Understanding the Problem: Finding the Number

The question asks us to find an unknown number based on a given condition. The condition states that 3/5 of 80 percent of this number is equal to 450. We need to translate this word problem into a mathematical equation and then solve for the unknown number.

Breaking Down the Problem

Let's represent the unknown number with a variable, say 'x'. Now, let's break down the condition step by step:

1. "80 percent of a number": This can be written as \(80\% \text{ of } x\), which is \( \frac{80}{100} \times x \).
2. "3/5 of 80 percent of a number": This means we take 3/5 of the expression from step 1. So, it becomes \( \frac{3}{5} \times \left( \frac{80}{100} \times x \right) \).
3. "is 450": This means the expression from step 2 is equal to 450.

Putting it all together, the equation is:

\( \frac{3}{5} \times \frac{80}{100} \times x = 450 \)

Solving the Equation to Find the Number

Now we need to solve the equation for 'x'.

\( \frac{3}{5} \times \frac{80}{100} \times x = 450 \)

First, simplify the fraction \( \frac{80}{100} \). Both 80 and 100 are divisible by 20:

\( \frac{80}{100} = \frac{80 \div 20}{100 \div 20} = \frac{4}{5} \)

Substitute this back into the equation:

\( \frac{3}{5} \times \frac{4}{5} \times x = 450 \)

Now, multiply the fractions on the left side:

\( \frac{3 \times 4}{5 \times 5} \times x = 450 \)

\( \frac{12}{25} \times x = 450 \)

To find 'x', we need to isolate it. Multiply both sides of the equation by the reciprocal of \( \frac{12}{25} \), which is \( \frac{25}{12} \):

\( x = 450 \times \frac{25}{12} \)

Now, we calculate the value:

\( x = \frac{450 \times 25}{12} \)

We can simplify by dividing 450 by 12. Let's do this in steps:

\( \frac{450}{12} = \frac{225}{6} \quad \text{(Dividing numerator and denominator by 2)} \)

\( \frac{225}{6} = \frac{75}{2} \quad \text{(Dividing numerator and denominator by 3)} \)

\( \frac{75}{2} = 37.5 \)

So, the equation becomes:

\( x = 37.5 \times 25 \)

Now, perform the multiplication:

\( 37.5 \times 25 \)

We can calculate this as \( (30 + 7 + 0.5) \times 25 \) or \( 37.5 \times (20 + 5) \).

\( 37.5 \times 20 = 750 \)

\( 37.5 \times 5 = 187.5 \)

\( x = 750 + 187.5 = 937.5 \)

So, the unknown number is 937.5.

Verifying the Result

Let's check if 3/5 of 80 percent of 937.5 is indeed 450.

• 80 percent of 937.5 = \( \frac{80}{100} \times 937.5 = 0.8 \times 937.5 = 750 \)
• 3/5 of 750 = \( \frac{3}{5} \times 750 \)

Calculate \( \frac{3}{5} \times 750 \):

\( \frac{3}{5} \times 750 = 3 \times \frac{750}{5} = 3 \times 150 = 450 \)

The result matches the given condition. So, the number is 937.5.

Revision Table: Key Concepts

Additional Information: Percentage and Fraction Relationships

Percentages and fractions are different ways of representing parts of a whole. Understanding how to convert between them is crucial for solving problems like this one. A percentage is a fraction out of 100. For example, 80% means 80 out of 100, or \( \frac{80}{100} \).

Fractions can be simplified to their lowest terms, like \( \frac{80}{100} \) simplifying to \( \frac{4}{5} \). This simplification often makes calculations easier.

When dealing with "a fraction of a percentage of a number," you are essentially multiplying these parts together: Fraction × Percentage (as a decimal or fraction) × Number.

In our problem, we used fractions throughout: \( \frac{3}{5} \times \frac{80}{100} \times x = 450 \). We could also have used decimals: \( 0.6 \times 0.8 \times x = 450 \), which simplifies to \( 0.48x = 450 \). Solving for x would be \( x = \frac{450}{0.48} \). Let's check this:

\( \frac{450}{0.48} = \frac{45000}{48} \)

Divide 45000 by 48:

\( \frac{45000}{48} = \frac{22500}{24} = \frac{11250}{12} = \frac{5625}{6} = 937.5 \)

Both methods yield the same result, confirming our answer. Using fractions can sometimes be more precise as it avoids rounding decimals prematurely.