Problem 34 - Basic College Mathematics

Question

Multiply and simplify. $$ \frac{1}{3} \cdot 120 $$

Step-by-Step Solution

Answer

40

Step 1: Identify the problem

The goal is to multiply the fraction \( \frac{1}{3} \) by 120.

Step 2: Set up the multiplication

Recognize that multiplying a fraction by a whole number involves multiplying the numerator of the fraction by the whole number. This gives us: \(\frac{1}{3} \times 120\)

Step 3: Perform the multiplication

Multiply the numerator of the fraction by the whole number: \[1 \times 120 = 120 \]

Step 4: Simplify the fraction

Place the result from the multiplication over the original denominator: \(\frac{120}{3}\). Now, simplify the fraction by dividing the numerator and the denominator by 3: \[ \frac{120 \div 3}{3 \div 3} = \frac{40}{1} = 40 \]

Key Concepts

Simplifying Fractions

Simplifying fractions is about making a fraction as simple as possible. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example, if you have the fraction \(\frac{120}{3}\), you want to find the largest number that both 120 and 3 can be divided by without a remainder. In this case, that number is 3.
You then divide both the numerator (120) and the denominator (3) by this number: \[ \frac{120 \div 3}{3 \div 3} = \frac{40}{1} = 40 \]Notice that the resulting fraction is \(\frac{40}{1} \), which simplifies to 40 since anything divided by 1 is itself. The fraction is now in its simplest form.

Numerator

The numerator is the top part of a fraction. It shows how many parts you have. For example, in the fraction \( \frac{1}{3} \), 1 is the numerator.
When multiplying fractions by whole numbers, you focus on multiplying the numerator by the whole number and keep the denominator the same. In our example: \(\frac{1}{3} \times 120 = \frac{1 \times 120}{3} \).
After performing the multiplication, the numerator becomes 120, giving us the fraction \( \frac{120}{3} \). This fraction still needs to be simplified.

Denominator

The denominator is the bottom part of a fraction. It indicates into how many equal parts the whole is divided. For instance, in the fraction \( \frac{1}{3} \), the denominator is 3.
In multiplying fractions by whole numbers, keep the denominator the same while multiplying. For example, \( \frac{1}{3} \times 120 \) results in \( \frac{120}{3} \).
After multiplication, always check if the fraction can be simplified. Here, the denominator helps us determine the GCD. Divide both the numerator and the denominator by this GCD to simplify the fraction.