Problem 40 - Algebra 1

Question

Simplify the expression. $$49 x \div 3 \frac{1}{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(14x\).

Step 1: Understand the Operation

The expression given is \(49x \div 3\frac{1}{2}\) or \(49x \div \frac{7}{2}\). This can be seen as 49x divided by 7/2.

Step 2: Convert Division into Multiplication

The operation of division by a fraction can be converted into a multiplication operation by multiplying by the reciprocal of the fraction. The reciprocal of \(3 \frac{1}{2} = \frac{7}{2}\) is \(\frac{2}{7}\). Therefore, the expression becomes \(49x \times \frac{2}{7}\).

Step 3: Perform the Multiplication

Performing the multiplication we get, \(49x \times \frac{2}{7} = 14x\).

Key Concepts

Division of Fractions

Fractions represent a part of a whole and often need to be divided, especially in algebraic expressions. Dividing fractions might seem tricky at first, but it can be made simple by using a straightforward method.
To divide one fraction by another, follow these steps:

• Identify the fraction that acts as the divisor. In our example, the fraction is \( \frac{7}{2} \).
• Understand that dividing by a fraction is the same as multiplying by its reciprocal. We'll explore reciprocals in more detail in another section.
• Convert the division operation to a multiplication operation by using the reciprocal.

For instance, in the expression \( 49x \div \frac{7}{2} \), division by a fraction \( \frac{7}{2} \) is converted into multiplication by its reciprocal, \( \frac{2}{7} \), transforming the expression to \( 49x \times \frac{2}{7} \). This technique simplifies the operation by creating an easier multiplication task rather than a division challenge.

Multiplication of Expressions

Multiplying expressions, especially those involving fractions, is fundamental in algebra. When you have an expression involving multiplication, you multiply the numerators together and the denominators together if the fractions are involved.
Consider the example we derived from the previous section: \( 49x \times \frac{2}{7} \).

• Express 49 as a fraction: \( \frac{49x}{1} \).
• Multiply the numerators: \( 49x \times 2 = 98x \).
• Multiply the denominators: \( 1 \times 7 = 7 \).

This gives us the fraction \( \frac{98x}{7} \), which is simplified by performing the division: \( 98x \div 7 = 14x \). The ability to transform different forms of expressions and simplifying them helps solve many algebraic problems efficiently.

Reciprocal of a Fraction

A reciprocal of a fraction is what you multiply by to get 1. It essentially flips the fraction upside down, swapping the numerator with the denominator.
Understanding reciprocals helps a lot in dividing fractions.

• If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
• The reciprocal of \( 3\frac{1}{2} \) first involves converting it to an improper fraction, \( \frac{7}{2} \), and then finding its reciprocal as \( \frac{2}{7} \).
• When you multiply a fraction by its reciprocal, the result is always 1: \( \frac{7}{2} \times \frac{2}{7} = 1 \).

Using the reciprocal makes dividing fractions much simpler and allows transformation of division problems into multiplication, which are easier to handle. Mastery of reciprocals is a crucial skill in algebra and will make solving expressions smoother and more intuitive.