Problem 52 - Fundamentals of Mathematics

Question

\(\frac{8}{9}\) of what number is \(\frac{2}{3} ?\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We want to find a number, which when multiplied by \(\frac{8}{9}\) will result in \(\frac{2}{3}\).

Step 2: Set Up the Equation

Let the unknown number be \(x\). The equation based on the problem description is: \(\frac{8}{9} \times x = \frac{2}{3}\).

Step 3: Solve for x

To isolate \(x\), divide both sides by \(\frac{8}{9}\). This gives \(x = \frac{2}{3} \div \frac{8}{9}\).

Step 4: Divide Fractions

Dividing by a fraction is the same as multiplying by its reciprocal. So, \(x = \frac{2}{3} \times \frac{9}{8}\).

Step 5: Multiply Fractions

Multiply the numerators and the denominators: \(x = \frac{2 \times 9}{3 \times 8} = \frac{18}{24}\).

Step 6: Simplify the Fraction

Simplify \(\frac{18}{24}\) by their greatest common divisor, which is 6. So, \(x = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}\).

Step 7: Conclusion

The number that satisfies the condition is \(\frac{3}{4}\).

Key Concepts

Solving Equations

When faced with the task of solving an equation, it's crucial to understand that the goal is to find the value of the unknown variable that makes the equation true. Consider the original problem: "\(\frac{8}{9}\) of what number is \(\frac{2}{3}\)?". Here, we need to determine a number, which we'll call \(x\), that when multiplied by \(\frac{8}{9}\) yields \(\frac{2}{3}\). So the first step is to translate the word problem into a mathematical equation: \(\frac{8}{9} \times x = \frac{2}{3}\). To solve for \(x\), you need to isolate it on one side of the equation. This involves performing operations that will "undo" the multiplication by \(\frac{8}{9}\). This is done by dividing both sides of the equation by \(\frac{8}{9}\). Remember, whatever you do to one side, you must do to the other to maintain balance.

• Equation setup: \(\frac{8}{9} \times x = \frac{2}{3}\)
• Isolate \(x\) by dividing by \(\frac{8}{9}\)
• Resulting equation: \(x = \frac{2}{3} \div \frac{8}{9}\)

Equations like these form the backbone of many math problems, focusing on the balance and logic to deduce unknown values.

Fraction Division

Dividing fractions may initially appear daunting, but it's a simple concept once you understand its logic. The division of fractions is actually done by multiplying by the reciprocal of the divisor. A reciprocal of a fraction is simply obtained by swapping its numerator and denominator. In our solution, we faced \( \frac{2}{3} \div \frac{8}{9} \). To divide these fractions, you must multiply \(\frac{2}{3}\) by the reciprocal of \(\frac{8}{9}\) which is \(\frac{9}{8}\). The equation then becomes \(\frac{2}{3} \times \frac{9}{8}\). Remember:

• Take the reciprocal: If dividing by \(\frac{8}{9}\), change to multiplying by \(\frac{9}{8}\)
• Multiply the fractions: Numerator with numerator and denominator with denominator

By applying this method, you maintain the integrity of the division operation while simplifying your calculations.

Simplifying Fractions

Once you have multiplied your fractions, like our intermediate result \( \frac{18}{24} \), the next critical step is to simplify it. Simplifying fractions involves reducing them to their smallest form; this makes them easier to understand and work with in subsequent calculations.Simplification requires finding the greatest common divisor (GCD) of the numerator and the denominator. In \(\frac{18}{24}\), the GCD is 6. Both 18 and 24 are divisible by 6. Simplifying correctly means that the fraction \(\frac{18}{24}\) becomes \(\frac{3}{4}\) after dividing both terms by 6.

• Find GCD of numerator and denominator
• Divide both terms by the GCD
• Simplified fraction: \(\frac{3}{4}\)

Remember, simplification does not change the value of the fraction; it just presents it in its most reduced form, making it more straightforward for further use or interpretation.