Problem 59 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$4 \frac{3}{25} \div 2 \frac{56}{75}$$

Step-by-Step Solution

Answer

\(\frac{309}{206}\)

Step 1: Convert Mixed Numbers to Improper Fractions

First, let's convert each mixed number into an improper fraction. To do this, multiply the whole number by the denominator and add the numerator.For \(4\frac{3}{25}\): \(4 \times 25 + 3 = 100 + 3 = 103\). So, \(4\frac{3}{25} = \frac{103}{25}\).For \(2\frac{56}{75}\): \(2 \times 75 + 56 = 150 + 56 = 206\). So, \(2\frac{56}{75} = \frac{206}{75}\).

Step 2: Division of Fractions

Now we will divide the improper fractions. To divide two fractions, multiply the first fraction by the reciprocal of the second.\[\frac{103}{25} \div \frac{206}{75} = \frac{103}{25} \times \frac{75}{206}\].

Step 3: Simplify the Fractions

Multiply the fractions:\(\frac{103 \times 75}{25 \times 206}\).Before multiplying, check for any common factors to simplify.\[103 \times 75 = 7725, \quad 25 \times 206 = 5150\], so:\[\frac{7725}{5150}\].

Step 4: Reduce the Final Fraction

We'll simplify \(\frac{7725}{5150}\) by finding the greatest common divisor (GCD) of the numerator and the denominator.The GCD of 7725 and 5150 is 25. Divide both by 25:\[\frac{7725 \div 25}{5150 \div 25} = \frac{309}{206}\].

Step 5: Check for Further Simplification

Check if \(\frac{309}{206}\) can be further reduced. Since 309 and 206 have no common divisors other than 1, the fraction is already in its simplest form. Thus, \(\frac{309}{206}\) is our final answer.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. For example, in "4 \(\frac{3}{25}\)," the number "4" is the whole number, and the fraction \(\frac{3}{25}\) is the fractional part. Mixed numbers are commonly used when the quantity is more than one whole item and some extra parts. To work with mixed numbers in equations, converting them to improper fractions often makes calculations easier.

• To convert, multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.

This gives us an improper fraction, which can be used more easily in mathematical calculations such as addition, subtraction, multiplication, or division.

Improper Fractions

An improper fraction has a numerator larger than or equal to its denominator. For example, \(\frac{103}{25}\) and \(\frac{206}{75}\) are improper fractions. These are common when converting mixed numbers for calculations.

• The term "improper" indicates that the value exceeds the whole represented by the denominator.
• Improper fractions are useful in operations, as they simplify arithmetic processes by eliminating the whole number part completely.

Improper fractions can either be converted back into mixed numbers after calculations or reduced to their simplest form if necessary.

Reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator. For instance, the reciprocal of \(\frac{206}{75}\) is \(\frac{75}{206}\). Using reciprocals is a crucial step when dividing fractions, as dividing by a fraction is equivalent to multiplying by its reciprocal.

• Change "division" to "multiplication" by using the reciprocal.
• This method maintains the equality of the equation and allows us to simplify fraction division.

Reciprocals simplify complex calculations by turning the division into a multiplication process, which is simpler to execute.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide two or more numbers without leaving a remainder. To simplify fractions effectively, finding the GCD of the numerator and the denominator helps in reducing the fraction to its simplest form. For our example, the GCD of 7725 and 5150 is 25, which allows \(\frac{7725}{5150}\) to be reduced to \(\frac{309}{206}\).

• The GCD is determined through methods such as the Euclidean algorithm.
• Finding the GCD is essential in ensuring a fraction is in its simplest form.

Utilizing the GCD in fraction reduction helps to make complex numbers more manageable, resulting in a cleaner numerical presentation.