Percentage Solver

Every percent question is really one equation wearing three different costumes. Once you label which of the three quantities is missing, the arithmetic is short. The three quantities are the part, the whole, and the rate.

• part: the amount you are talking about
• whole: the full amount the part comes from
• rate: the percent, written as a decimal in equations

The Formula and Why It Holds

The single relationship behind every percent problem is

$$\text{part} = \text{rate} \times \text{whole}$$

where

$$\text{rate} = \frac{\text{percent}}{100}$$

Why does one equation cover everything? Because "a percent of a whole" is a multiplication. Saying "$30\%$ of $60$" means taking $\frac{30}{100}$ of $60$, which is exactly $\text{rate} \times \text{whole}$. The three famous question types are just this equation solved for a different letter:

1. What is $p\%$ of $W$? — the part is missing, so multiply.
2. $P$ is what percent of $W$? — the percent is missing, so divide the part by the whole.
3. $P$ is $p\%$ of what number? — the whole is missing, so divide the part by the rate.

The Three Solved Forms

When the whole and percent are known:

$$\text{part} = \frac{p}{100} \times \text{whole}$$

When the part and whole are known:

$$\text{percent} = \frac{\text{part}}{\text{whole}} \times 100\%$$

This requires $\text{whole} \ne 0$.

When the part and percent are known:

$$\text{whole} = \frac{\text{part}}{p/100}$$

This requires $p \ne 0$.

Worked Example: $18$ Is $30\%$ of What Number?

Here the part is $18$ and the percent is $30\%$. The whole is missing.

Convert the percent to a decimal:

$$30\% = 0.30$$

Use the "whole is missing" form:

$$\text{whole} = \frac{\text{part}}{\text{rate}} = \frac{18}{0.30} = 60$$

So the whole is $60$.

Check by going backward:

$$30\% \text{ of } 60 = 0.30 \times 60 = 18$$

This reverse check is worth doing every time, because percentage mistakes usually come from swapping the part and the whole.

Practice It Yourself

Solve this and verify with the reverse check: $45$ is what percent of $180$?

Start from

$$\text{percent} = \frac{\text{part}}{\text{whole}} \times 100\%$$

Sanity check: $45$ is one quarter of $180$, so your answer should land near $25\%$ and well under $100\%$. If it does not, you likely flipped the part and the whole.

Where the Setup Trips People

• Using $25$ instead of $0.25$ in the equation. In the multiply/divide forms, the rate must be a decimal unless you keep it as $25/100$.
• Mixing up the part and the whole. Reverse them and a "what percent" answer comes out far too large or far too small.
• Forgetting the denominator condition: $\text{percent} = \frac{\text{part}}{\text{whole}} \times 100\%$ only works when the whole is not $0$.

Identifying the Whole

The whole is the base amount before the percent is taken. In "15 is $25\%$ of 60," the whole is $60$ because the $25\%$ refers to 60. In word problems it is often the original price, total items, full test score, or total population. When the wording is murky, ask "percent of what?" — the answer is almost always the whole.

The same part-rate-whole structure runs through discounts, tax and tip, test scores, and percentage-change problems. That is the real payoff: stop memorizing separate tricks and just find the missing quantity.