Problem 15 - Prealgebra

Question

Write your answers as proper fractions or mixed numbers, not as improper fractions. Find the following products. (Multiply.) $$\frac{3}{4} \cdot 7 \cdot 1 \frac{4}{5}$$

Step-by-Step Solution

Answer

The product is 9 \(\frac{9}{20}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number to an improper fraction. The mixed number is 1 \(\frac{4}{5}\). To convert it, multiply the whole number 1 by the denominator 5 and add the numerator 4: \((1 \times 5) + 4 = 9\). So, 1 \(\frac{4}{5}=\frac{9}{5}\).

Step 2: Write the Entire Expression with Common Denominator

Now our expression is \(\frac{3}{4} \cdot 7 \cdot \frac{9}{5}\). To ease the multiplication, express 7 as a fraction: \(7 = \frac{7}{1}\). Now the expression is \(\frac{3}{4} \cdot \frac{7}{1} \cdot \frac{9}{5}\).

Step 3: Multiply the Numerators

Multiply the numerators of the fractions: \(3 \times 7 \times 9 = 189\).

Step 4: Multiply the Denominators

Multiply the denominators of the fractions: \(4 \times 1 \times 5 = 20\).

Step 5: Write the Product as an Improper Fraction

The product of the fractions is \(\frac{189}{20}\). This is an improper fraction.

Step 6: Convert Improper Fraction to Mixed Number

Divide 189 by 20. The quotient is 9 and the remainder is 9, so we write the mixed number as \(9 \frac{9}{20}\).

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a proper fraction. It represents the sum of these two components.
For example, in the mixed number 1 \(\frac{4}{5}\), the 1 is the whole number, and \(\frac{4}{5}\) is the fraction.
Mixed numbers are often used because they provide more intuitive representation of quantities, especially in daily life scenarios.

• To read a mixed number, simply interpret it as "1 whole and \(\frac{4}{5}\) of another whole" in this case.
• Mixed numbers are beneficial when relating to actual counts or measurements.

Understanding mixed numbers and how to manipulate them is essential for various mathematical operations such as addition, subtraction, multiplication, and division.

Converting Fractions

Converting fractions often involves changing a fraction from one form to another for ease of calculation. The most common conversions are between mixed numbers and improper fractions.
Converting between these forms lets us handle different types of arithmetic efficiently.

• To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction and add the numerator.
• This gives you the new numerator, while the denominator remains unchanged.
• For example, to convert 1 \(\frac{4}{5}\) to an improper fraction: \(1 \times 5 + 4 = 9\), so it becomes \(\frac{9}{5}\).

Conversely, converting an improper fraction back to a mixed number means dividing the numerator by the denominator to find the whole number, with any remainder becoming the new numerator.

Improper Fractions

An improper fraction has a numerator greater than or equal to its denominator. These fractions indicate more than a whole and are common in various calculations and measurements.
They are particularly useful in multiplications or when simplifying or combining fractions.

• In \(\frac{189}{20}\), the numerator (189) is much larger than the denominator (20), indicating a fraction that exceeds one whole.
• Improper fractions can be easily converted to mixed numbers to provide a clearer idea of the size of the number, especially in practical contexts.
• For example, dividing 189 by 20 gives 9 whole parts with a remainder of 9, resulting in the mixed number \(9 \frac{9}{20}\).

While improper fractions are mathematically efficient, mixed numbers are often more understandable in application settings.