Percent problems: calculating a discount, a markup, and a percent of increase

The main idea of percent problems is a single one: first determine which value the percent is taken from. For a discount and a markup, the new value is found using a coefficient; for a percent of increase, the change is divided by the old value.

Below, these rules are applied step by step in examples of a discount, an inverse problem, a markup, and a successive change.

Basic rules for finding a percent

A percent is one hundredth of a number: $1\% = \frac{1}{100} = 0{,}01$. From this follow three basic operations:

1. Finding a percent of a number: for the number $a$, its $p\%$ equals $a \cdot \frac{p}{100}$. For example, for $48$, its $25\%$ is $48 \cdot 0{,}25 = 12$.
2. Increasing by a percent (markup, increase): the new value is $a \cdot \left(1 + \frac{p}{100}\right)$.
3. Decreasing by a percent (discount): the new value is $a \cdot \left(1 - \frac{p}{100}\right)$.

There is also an inverse question: "The number $12$ is what percent of $48$?" Here we find the share and multiply by $100\%$: $\frac{12}{48} \cdot 100\% = 25\%$. That is, finding a percent actually means finding "the ratio of the part to the whole".

In practice, the most convenient way is to convert the percent into a coefficient. A discount of $20\%$ means multiplying the price by $0{,}8$, and a markup of $35\%$ means multiplying by $1{,}35$. The coefficient method makes it possible to solve percent problems in one step and, what matters, it also works without error on inverse problems.

Discount problems

1-example. The price of the jacket is 250 000 soums. The shop announced a discount of $20\%$. What is the new price?

Solution. With a discount of $20\%$, the part $0{,}2$ of the price is removed, so the part $0{,}8$ remains:

$250\,000 \cdot 0{,}8 = 200\,000$ soums.

The amount of the discount can also be found at once: $250\,000 - 200\,000 = 50\,000$ soums.

Now we look at the inverse problem: here the original price is recovered from the value after the discount.

2-example. After a discount of $20\%$, the bag came to 144 000 soums. What was its original price?

Solution. If we let the original price be $x$, the price after the discount equals $0{,}8x$:

$0{,}8x = 144\,000$, hence $x = \dfrac{144\,000}{0{,}8} = 180\,000$ soums.

We check: $180\,000 \cdot 0{,}8 = 144\,000$ — the condition is satisfied. Note: here one cannot multiply $144\,000$ by $1{,}2$, because the discount is taken from the original price, not from the later price.

Finding a markup and a percent of increase

3-example. The shop buys the product for 80 000 soums and sells it with a markup of $35\%$. What is the selling price?

Solution. In one step using the coefficient:

$80\,000 \cdot 1{,}35 = 108\,000$ soums.

From here one can also see that the markup itself is $80\,000 \cdot 0{,}35 = 28\,000$ soums.

4-example. The price of the product rose from 120 000 soums to 138 000 soums. By what percent did the price increase?

Solution. To find the percent of increase, the change is always divided by the old value:

$\dfrac{138\,000 - 120\,000}{120\,000} = \dfrac{18\,000}{120\,000} = 0{,}15 = 15\%$.

General formula: the percent of change $= \dfrac{\text{yangi} - \text{eski}}{\text{eski}} \cdot 100\%$. If the result is positive, we are speaking of an increase; if it is negative, of a decrease.

Successive percent changes

The case that causes the most confusion on tests is the successive application of two percents.

5-example. The price was first increased by $20\%$, then a discount of $20\%$ was taken from the new price. How did the price change relative to the original?

Solution. Most people answer "it did not change", but that is a mistake. In successive changes the coefficients are multiplied:

$1{,}2 \cdot 0{,}8 = 0{,}96$,

that is, the price fell to $96\%$ of the original — in total it decreased by $4\%$. The reason is simple: the second $20\%$ is taken relative to a larger number (the increased price), so the drop in soums is greater than the increase in soums.

By the same rule, increasing two times by $10\%$ gives not $20\%$, but an increase of $21\%$: $1{,}1 \cdot 1{,}1 = 1{,}21$. If the same positive percent is applied in succession, a coefficient is written for each step and the coefficients are multiplied.

Independent exercise

After a discount of $15\%$ the product cost 255 000 soums. Find the original price. Short answer: $0{,}85x=255\,000$, so $x=300\,000$ soums.

Typical mistakes and how to avoid them

1. Taking a percent from the wrong base. in the 2-example, calculating as $144\,000 \cdot 1{,}2 = 172\,800$ is a classic mistake. The discount was taken from the original price, so one must divide by $0{,}8$, not multiply by $1{,}2$.
2. Adding successive percents together. Changing the price successively by $+20\%$ and by $-20\%$ does not restore the original price. Convert each change into a coefficient and multiply.
3. Dividing the amount of increase by the new value. in the 4-example, one gets $\dfrac{18\,000}{138\,000} \approx 13\%$ — this is an incorrect option placed deliberately in many tests. The old value always stands in the denominator.
4. Mixing up a percent and soums. Whether the question asks for a percent or for a unit of money, read the problem statement to the end and write the answer in exactly the form requested.

Conclusion and the next step

• To find a number's $p\%$, multiply it by $\frac{p}{100}$.
• For an increase by $p\%$, use the coefficient $1+\frac{p}{100}$, and for a decrease, the coefficient $1-\frac{p}{100}$.
• When calculating the percent of increase, divide the change by the old value.
• Do not add successive percents; multiply their coefficients.
• Solve topical tests on percents or test yourself on the daily problem.