Problem 37 - Prealgebra

Question

Number Problem Find \(\frac{3}{4}\) of \(1 \frac{7}{9}\). (Remember that of means multiply.)

Step-by-Step Solution

Answer

The solution is \(1 \frac{1}{3}\).

Step 1: Convert the Mixed Number to an Improper Fraction

First, we need to convert the mixed number \(1 \frac{7}{9}\) into an improper fraction. To do this, multiply the whole number \(1\) by the denominator \(9\) and then add the numerator \(7\). This gives \(1 \times 9 + 7 = 16\). Thus, \(1 \frac{7}{9}\) is \(\frac{16}{9}\).

Step 2: Multiply the Fractions

Next, multiply the fraction \(\frac{3}{4}\) by the improper fraction \(\frac{16}{9}\). The multiplication of these fractions can be calculated as follows: \(\frac{3}{4} \times \frac{16}{9} = \frac{3 \times 16}{4 \times 9}\). This becomes \(\frac{48}{36}\).

Step 3: Simplify the Fraction

To simplify \(\frac{48}{36}\), find the greatest common divisor (GCD) of 48 and 36, which is 12. Divide both the numerator and denominator by 12: \(\frac{48}{36} = \frac{48 \div 12}{36 \div 12} = \frac{4}{3}\). Thus, the simplified fraction is \(\frac{4}{3}\).

Step 4: Convert to a Mixed Number

Since \(\frac{4}{3}\) is an improper fraction, convert it to a mixed number by dividing the numerator 4 by the denominator 3. This gives 1 whole and a remainder of 1. So, \(\frac{4}{3}\) as a mixed number is \(1 \frac{1}{3}\).

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator (top number) is greater than or equal to the denominator (bottom number). For example, in the fraction \( \frac{16}{9} \), 16 is the numerator that is larger than the denominator, 9. This means more parts are described by the fraction than are needed to make a whole. Improper fractions can be helpful because they simplify calculations, especially when multiplying or dividing fractions. They represent values greater than or equal to one whole, which is easier to manage in arithmetic operations because everything stays in fraction form.

Mixed Numbers

Mixed numbers combine whole numbers and fractions, like \( 1 \frac{7}{9} \). This representation is useful for easily understanding quantities larger than one. However, when performing operations like multiplication or division, it is more efficient to convert mixed numbers to improper fractions. Why? Because it allows us to handle the entire number with a single arithmetic rule, conserving the integrity of our mathematical calculations. So, for multiplying \( 1 \frac{7}{9} \) by another fraction, we first transform it to an improper fraction, \( \frac{16}{9} \), keeping everything consistent.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two numbers without leaving a remainder. It's crucial when simplifying fractions, as it helps in reducing a fraction to its simplest form. Consider simplifying \( \frac{48}{36} \). First, find the GCD of 48 and 36.

• List the divisors of each number: For 48, they are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. For 36, they are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
• The largest common number in both lists is 12.

We divide both the numerator and the denominator by this GCD to simplify the fraction, turning \( \frac{48}{36} \) into \( \frac{4}{3} \). This reduction makes the fraction easier to interpret and utilize.

Fraction Multiplication

When you multiply fractions, you're essentially multiplying the numerators together and the denominators together. This forms a new fraction. Consider multiplying \( \frac{3}{4} \) by \( \frac{16}{9} \). You multiply the numerators: 3 and 16, giving 48. Then multiply the denominators: 4 and 9, resulting in 36. You get a new fraction, \( \frac{48}{36} \).
To streamline calculations and arrive at simpler forms quickly, you can simplify fractions before multiplication. As you see, we previously simplified \( \frac{48}{36} \) after multiplication, but pre-simplifying numerators and denominators when possible can save time. This is a useful skill for future calculations, allowing quick work with complex numbers while maintaining accuracy.