Problem 19 - Fundamentals of Mathematics

Question

\(1 \frac{5}{9}\) of \(2 \frac{4}{7}\) is what number?

Step-by-Step Solution

Answer

The number is 4.

Step 1: Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers into improper fractions. For \(1 \frac{5}{9}\), multiply the whole number 1 by the denominator 9, then add the numerator 5 to get \(\frac{14}{9}\). For \(2 \frac{4}{7}\), multiply the whole number 2 by the denominator 7, then add the numerator 4 to get \(\frac{18}{7}\).

Step 2: Multiply the Improper Fractions

Multiply the improper fractions obtained. The operation is \(\frac{14}{9} \times \frac{18}{7}\). Multiply the numerators: \(14 \times 18 = 252\) and the denominators: \(9 \times 7 = 63\). The result is \(\frac{252}{63}\).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{252}{63}\) by finding the greatest common divisor (GCD) of 252 and 63. The GCD is 63. Divide the numerator and the denominator by 63: \(\frac{252 \div 63}{63 \div 63} = \frac{4}{1}\).

Step 4: Convert Improper Fraction to Mixed Number

If needed, convert \(\frac{4}{1}\) into a mixed number. Since \(\frac{4}{1}\) is equivalent to 4, the result is simply 4.

Key Concepts

Improper Fractions

Improper fractions have a numerator (top number) larger than or equal to the denominator (bottom number). They are essential in solving problems involving mixed numbers because they simplify the multiplication process.

To convert a mixed number into an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• Place this sum over the original denominator.

For example, for the mixed number \(2 \frac{4}{7}\):- Multiply the whole number 2 by 7 (denominator) to get 14.- Add 4 (numerator) to get 18.- The improper fraction becomes \(\frac{18}{7}\).

Understanding improper fractions helps us in calculating products or sums without switching between different number formats, making calculations more straightforward.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common divisors other than 1.

To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by this number. Here's how you can do it:

• List the factors of the numerator and the denominator.
• Find the largest factor they share, which is their GCD.
• Divide both by the GCD to arrive at the simplified fraction.

For the fraction \(\frac{252}{63}\), the GCD is 63.- Divide 252 by 63 to get 4.- Divide 63 by 63 to get 1.- The simplified fraction is \(\frac{4}{1}\), which equals 4.

Simplifying fractions is crucial for making answers clearer and easier to understand, especially in final responses.

Mixed Numbers

Mixed numbers combine a whole number and a fraction. They are commonly used in situations where an improper fraction might be less intuitive.

To convert an improper fraction to a mixed number, you can use the following steps:

• Divide the numerator by the denominator to find the whole number.
• The remainder becomes the new numerator, while the denominator stays the same.
• Write the result as "whole number \(\frac{remainder}{denominator}\)".

For instance, if you have \(\frac{4}{1}\), it simplifies simply to 4 since there is no remainder, hence no fractional part.

Mixed numbers offer a more comprehensible format for expressing parts of a whole, useful in real-world scenarios like recipe measurements or when dealing with mixed quantities.