Problem 6 - Fundamentals of Mathematics

Question

What part of \(\frac{3}{5}\) is \(\frac{9}{10} ?\)

Step-by-Step Solution

Answer

\( \frac{9}{10} \) is \( \frac{3}{2} \) (or 1.5 times) of \( \frac{3}{5} \).

Step 1: Set up the Problem

To find what part of \( \frac{3}{5} \) is \( \frac{9}{10} \), we can express this as \( x \times \frac{3}{5} = \frac{9}{10} \). We need to solve for \( x \).

Step 2: Isolate the Variable

Divide both sides of the equation by \( \frac{3}{5} \) to isolate \( x \): \[ x = \frac{9}{10} \times \frac{5}{3} \].

Step 3: Simplify the Expression

Multiply \( \frac{9}{10} \) by \( \frac{5}{3} \):\[ x = \frac{9 \times 5}{10 \times 3} = \frac{45}{30} \].

Step 4: Reduce the Fraction

Reduce \( \frac{45}{30} \) by finding the greatest common divisor of 45 and 30, which is 15.This gives us:\[ x = \frac{45 \div 15}{30 \div 15} = \frac{3}{2} \].

Key Concepts

Understanding Part of a Fraction

When dealing with fractions, it's fundamental to comprehend what it means to find "part of a fraction." This involves expressing a fraction as a portion of another fraction.
For example, in the exercise provided, we need to determine what part of \( \frac{3}{5} \) is \( \frac{9}{10} \). This means we are asking, "How many times does \( \frac{3}{5} \) fit into \( \frac{9}{10} \) or conversely, how is \( \frac{9}{10} \) a fraction of \( \frac{3}{5} \)?"
To address this type of problem, we consider using the equation \( x \times \text{part} = \text{whole} \). In this context, \( \frac{9}{10} \) is the whole value, and \( \frac{3}{5} \) is the part, leading us to solve for \( x \). Understanding these terms helps set up our equation.

Solving Equations Involving Fractions

Solving equations with fractions might seem a bit tricky at first, but it's all about following a clear process. The goal is to find the unknown variable. Here’s a simple approach to these types of problems:
Begin by setting up an equation where your fraction represents one part of a whole. For the exercise, this is represented as \( x \times \frac{3}{5} = \frac{9}{10} \).

• Isolating the Variable: This involves getting the variable \( x \) on its own. In the equation \( x \times \frac{3}{5} = \frac{9}{10} \), you'll divide both sides by \( \frac{3}{5} \) to isolate \( x \).


• Solving for \( x \): Perform the division: \( x = \frac{9}{10} \times \frac{5}{3} \). Multiplying by the reciprocal is key here, allowing cancellation and easier multiplication.

This method simplifies the task and brings you one step closer to the solution.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form while keeping their original value identical.
This is done by finding the greatest common divisor (GCD) of the numerator and the denominator.
Using our exercise example:

• We endeavored to simplify \( \frac{45}{30} \).


• First, we identified the GCD of 45 and 30, which is 15.


• Then, divide both 45 and 30 by 15, resulting in \( \frac{3}{2} \).

By doing so, the fraction is easier to understand, use, and compare. Simplifying fractions helps in solving equations faster and makes the numbers easier to work with. It's a critical skill that enhances your mathematical intuition and precision.