Understanding the Fraction Problem

The question asks us to find a specific fraction of a number, given another fraction of the same number. We are told that three-fifths of a number is 54. We need to determine what two-ninth of that number is.

Let the unknown number be represented by the variable $x$.

Setting up the Equation from the Given Information

The first piece of information given is "three-fifths of a number is 54". We can translate this statement into a mathematical equation:

$$ \frac{3}{5} \times x = 54 $$

This equation means that multiplying the number $x$ by the fraction $\frac{3}{5}$ gives us 54.

Solving for the Unknown Number

To find the value of the number $x$, we need to isolate $x$ in the equation $\frac{3}{5}x = 54$. We can do this by performing inverse operations:

1. Multiply both sides of the equation by 5 to get rid of the denominator:
$$ 5 \times \left(\frac{3}{5}x\right) = 5 \times 54 $$ $$ 3x = 270 $$
2. Divide both sides of the equation by 3 to solve for $x$:
$$ \frac{3x}{3} = \frac{270}{3} $$ $$ x = 90 $$

So, the unknown number is 90.

Calculating Two-Ninth of the Number

Now that we know the number is 90, the question asks for "two-ninth of it". This means we need to calculate $\frac{2}{9}$ of 90.

$$ \text{Two-ninth of 90} = \frac{2}{9} \times 90 $$

To calculate this, we can multiply the numerator by 90 and divide by the denominator, or simplify first:

$$ \frac{2}{9} \times 90 = 2 \times \left(\frac{90}{9}\right) $$

Since $90 \div 9 = 10$, we have:

$$ 2 \times 10 = 20 $$

Therefore, two-ninth of the number is 20.

Step-by-Step Solution Summary

Here is a summary of the steps taken to solve the problem:

• Represent the unknown number with a variable (e.g., $x$).
• Translate the first statement ("three-fifths of a number is 54") into an equation: $\frac{3}{5}x = 54$.
• Solve the equation for $x$ to find the number: $x = 90$.
• Translate the second statement ("what is two-ninth of it?") into a calculation: $\frac{2}{9} \times 90$.
• Perform the calculation: $\frac{2}{9} \times 90 = 20$.

Comparing with Options

The calculated value is 20. Let's look at the given options:

Our calculated value of 20 matches Option 2.

Revision Table: Key Concepts

Additional Information: Solving Fraction Word Problems

Fraction word problems often involve setting up and solving equations. Here are some tips for tackling such problems:

• Read the problem carefully to understand what is given and what needs to be found.
• Identify the unknown quantity and represent it with a variable.
• Translate the sentences describing relationships between quantities into mathematical expressions or equations. Keywords like "of" often indicate multiplication, "is" indicates equality.
• Solve the equation for the unknown variable.
• Use the value of the unknown variable to answer the specific question asked.
• Always check your answer to see if it makes sense in the context of the original problem.

In this specific problem, we used the given fraction and its value to find the whole number first, and then used that whole number to find the value of a different fraction of it.