Question

Identify the unknown amount as the percent, part, or whole. \(15 \%\) of what number is \(72 ?\)

Short answer

Answer: The whole number is 480.

Step-by-step solution

Understand the problem

We are given that 15% of a certain number is 72, and we need to find the whole number.

Write the equation

We can represent the problem with the equation: \(\frac{15}{100}W = 72\), where W is the whole number we want to find.

Solve for W

To solve for W, we will first multiply both sides of the equation by 100: \(15W = 7200\) Now, we will divide both sides by 15: \(W = \frac{7200}{15}\) Calculating the value for W: \(W = 480\) So, 15% of 480 is 72.

Key concepts

Unknown Quantity

In any mathematical problem, identifying the unknown quantity is the key to finding a solution. The unknown quantity is the value you are trying to find. You may see it symbolized as a variable like "W" or "X". In this exercise, our unknown is the whole number ("W") that 15% equals 72. Sometimes, understanding what the unknown quantity represents can take a bit of practice. It either could relate to a percentage, part, or whole in percent problems. In practical terms, you should ask three crucial questions:

• Is the unknown a part of the total amount?
• Is the unknown the total or whole amount itself?
• Or perhaps, is the unknown the percentage involved?

By answering these questions, you can determine how to approach the problem best and set up your equations.

Equation Solutions

Solving equations requires a clear and systematic approach to find the value of the unknown. In our original problem, the equation we derived is \( \frac{15}{100}W = 72 \). This equation tells us that 15% of "W" equals 72. To solve this, we first multiply both sides by 100 to get rid of the fraction:

1. This results in \( 15W = 7200 \).
2. Next, divide both sides by 15. This isolates "W" on one side of the equation.

In detailed steps:

• First, \( 15W = 7200 \) so divide both sides by 15.
• This yields \( W = \frac{7200}{15} \).
• Perform the division, \( W = 480 \).

Thus, the whole number you were looking for is 480. Mastering the skills of solving equations is vital as it applies to numerous mathematical problems beyond simple percent scenarios.

Proportional Reasoning

Proportional reasoning is all about using the concept of ratios and understanding that two sets of numbers are connected in a way that you can predict one if you know the other. In percent problems, mastering this skill can help solve problems like the one in our exercise.Here, 15% represents a part of the whole number. Proportional reasoning helps us see that

• "part/whole = percent/100". It's about linking parts to a known whole via a percentage relation.
• You can set up a proportion: \( \frac{15}{100} = \frac{72}{W} \).

By rearranging the proportion, you double-check:

• \( 15W = 72 \times 100 \).
• Understanding these links confirms our earlier steps showing that 7200 divided by 15 gives "W".

Using proportional reasoning simplifies many percent problems considerably. It's a mathematical bridge between concrete amounts and abstract percentages.