{100 \times 25} = \frac{475}{2500} \)
06
Simplify the Resulting Fraction
Divide the numerator and the denominator by their greatest common divisor (GCD), which is 25: \( \frac{475 \textdiv 25}{2500 \textdiv 25} = \frac{19}{100}\)
07
Convert to Decimal Form
Convert \( \frac{19}{100} \) back to decimal form: 0.19
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fraction Simplification
Simplifying fractions is a key skill in prealgebra. It makes fractions easier to understand and work with. To simplify a fraction, you divide both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).
For example, if you have the fraction \( \frac{475}{2500} \), first find the GCD of 475 and 2500, which is 25. Then, divide both numbers by 25:
For example, if you have the fraction \( \frac{475}{2500} \), first find the GCD of 475 and 2500, which is 25. Then, divide both numbers by 25:
- \( 475 \text{ divided by } 25 = 19 \)
- \( 2500 \text{ divided by } 25 = 100 \)
Decimal to Fraction Conversion
Converting decimals to fractions is a useful skill, especially when working with division problems. Let's convert 4.75 to a fraction. Here鈥檚 how you do it easily.
First, write down the decimal number over its place value. The number 4.75 means 4 and 75 hundredths, or \( \frac{475}{100} \):
First, write down the decimal number over its place value. The number 4.75 means 4 and 75 hundredths, or \( \frac{475}{100} \):
- 4.75 = \( 4 + 0.75 \)
- 0.75 = \( \frac{75}{100} \)
- 4 = \( \frac{400}{100} \)
- So, \( 4.75 = \frac{400}{100} + \frac{75}{100} = \frac{475}{100} \)
Greatest Common Divisor
The greatest common divisor (GCD) is essential for simplifying fractions. It is the largest number that can divide both the numerator and denominator without leaving a remainder.
To find the GCD of two numbers, like 475 and 2500 in our exercise, you can use several methods:
To find the GCD of two numbers, like 475 and 2500 in our exercise, you can use several methods:
- **Prime Factorization**: Break each number down into its prime factors and find the highest factors they have in common.
- **Euclidean Algorithm**: Apply repeated division.*Example:* Divide 2500 by 475. The remainder is used in the next division step, until the remainder is 0. The last non-zero remainder is the GCD.
- **List of Divisors**: Write down all divisors of each number and find the highest one they share.
