Problem 39 - Algebra 1

Question

Simplify the expression. $$33 x \div \frac{3}{11}$$

Step-by-Step Solution

Answer

The simplified expression is \(121x\).

Step 1: Convert the division operation to multiplication

The expression should first be rewritten to: \(33x \cdot \frac{11}{3}\). This step is based on the fact that any number divided by a fraction is equivalent to the number multiplied by the reciprocal of the fraction.

Step 2: Simplify the multiplication

Now you can carry out the multiplication. The coefficient of the variable x, 33, is multiplied by \(\frac{11}{3}\), which equals \(121\). So the expression transforms into \(121x\).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while preserving their original value. It's like cleaning up a messy room so you can find everything you need more easily. In algebra, simplifying often means performing all the operations to produce a single, cleaner form of the equation or expression.

• Combine like terms.
• Perform arithmetic operations such as addition, subtraction, multiplication, and division.
• Use properties of numbers, such as the distributive, associative, and commutative properties.

By simplifying, you make the expression more manageable and easier to work with for further calculations or to solve an equation. In our original exercise, the initial expression was simplified to obtain a clearer result.

Division of Fractions

Dividing fractions might seem tricky at first, but it can be easily handled once you understand the process. When faced with a division of fractions, the crucial step is to take the reciprocal of the divisor and multiply instead. This is because dividing by a number is equivalent to multiplying by its reciprocal.
For instance, in the expression \(33x \div \frac{3}{11}\), you switch the division to multiplication and flip the fraction, resulting in \(33x \cdot \frac{11}{3}\).

• Find the reciprocal of the divisor (flip the second fraction).
• Change the division sign to a multiplication sign.
• Proceed with the multiplication of the fractions.

Multiplication of Fractions

Once you've converted division into multiplication, the next step is to multiply the fractions. This involves multiplying the numerators with each other and the denominators with each other. Here's how to do it:

• Multiply the numerator of the first fraction by the numerator of the second fraction to get the new numerator.
• Multiply the denominator of the first fraction by the denominator of the second fraction to get the new denominator.

In the example \(33x \cdot \frac{11}{3}\), the multiplication results in \((33 \times 11) / 3 = 121\), so the expression becomes \(121x\). When multiplying fractions, ensure that the computation is simplified, if possible, by reducing any common factors.

Reciprocal of a Fraction

The reciprocal of a fraction is simply flipping the fraction upside down. This switch is a powerful tool, especially in division, because it transforms a division problem into a multiplication one. For any fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).

• To find the reciprocal, exchange the numerator with the denominator.
• When you multiply by a reciprocal, you effectively perform a division.

In our exercise, finding the reciprocal of \(\frac{3}{11}\) turned it into \(\frac{11}{3}\). This allowed us to convert the division operation in our expression to multiplication, simplifying the process considerably.