Problem 51 - Fundamentals of Mathematics

Question

What part of \(\frac{23}{40}\) is \(3 \frac{9}{20} ?\)

Step-by-Step Solution

Answer

It is 6 times.

Step 1: Convert Mixed Number

First, we need to convert the mixed number \(3 \frac{9}{20}\) into an improper fraction. Multiply the whole number 3 by the denominator 20 and then add the numerator 9: \((3 \times 20) + 9 = 60 + 9 = 69\). Thus, \(3 \frac{9}{20} = \frac{69}{20}\).

Step 2: Setup Division

To find what part of \(\frac{23}{40}\) is \(3 \frac{9}{20}\), we need to divide \(\frac{69}{20}\) by \(\frac{23}{40}\). This means computing \( \frac{69}{20} \div \frac{23}{40} \).

Step 3: Division of Fractions

Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second. Thus, we compute: \( \frac{69}{20} \times \frac{40}{23} \).

Step 4: Multiply the Fractions

Multiply the numerators and the denominators: \(69 \times 40 = 2760\) and \(20 \times 23 = 460\). So, \( \frac{2760}{460} \).

Step 5: Simplify the Fraction

Simplify \( \frac{2760}{460} \) by finding the greatest common divisor (GCD) of 2760 and 460. The GCD is 20. Divide both the numerator and the denominator by 20: \( \frac{2760 \div 20}{460 \div 20} = \frac{138}{23} \). Since 138 divided by 23 equals 6, the simplified fraction is 6.

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is often used to express values that are greater than 1 but not whole. For example, the mixed number \(3 \frac{9}{20}\) consists of the whole number 3 and the fraction \(\frac{9}{20}\). To work with mixed numbers, it is important to know how to convert them into improper fractions. This involves multiplying the whole number by the denominator and adding the numerator. This method makes calculations easier. For instance, for \(3 \frac{9}{20}\), we multiply 3 by 20 and then add 9, resulting in \(\frac{69}{20}\). Converting mixed numbers to improper fractions is an essential skill in fraction arithmetic.

Improper Fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. This means the value of the fraction is greater than or equal to 1. Improper fractions are useful because they simplify the process of performing mathematical operations, like addition or division, over mixed numbers.

• Example: \(\frac{69}{20}\)
• The numerator (69) is larger than the denominator (20).

To convert a mixed number into an improper fraction, you multiply the whole number by the denominator and add this result to the numerator. This simplifies calculations by representing the entire quantity as a single fraction.

Division of Fractions

Dividing fractions might sound complex but it's simpler than it seems! To divide one fraction by another, you multiply the first fraction by the reciprocal of the second fraction.

• This is often remembered as "flip and multiply".

The reciprocal of a fraction is created by swapping its numerator and denominator. For instance, to divide \(\frac{69}{20}\) by \(\frac{23}{40}\), we take the reciprocal of \(\frac{23}{40}\), resulting in \(\frac{40}{23}\). Then, multiply \(\frac{69}{20}\) by \(\frac{40}{23}\). This gives: \[ \frac{69}{20} \times \frac{40}{23} = \frac{2760}{460} \]Understanding this process is crucial for solving problems involving fraction division effectively.

Simplifying Fractions

After performing operations like multiplication or division, you often need to simplify the resulting fraction. Simplifying a fraction means reducing it to its simplest form where the numerator and denominator have no common factors other than 1. To simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by this GCD.

For the fraction \(\frac{2760}{460}\), the GCD is 20. By dividing both terms by 20, we simplify the fraction to \(\frac{138}{23}\). Further simplification shows that \(138 \div 23 = 6\), so the fraction simplifies completely to 6. Mastering simplification ensures your answers are always in their simplest form.