Problem 22 - Algebra 1

Question

Find the product. $$(-15)\left(\frac{3}{5}\right)$$

Step-by-Step Solution

Answer

The product of (-15) and \( \frac{3}{5} \) is -9.

Step 1: Apply Multiplication

Multiply the number -15 with the fraction \( \frac{3}{5} \). The calculation becomes \( -15 * \frac{3}{5} \) = -9.

Step 2: Simplify if Needed

In this case, no further simplification is needed as -9 is already a simplified integer.

Key Concepts

Applying Multiplication

When multiplying a whole number and a fraction, it's useful to think of the whole number as a fraction itself. This allows us to keep things simple. Imagine we're multiplying \(-15\) by \(\frac{3}{5}\). We can see this as multiplying by \(\frac{15}{1}\) instead. This sets us up to just multiply straight across.
Take the numerators from both: \(-15\) and \(3\), and multiply them together to get \(-45\). Then, do the same for the denominators: \(1\) and \(5\), which gives \(5\). This results in \(\frac{-45}{5}\). By treating \(-15\) as \(\frac{15}{1}\), tasks become more straightforward. Simple, right?

• Convert whole numbers to fractions.
• Multiply numerators together.
• Multiply denominators together.
• Resulting fraction is \(\frac{-45}{5}\).

Just by taking these steps, you can easily multiply fractions with integers.

Simplifying Fractions

Simplification is the process of reducing a fraction to its smallest form. In our multiplication, we arrived at the fraction \(\frac{-45}{5}\). Simplifying means finding a form where the numerator and the denominator have no numbers in common other than \(1\). Here, we divide both the top and bottom by the greatest common divisor, which is the largest number that fits into both evenly.
For \(\frac{-45}{5}\), the greatest common divisor is \(5\). By dividing both by \(5\), we simplify it to \(-9\). The idea is to ensure the fraction is as condensed as possible, making calculations easier. Simplifying changes how it looks, not what it is. This means \(\frac{9}{1} = 9\), but since we have already a simplified single value here, no change is needed further.

• Simplification means reducing a fraction to its simplest form.
• Find the greatest common divisor of the numerator and the denominator.
• Divide both numerator and denominator by this number.
• End result for our example is truly \(-9\).

It's all about making numbers simpler and easier to work with.

Integer Multiplication

Multiplying integers is a basic yet crucial mathematical skill. It involves combining whole numbers, which can be both positive and negative. A key point with integer multiplication is understanding the sign rules. When multiplying two numbers:

• If both integers have the same sign (both positive, or both negative), the result is positive.
• If the integers have different signs (one positive, one negative), the result is negative.

In our problem, we had \(-15\), a negative integer, and \(\frac{3}{5}\), a positive fraction. When performing integer multiplication here, remember that \(\frac{-15 \,*\, 3}{5} = -9\). Understanding integer multiplication is vital as it often sets the groundwork for various algebraic concepts.
Keeping these principles in mind helps enhance number sense and reinforces the simplicity of multiplying and understanding the relationships between integers and their signs.