Problem 26 - Fundamentals of Mathematics

Question

\(\frac{10}{3}\) of what number is \(\frac{5}{9} ?\)

Step-by-Step Solution

Answer

The number is \( \frac{1}{6} \).

Step 1: Define the unknown

Let the unknown number be represented by the variable \( x \).

Step 2: Set up the equation

The phrase '\( \frac{10}{3} \) of what number is \( \frac{5}{9} \)' translates to the equation: \( \frac{10}{3} \times x = \frac{5}{9} \).

Step 3: Solve for x

To solve for \( x \), divide both sides of the equation by \( \frac{10}{3} \). The equation becomes:\[x = \frac{5}{9} \div \frac{10}{3}\]This is equivalent to multiplying by the reciprocal:\[x = \frac{5}{9} \times \frac{3}{10}\]Calculating this gives:\[x = \frac{5 \times 3}{9 \times 10} = \frac{15}{90}\]Simplifying \( \frac{15}{90} \) gives \( \frac{1}{6} \).

Step 4: Simplify the answer

After performing the multiplication and simplification, the value of \( x \) is found to be \( \frac{1}{6} \).

Key Concepts

Equation Solving

Solving equations is like solving puzzles. It's about finding the value of an unknown that makes an equation true. In our problem, the unknown number is represented by the variable \( x \). When we see phrases like "of what number," it hints at multiplication. To set up the problem, we translate the phrase into an equation: \( \frac{10}{3} \times x = \frac{5}{9} \). Here, \( \frac{10}{3} \) and \( \frac{5}{9} \) are fractions, and we're multiplying \( x \) to find what makes both sides equal. Solving for \( x \) means isolating \( x \) on one side of the equation. We do this by performing the same operation on both sides until \( x \) is by itself, giving us the solution.

Fractions

Fractions represent a part of a whole. They are composed of a numerator (the top number) and a denominator (the bottom number).

• The fraction \( \frac{10}{3} \) means 10 parts of something that is divided into 3 equal parts.
• The fraction \( \frac{5}{9} \) means 5 parts of something that is divided into 9 equal parts.

Fractions can be multiplied, divided, added, and subtracted, just like whole numbers. To multiply fractions, multiply the numerators together and the denominators together. For division, multiply by the reciprocal of the divisor. Simplifying fractions is also important to make them easier to understand and work with. A fraction is simplified when you divide the numerator and the denominator by their greatest common divisor.

Reciprocals

Reciprocals are the magical flip of fractions. To find the reciprocal of a fraction, we switch its numerator and denominator. For instance, the reciprocal of \( \frac{10}{3} \) is \( \frac{3}{10} \). This is because the number multiplied by its reciprocal will always equal 1.Reciprocals are especially useful in division. Dividing by a fraction is the same as multiplying by its reciprocal. In our problem, we divided \( \frac{5}{9} \) by \( \frac{10}{3} \) by multiplying \( \frac{5}{9} \) by the reciprocal \( \frac{3}{10} \). This process helps simplify the equation and find the accurate value of \( x \).