Problem 30 - Prealgebra

Question

Multiply each of the following. Be sure all answers are written in lowest terms. $$\frac{3}{5} \cdot 15$$

Step-by-Step Solution

Answer

The product is 9.

Step 1: Break Down the Problem

To multiply a fraction by a whole number, convert the whole number into a fraction. The whole number 15 can be written as \( \frac{15}{1} \). Now the problem is \( \frac{3}{5} \cdot \frac{15}{1} \).

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \( 3 \times 15 = 45 \). The new numerator is 45.

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \( 5 \times 1 = 5 \). The new denominator is 5.

Step 4: Form a New Fraction

Combine the results from Steps 2 and 3 to form a new fraction: \( \frac{45}{5} \).

Step 5: Simplify the Fraction

Divide both the numerator and the denominator by their greatest common divisor, which is 5. So, \( \frac{45}{5} \) simplifies to \( 9 \).

Key Concepts

Multiplying Fractions

When it comes to multiplying fractions, it's pretty straightforward! Here’s the gist of it: you multiply the top numbers (numerators) together and the bottom numbers (denominators) together. If you have fractions like \( \frac{a}{b} \times \frac{c}{d} \), you'll simply do the math:

• First, multiply the numerators: \( a \times c \)
• Next, multiply the denominators: \( b \times d \)

This will give you a new fraction: \( \frac{a \times c}{b \times d} \). Easy as pie! However, if you’re multiplying a fraction by a whole number, don’t sweat it! Just convert that whole number into a fraction by placing it over 1. For example, the number 15 becomes \( \frac{15}{1} \). Then, follow the same steps as usual. Just as in the given problem, multiplying \( \frac{3}{5} \cdot \frac{15}{1} \) is just like any other fraction multiplication.

Simplifying Fractions

Simplifying fractions makes our numbers extra tidy! It means reducing the fraction to its smallest form, so it’s easier to read and compare. To simplify the fraction \( \frac{45}{5} \), you find the greatest common divisor (GCD) of the numerator (45) and the denominator (5). The GCD is simply the biggest number that divides both numbers without leaving a remainder. In our example, 5 is the GCD. Simplification works like this:

• Divide the numerator by the GCD: \( 45 \div 5 = 9 \)
• Divide the denominator by the GCD: \( 5 \div 5 = 1 \)

So, \( \frac{45}{5} \) simplifies to \( \frac{9}{1} \) or just 9, since any number over 1 is simply the number itself. Remember, simplifying fractions is like tidying up a room – you bring everything down to the basic essentials!

Numerator and Denominator

Fractions are all about the relationship between two numbers. When you look at a fraction, like \( \frac{3}{5} \), you see two parts. The top number, 3, is called the numerator. It tells you how many parts you have. Meanwhile, the bottom number, 5, is the denominator. It shows the total number of equal parts something is divided into. Think of slicing a pizza:

• If you have 3 slices out of 5, then \( \frac{3}{5} \) means you have 3 slices (numerator) of a pizza that’s divided into 5 slices (denominator).

This way of thinking makes fractions easier to understand; plus, when multiplying or dividing fractions, you'll know what part you’re working with. Remember:

• Numerator = How many parts you have
• Denominator = Total parts available

This concept will make fractions your new best friend in math!