Mastering Percentage Questions on the SAT

Percentages are a fundamental concept in math, representing a part-to-whole relationship out of 100. On the Digital SAT exam, these questions show up on almost every single exam. The questions can vary greatly in complexity and difficulty, but they all test the same concepts. Let's dive into how to master these questions.

A percentage is essentially a ratio that expresses how much out of 100. The term "percent" means "per hundred." For example, 25% means 25 out of 100. This guide will help you calculate percentages, switch between equivalent forms, and understand percent changes.

Calculate Percentages

To calculate a percentage, you need two values: the part and the whole. Use the formula: $\text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100$.

Finding a Percentage

Let's say you got 18 out of 20 questions correct on a quiz. The part is 18, and the whole is 20. To find the percentage:

$\left(\frac{18}{20}\right) \times 100 = 90\%$

Another example: If you have 45 apples out of a total of 60 apples, the percentage of apples you have is:

$\left(\frac{45}{60}\right) \times 100 = 75\%$

Finding Complementary Percentages

If you know the percentage of one part of a whole, you can find the remaining percentage. For example, if 30% of the marbles in a bag are blue, then 70% must be red.

Example: In a bag of 100 marbles, if 30% are blue, then there are 70 red marbles.

Another example: A store sold 40% of its 200 items. How many items were not sold? The unsold items are 60%, so:

$\left(\frac{60}{100}\right) \times 200 = 120 \text{ items}$

Complete Example

Of the juniors at South High School, 60% played for a school sports team. If 300 juniors played, how many students are in the junior class?

We need to solve for the whole. The part is 300 and the percentage is 60%. So, $300 = 0.60 \times \text{Whole}$, solving this gives $\text{Whole} = 500$.

Switch Between Equivalent Forms

Percentages can be expressed as fractions or decimals. For example, 25% is equivalent to 0.25 or 1/4.

Converting Percentages

To convert a percentage to a decimal, remove the percent symbol and move the decimal point two places to the left. To convert a percentage to a fraction, write the percentage as a ratio over 100 and simplify if possible.

Example: 75% is $0.75$ as a decimal and $\frac{75}{100} = \frac{3}{4}$ as a fraction.

Another example: Convert 45% to a decimal and a fraction. As a decimal, it's $0.45$ and as a fraction, it's $\frac{45}{100} = \frac{9}{20}$.

Complete Example

In 2020, 25% of Major League Baseball starting pitchers were left-handed. The equivalent fraction is $\frac{25}{100} = \frac{1}{4}$ and the equivalent decimal is $0.25$.

Calculate Percent Change

To calculate the percent change from an initial value to a final value, find the difference, divide by the initial value, and multiply by 100.

Calculating Percent Change

Example: The price of a vacuum was reduced from $80 to $60. The percent reduction is:

$\left(\frac{80 - 60}{80}\right) \times 100 = 25\%$

Another example: A population increased from 1,000 to 1,200. The percent increase is:

$\left(\frac{1200 - 1000}{1000}\right) \times 100 = 20\%$

Complete Example

The price of a pair of shoes is $45 after a 10% discount. What was the original price?

Let $x$ be the original price. Then, $0.90x = 45$, solving this gives $x = 50$.