Problem 36 - Fundamentals of Mathematics

Question

\(1 \frac{9}{12}\) of what number is \(5 \frac{1}{4} ?\)

Step-by-Step Solution

Answer

The number is 3.

Step 1: Understand the Problem

We need to find a number such that when multiplied by \(1 \frac{9}{12}\), we get \(5 \frac{1}{4}\). We can rephrase the problem as: "\(x \times 1\frac{9}{12} = 5\frac{1}{4}\), find \(x\)."

Step 2: Convert Mixed Numbers to Improper Fractions

Convert \(1 \frac{9}{12}\) to an improper fraction: \(1 + \frac{9}{12} = \frac{12}{12} + \frac{9}{12} = \frac{21}{12}\). \ Convert \(5 \frac{1}{4}\) to an improper fraction: \(5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4}\).

Step 3: Set Up the Equation

We have the equation \(x \times \frac{21}{12} = \frac{21}{4}\). We need to solve for \(x\).

Step 4: Solve the Equation

To find \(x\), divide both sides of the equation by \(\frac{21}{12}\). This is equivalent to multiplying by its reciprocal. \[ x = \frac{21}{4} \times \frac{12}{21} \]

Step 5: Simplify the Expression

Simplify the fraction: \[ x = \frac{21 \times 12}{4 \times 21} = \frac{12}{4} \]The 21's cancel out from numerator and denominator, leaving \(x = \frac{12}{4} = 3\).

Key Concepts

Understanding Mixed Numbers

In mathematics, mixed numbers are a combination of a whole number and a proper fraction. You'll often see them in day-to-day scenarios like when discussing hours and minutes, such as 1 hour and 1/4 hour, noting that 1 hour is the whole number and the fraction represents the fraction of an hour. Mixed numbers can be an easier way to represent numbers greater than one, making them clearer in real-life applications.

• To convert mixed numbers to improper fractions: Multiply the whole number by the fraction's denominator.
• Add the result to the numerator of the fraction to get the new numerator.
• Place this new numerator over the original denominator.

Converting mixed numbers is essential, especially when you need to perform calculations like multiplication or division. For instance, in our problem, we converted the mixed number \(1 \frac{9}{12}\) to the improper fraction \(\frac{21}{12}\). This step helps simplify our calculations.

Improper Fractions Explained

Improper fractions are those tricky fractions where the numerator is larger than the denominator. This type of fraction can sometimes be confusing, but they are quite straightforward once understood. Improper fractions are essentially an alternative way to represent whole numbers and portions:

• The numerator is the total number of parts you have.
• The denominator is the number of parts the whole is divided into.

These fractions are especially handy in mathematical operations like multiplication and division. For example, converting a mixed number like \(5 \frac{1}{4}\) into an improper fraction gets us \(\frac{21}{4}\). This fraction tells us we have 21 parts, each a quarter of a whole. Understanding this representation makes solving equations involving fractions much simpler and more straightforward.

The Process of Fraction Multiplication

When it comes to fraction multiplication, the process is refreshingly simple compared to other operations involving fractions. The beauty lies in its straightforwardness: multiplying fractions is a matter of multiplying across.

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.

If you're using improper fractions, the multiplication becomes even more straightforward. For example, in our exercise, multiplying \(\frac{21}{4} \times \frac{12}{21}\) involves merely multiplying the numerators 21 and 12, and the denominators 4 and 21, which simplifies to \(\frac{12}{4}\) after canceling out the equivalent terms. This simplification step is critical and often streamlines complex calculations with fractions.

Solving Equations with Fractions

Equations with fractions often feel more complex, but once you have a strategy, they become much more manageable. The goal in such equations is to isolate the variable you're solving for. In our exercise, fraction use slightly complicates matters, but with a step-by-step approach, it's feasible:

• First, convert any mixed numbers to improper fractions.
• Set up the equation appropriately, aligning terms for clarity.
• Most importantly, use reciprocal multiplication to isolate the variable.

In essence, solving the equation \(x \times \frac{21}{12} = \frac{21}{4}\), involves multiplying both sides by the reciprocal of \(\frac{21}{12}\), leading to the simplified equation where we could deduce \(x = 3\). This approach of flipping the fraction you multiply by helps eliminate the fraction and solve for the variable effectively.