Problem 73 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{18}{19} \cdot 38$$

Step-by-Step Solution

Answer

The product simplifies to 36.

Step 1: Multiply the Numerator

First, multiply the numerator of the fraction by the whole number. Here, we have \(18\) (numerator of the fraction) and \(38\) (whole number). Multiply them:\[18 \times 38 = 684\] This result becomes the new numerator of the fraction.

Step 2: Use the Denominator of the Original Fraction

The denominator of the original fraction is \(19\). The product will use this same denominator, giving us the fraction:\[\frac{684}{19}\].

Step 3: Simplify the Fraction to Lowest Terms

To reduce the fraction, check if the numerator and the denominator share any common factors. Divide both by the greatest common divisor (GCD). Here, we need to find \(684 \div 19\).Perform the division:\(684 \div 19 = 36\). No remainder suggests the fraction simplifies to a whole number.

Step 4: Verify the Simplification

The calculation shows \(\frac{684}{19} = 36\), thus confirming the numerator is evenly divisible by the denominator. Therefore, the fraction simplifies to \(36\).

Key Concepts

Reducing Fractions

Reducing fractions, also known as simplifying fractions, is a key math concept. It involves making the fraction as simple as possible. When a fraction is reduced to its simplest form, the numerator and denominator will have no common factors other than 1.
To simplify a fraction, you must check if the numerator and denominator have a common divisor. This divides exactly into both numbers without leaving a remainder.
For example, if you have a fraction like \( \frac{684}{19} \), you try to divide both numbers by the same value. Here, both can be divided perfectly resulting in a whole number. It becomes crucial in math because it makes complex calculations more manageable.

• The simplified form helps in easier math operations, saving time and effort.
• It increases the accuracy of calculations as simpler numbers have fewer chances of error.

Practice is key in mastering fraction reduction, as recognizing common factors quickly is a valuable skill.

Numerator and Denominator

In a fraction, understanding the roles of the numerator and denominator is fundamental. The numerator is the top part of the fraction, and the denominator is the bottom part.
When you multiply fractions by a whole number, as in the exercise, you primarily focus on the numerator. You multiply it by the whole number to find a new value.
Here’s how the process works:

• Start by multiplying the numerator (top number) by the whole number.
• Use the existing denominator (bottom number) since the whole number is not a fraction.

This will create a new fraction where only the numerator changes, as demonstrated in the solution where \( 18 \times 38 = 684 \).
After this multiplication, you continue with simplifying the fraction using the denominator.

Greatest Common Divisor

The greatest common divisor (GCD) is a critical concept when simplifying fractions. It helps identify which number can perfectly divide both the numerator and the denominator.
The GCD is the largest number that both numbers can be divided by without leaving a remainder. This number is essential when seeking to reduce a fraction to its lowest terms.

• Finding the GCD involves listing the factors of both numbers.
• Then, select the largest factor that appears in both lists.
• Use this factor to divide the numerator and the denominator.

As in the exercise, once you find that \( 684 \) can be divided by \( 19 \) without any remainder, the GCD is \( 19 \). It allows simplifying the fraction easily because the division results in a whole number.
This simplification confirms whether the fraction can be expressed more simply, enhancing understanding and ease of use.