Problem 4 - Fundamentals of Mathematics

Question

\(\frac{3}{5}\) of what number is \(\frac{9}{20} ?\)

Step-by-Step Solution

Answer

The number is \( \frac{3}{4} \).

Step 1: Understand the Problem

We are given that \( \frac{3}{5} \) of a certain number equals \( \frac{9}{20} \). We need to find this unknown number, which we can represent as \( x \).

Step 2: Set up the Equation

Translate the problem into a mathematical equation: \( \frac{3}{5} \times x = \frac{9}{20} \). Here, \( x \) represents the number we are looking to find.

Step 3: Solve for x by Isolating the Variable

To find \( x \), divide both sides of the equation by \( \frac{3}{5} \):\[x = \frac{9}{20} \div \frac{3}{5}\]

Step 4: Simplify the Division of Fractions

To divide by a fraction, multiply by its reciprocal:\[x = \frac{9}{20} \times \frac{5}{3}\]

Step 5: Multiply the Fractions

Perform the multiplication:\[x = \frac{9 \times 5}{20 \times 3} = \frac{45}{60}\]

Step 6: Simplify the Result

Simplify \( \frac{45}{60} \) by finding the greatest common divisor (GCD), which is 15:\[x = \frac{45 \div 15}{60 \div 15} = \frac{3}{4}\]

Step 7: Verify the Solution

Check the solution by substituting \( x = \frac{3}{4} \) back into the original problem to ensure \( \frac{3}{5} \times \frac{3}{4} = \frac{9}{20} \). Calculate:\[\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}\]This confirms that the solution is correct.

Key Concepts

Equation Solving

Solving equations is all about finding unknown values that make an equation true. When you deal with fractions, this process is quite similar to equations with regular numbers, but with an added layer of understanding the rules of fractions. To solve an equation like \(\frac{3}{5} \times x = \frac{9}{20}\), follow these basic steps:

• Translate the Problem: Convert the word problem into a mathematical expression.
• Isolate the Variable: Rearrange the equation to get the unknown variable (\(x\) in our case) alone on one side. This is typically done by performing operations that reverse what’s been done to it. If \(x\) is multiplied by a fraction, you would divide by the same fraction.
• Verify the Solution: Substitute your found variable back into the original equation to check your work.

Understanding the roots of solving equations with fractions greatly aids in ensuring the breakdown of complex problems into easier steps. It is just as important to ensure any solution you derive is logical and holds true when checked against the initial problem statement.

Reciprocal Multiplication

Reciprocal multiplication is a useful technique when dealing with division in equations, especially when fractions are involved. A reciprocal is simply a flipped version of a fraction. For instance, the reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\). Why does this matter? It's because dividing by a fraction is equivalent to multiplying by its reciprocal. Let's look into this:

• When you need to divide \(\frac{9}{20}\) by \(\frac{3}{5}\), instead of actually dividing, you multiply by its reciprocal: \(\frac{9}{20} \times \frac{5}{3}\).
• This formula simplifies solving equations, removing the hassle of complex division of fractions.

Reciprocal multiplication streamlines solving equations since it converts complex division into a simpler multiplication, which most students find less intimidating. This concept is foundational in understanding how to manipulate fractions effectively.

Greatest Common Divisor

Simplifying fractions is often necessary to express results in their simplest form. The greatest common divisor (GCD) plays a crucial role in this process. The GCD of two numbers is the largest integer that divides both numbers without any remainder. In fraction terms, it’s what you use to reduce the fraction:

• For example, to simplify \(\frac{45}{60}\), you first find the GCD of 45 and 60, which is 15.
• Then, divide the numerator and the denominator by this GCD: \(\frac{45 \div 15}{60 \div 15} = \frac{3}{4}\).

Understanding the GCD is vital because it not only simplifies fractions during your calculations but also helps reaffirm your solutions by ensuring they are presented in their simplest form. This makes comparisons easier and ensures clarity in mathematical communication.