Problem 48 - Basic College Mathematics

Question

Use \(=\) or \(\neq\) for \(\square\) to write a true sentence. $$ \frac{700}{1000} \square \frac{70}{100} $$

Step-by-Step Solution

Answer

The correct symbol is \( = \).

Step 1: Simplify the Fractions

Simplify each fraction to its lowest terms. For the first fraction, \( \frac{700}{1000} \), divide the numerator and the denominator by their greatest common divisor, which is 100. This gives: \[ \frac{700 \div 100}{1000 \div 100} = \frac{7}{10} \]. For the second fraction, \( \frac{70}{100} \), similarly divide the numerator and the denominator by their greatest common divisor, which is also 10. This gives: \[ \frac{70 \div 10}{100 \div 10} = \frac{7}{10} \].

Step 2: Compare the Simplified Fractions

Now, compare the simplified fractions. Both fractions simplify to \( \frac{7}{10} \). Therefore, \( \frac{700}{1000} \) is equal to \( \frac{70}{100} \).

Step 3: Write the True Sentence

Since both fractions are equal, the correct symbol to use is \( = \). So, the true sentence is: \[ \frac{700}{1000} = \frac{70}{100} \].

Key Concepts

Equivalent Fractions

When two fractions represent the same part of a whole, they are known as equivalent fractions.
For example, in the exercise \(\frac{700}{1000} \) and \(\frac{70}{100} \) are equivalent fractions.
Even though the numbers (numerators and denominators) are different, the fractions represent the same value.

Equivalent fractions are obtained by either multiplying or dividing both the numerator and the denominator by the same non-zero number.
For example, multiplying the numerator and denominator of \(\frac{7}{10} \) by 10 gives \(\frac{70}{100} \). Similarly, dividing both by 10 brings us back to \(\frac{7}{10} \).
So, \(\frac{700}{1000} \), \(\frac{70}{100} \), and \(\frac{7}{10} \) are all equivalent fractions.
Identifying equivalent fractions helps in comparing and simplifying them.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form by dividing the numerator and denominator by their greatest common divisor (GCD).
In the given problem, \(\frac{700}{1000} \) was simplified by dividing both the numerator and the denominator by 100.
This gave: \[ \frac{700 \div 100}{1000 \div 100} = \frac{7}{10} \].
Similarly, \(\frac{70}{100} \) was simplified by dividing both by 10, resulting in \[ \frac{70 \div 10}{100 \div 10} = \frac{7}{10} \].
Simplification helps in comparing fractions easily and in performing arithmetic operations like addition and subtraction.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest integer that divides both the numerator and the denominator without leaving a remainder.
Identifying the GCD is key for simplifying fractions.
To find the GCD of two numbers, you list out the factors of each number and then determine the largest factor common to both.
For example, the GCD of 700 and 1000 is 100 because:

• Factors of 700: 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700
• Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000

Among these, 100 is the largest common factor.
Therefore, dividing both 700 and 1000 by their GCD of 100 simplifies the fraction to \(\frac{7}{10} \).
This makes the fractions easier to work with and understand.