Problem 50 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{15 a^{2}}{14 d} \div \frac{a}{d} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{15 a}{14} \).

Step 1: Rewrite the Division as Multiplication

To simplify the given expression, start by rewriting the division as multiplication of the reciprocal. The given expression is \[ \frac{15 a^{2}}{14 d} \div \frac{a}{d} \] which can be rewritten as \[\frac{15 a^{2}}{14 d} \times \frac{d}{a} \].

Step 2: Multiply the Numerators and Denominators

Next, multiply the numerators and the denominators of the fractions: \[ \frac{15 a^{2} \times d}{14 d \times a} \].

Step 3: Simplify the Expression

Simplify the expression by canceling out common factors in the numerator and the denominator. The common factors are \(d\) and \(a\). After canceling, the expression simplifies to: \[ \frac{15 a}{14} \].

Key Concepts

Division of Fractions

When dividing fractions, you need to understand that division is the same as multiplying by the reciprocal of the fraction.
The reciprocal of a fraction is simply flipping the numerator and the denominator.
For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
So, when you see a division like \(\frac{15a^{2}}{14d} \div \frac{a}{d}\), it can be rewritten as \(\frac{15a^{2}}{14d} \times \frac{d}{a}\).
This makes it much easier to work with.

Multiplication of Fractions

Multiplying fractions involves two simple steps:

• Multiply the numerators together


• Multiply the denominators together

For example, with \(\frac{15a^{2}}{14d} \times \frac{d}{a}\),
multiply the numerators: 15a\textsuperscript{2} \( \times \) d,
and multiply the denominators: 14d \( \times \) a.
This gives us \(\frac{15a^{2} \times d}{14d \times a} \).
Working step by step makes the multiplication clear and simple.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means reducing the expression to its simplest form.
This often involves canceling out common factors in the numerator and denominator.

• In our example, \(\frac{15a^{2} \times d}{14d \times a} \)

we can cancel out 'd' and 'a' because they appear in both the numerator and the denominator.
After canceling, we get \(\frac{15a}{14}\).

It's important to always look for common factors to make your expressions as simple as possible.

Reciprocal

A reciprocal is essentially flipping a fraction.
It turns the numerator into the denominator and the denominator into the numerator.
For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
When solving problems that involve division of fractions,
like \(\frac{15a^{2}}{14d} \div \frac{a}{d}\), the division turns into multiplication with the reciprocal.
This transforms the problem into \(\frac{15a^{2}}{14d} \times \frac{d}{a}\),
making the calculation simpler.