Problem 98 - Fundamentals of Mathematics

Question

What part of \(3 \frac{1}{5}\) is \(1 \frac{7}{25}\) ?

Step-by-Step Solution

Answer

\(1 \frac{7}{25}\) is \(\frac{2}{5}\) of \(3 \frac{1}{5}\).

Step 1: Convert Mixed Numbers to Improper Fractions

We start by converting the mixed numbers into improper fractions. For \(3 \frac{1}{5}\), multiply the whole number 3 by the denominator 5 and then add the numerator: \(3 \times 5 + 1 = 16\). So, \(3 \frac{1}{5} = \frac{16}{5}\). Likewise, for \(1 \frac{7}{25}\), multiply 1 by 25 and add 7: \(1 \times 25 + 7 = 32\). Hence, \(1 \frac{7}{25} = \frac{32}{25}\).

Step 2: Divide the Improper Fractions

To find out what part \(1 \frac{7}{25}\) is of \(3 \frac{1}{5}\), we divide \(\frac{32}{25}\) by \(\frac{16}{5}\). This is performed by multiplying \(\frac{32}{25}\) by the reciprocal of \(\frac{16}{5}\), which is \(\frac{5}{16}\). The calculation is \(\frac{32}{25} \times \frac{5}{16}\).

Step 3: Simplify the Multiplication

Now, multiply the fractions: \(\frac{32}{25} \times \frac{5}{16}\). Multiply the numerators: \(32 \times 5 = 160\) and the denominators: \(25 \times 16 = 400\), resulting in \(\frac{160}{400}\).

Step 4: Reduce the Fraction to Simplest Terms

To simplify \(\frac{160}{400}\), we find the greatest common divisor (GCD) of 160 and 400, which is 80. Divide both the numerator and the denominator by 80: \(\frac{160 \div 80}{400 \div 80} = \frac{2}{5}\). Thus, the fraction \(\frac{160}{400}\) simplifies to \(\frac{2}{5}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are commonly used to represent numbers that are greater than whole numbers but less than the next whole number. For instance, in the exercise given, you had numbers like \(3 \frac{1}{5}\) and \(1 \frac{7}{25}\). Understanding mixed numbers is vital as it allows easier interpretation of large number figures, particularly in everyday contexts such as cooking or measuring materials.
To convert a mixed number to 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.
• Write the result over the original denominator.

This converts the mixed number into a single improper fraction, making it simpler to perform arithmetic operations like addition, subtraction, multiplication, or division of fractions.

Improper Fractions

Improper fractions are fractions where the numerator (the top part of the fraction) is greater than or equal to the denominator (the bottom part). They are often considered less neat than mixed numbers but can be more convenient for calculations.
In our exercise, we converted \(3 \frac{1}{5}\) into \(\frac{16}{5}\) and \(1 \frac{7}{25}\) into \(\frac{32}{25}\). This facilitated the arithmetic needed to solve the problem.
Improper fractions allow for seamless multiplication and division of fractions. To manipulate them further, having them in this form can help when simplifying or reducing fractions to their lowest terms.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means expressing the fraction such that the numerator and denominator have no common factors other than 1.
You simplify a fraction by:

• Finding the greatest common divisor (GCD) of the numerator and denominator.
• Dividing both the numerator and the denominator by the GCD.

In the exercise, we simplified \(\frac{160}{400}\) to \(\frac{2}{5}\) by dividing both 160 and 400 by their GCD, which is 80.
Simplifying fractions makes them easier to understand and compare, and it's a crucial skill in both basic and advanced mathematics.

Greatest Common Divisor

The greatest common divisor, or GCD, is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. It's an essential tool for simplifying fractions.
To find the GCD, you can use several methods:

• Listing Factors: Enumerate all factors of both numbers and identify the largest common one. This is simple but can be time-consuming.
• Prime Factorization: Break down both numbers into their prime factors, and multiply the common prime factors together.
• Euclidean Algorithm: This involves repeated division, focusing on remainders, until reaching a remainder of zero.

Understanding the greatest common divisor helps in reducing any fraction to its simplest form, which is particularly helpful not only for simplifying fractions but also for solving other mathematical problems like those involving ratios and proportions.