Problem 54 - Prealgebra

Question

Find 3 times the difference of \(1 \frac{7}{9}\) and \(\frac{2}{9}\).

Step-by-Step Solution

Answer

The result is \(\frac{14}{3}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(1 \frac{7}{9}\) into an improper fraction. Multiply the whole number (1) by the denominator (9) and add the numerator (7), which gives you \(9 + 7 = 16\). Thus, \(1 \frac{7}{9}\) is equal to \(\frac{16}{9}\).

Step 2: Calculate the Difference

Next, subtract the fractions \(\frac{16}{9}\) and \(\frac{2}{9}\). Since these fractions have the same denominator, you can subtract the numerators directly: \(16 - 2 = 14\). Hence, \(\frac{16}{9} - \frac{2}{9} = \frac{14}{9}\).

Step 3: Multiply the Difference by 3

Finally, find 3 times the difference we calculated. Multiply \(\frac{14}{9}\) by 3: \(3 \times \frac{14}{9} = \frac{42}{9}\). Simplify this fraction by dividing the numerator and the denominator by their greatest common divisor, which is 3: \(\frac{42 \div 3}{9 \div 3} = \frac{14}{3}\).

Key Concepts

Mixed Numbers

A mixed number combines a whole number with a fraction. It is often used in everyday life because it represents numbers greater than one without having to use improper fractions, which can be a bit clunkier to interpret initially. In a mixed number like \(1 \frac{7}{9}\), the "1" is the whole number and the "\frac{7}{9}\" is the fractional part. This way of representing numbers is useful because it is visually simpler and more intuitive in many situations. However, for mathematical operations, converting mixed numbers to improper fractions can simplify calculations.

Improper Fractions

Improper fractions are fractions where the numerator, the number above the line, is greater than or equal to the denominator, which is the number below the line. For example, \(\frac{16}{9}\) is an improper fraction since 16 is greater than 9. They are very useful in mathematical operations like multiplication and division because they provide a straightforward way to handle parts of a whole. Converting mixed numbers to improper fractions involves:

• Multiplying the whole number by the denominator
• Adding the numerator to this product

So for \(1 \frac{7}{9}\), it becomes \(\frac{16}{9}\) by calculating \(1 \times 9 + 7 = 16\). All you need to do after that is place this number over the original denominator.

Fraction Multiplication

Multiplying fractions is straightforward once you catch the rhythm of it. To multiply fractions, simply multiply the numerators together and the denominators together. For example, to multiply \(\frac{14}{9}\) by 3, you can visualize the 3 as \(\frac{3}{1}\). This turns the multiplication into:

• Numerator: \(14 \times 3 = 42\)
• Denominator: \(9 \times 1 = 9\)

Thus, \(\frac{14}{9} \times 3 = \frac{42}{9}\). You should simplify your result whenever possible. In this case, simplify \(\frac{42}{9}\) by dividing both the numerator and the denominator by 3, resulting in \(\frac{14}{3}\). Simplicity is key in your final expression.

Fraction Subtraction

Subtracting fractions involves making sure the fractions have the same denominator, allowing you to subtract numerators directly. In this exercise, both fractions, \(\frac{16}{9}\) and \(\frac{2}{9}\), already have the same denominator of 9. This makes subtraction straightforward:

• Simply subtract the numerators: \(16 - 2 = 14\).
• Keep the denominator the same: \(\frac{14}{9}\).

If fractions do not initially share the same denominator, you'll need to find a common denominator before you can subtract them. Once you achieve this, it's as simple as subtracting the numerators and keeping the denominator as is. Always remember to check if your answer can be simplified.