# Understanding Percents

## Percents and Decimals

Fundamentally, a percent is a fraction out of 100 — it is *per centum* (Latin for "per 100"). It's easy to change a percent to a decimal. For example, 37{\%} means 37 parts out of one hundred, or \dfrac{37}{100}. As a decimal, that's just 0.37. Changing a percent to a decimal simply involves sliding the decimal to the left two places.

## Percent Changes as Multipliers

This is one of the **BIG** math ideas for QUANT. A multiplier is a factor by which you multiply a number to get a desired result. There are three percent-related multipliers you will need to understand

### a) X{\%} of a number

Suppose I have \${400} in an account, and need to know what 30{\%} of this account is. The multiplier = the percent as a decimal. 30{\%} as a decimal is 0.30, and \${400}(0.30) = \${120}, so \${120} is 30{\%} of \${400}.

### b) an X{\%} increase

Suppose I have \${400} in an account, over time period, I am going to get an additional 5{\%} of interest; in other word, my account is going to increase by 5{\%}. Here, the **multiplier \bold{= 1 +} (the percent as a decimal)**. Thus, \${400}(1.05) = \${420}, so that's the amount I would have after a 5{\%} increase

### c) an X{\%} decrease

Suppose I have \${400} in an account, and because of some kind of penalty, I am going to be nailed with a 15{\%} deduction; in other words, my account will decrease by 15{\%}. Here, the **multiplier \bold{= 1 -} (the percent as a decimal)**. In this case, the multiplier = 1 - 0.15 = 0.85, and the result after the deduction is (\${400})(0.85) = \${340}.

## Calculating a percent change

Basically, a percent is a simple \dfrac{part}{whole} ratio times 100. The QUANT will ask you to calculate percent changes, and here you have to be very careful with order, i.e., what’s the starting number and what’s the ending number. IMPORTANT: **in a percent change, the starting number is always \bold{100{\%}}**. Thus, we can say:

\text{\it{percent change}} = {\dfrac{\text{\it{amount of change}}}{\text{\it{starting amount}}}} \times 100

Here are a couple examples

a) *Price increases from \${400} to \${500}; find the percentage increase*.

Of course, that’s a change of \${100}, so \${100} divided by starting value of \${400} is 0.25, times 100 is 25\%. A move from \${400} to \${500} is a 25\% increase.

b) *Price decreases from \${500} to \${400}; find the percentage decrease*.

Change is still \${100}, but now the starting value is \${500}, and \dfrac{\${100}}{\${500}} = 0.20, times 100 is 20\%. A move from \${500} to \${400} is a 20\% decrease.

**BIG IDEA**: Order matters. When you change from one value to another and want the percentage change, it matters which value was the starting value.

## A Series of Percentage Changes

Example: “Profits increased by 40\% in January, then decreased by 30\% in February, then increased by 20\% in March. Express the change over the entire first quarter as a single percentage.” This may seem like a nightmare problem, but it’s quite approachable with multipliers. First Caution: **NEVER** add a series of percent. That’s what many people will do, and on multiple choice, it’s always an answer choice — here, that would be 40 - 30 + 20 = 30. That is **not** the way to go about answering the question.

The way to attack this question is with a series of multipliers:

In January, a 40\% increase \to multiplier = 1.40

In February, a 30\% decrease \to multiplier = 0.70

In March, a 20\% increase \to multiplier = 1.20

Aggregate change = (1.40)(0.70)(1.20) = 1.176 \to that’s a 17.6\% increase for the quarter.

**BIG IDEA**: For a series of percentage changes, simply multiply the respective multipliers.

## The Increase – Decrease Trap

This is a predictable QUANT Math trap: the result of a percentage increase, followed by a percentage decrease of the same numerical value. For example, “The price of the appliance increase 20\%, and then decreased 20\%. The final price is what percent of the original price.” Every single time that question is asked on multiple choice, the incorrect answer of 100\% will be an answer choice, and every single time, a large portion of folks who take the QUANT will select it. You have a leg up if you simply recognize and remember that this is a trap.

In fact, solving this problem is just an extension of the previous item:

a 20\% increase \to multiplier = 1.20

a 20\% decrease \to multiplier = 0.80

total change = (1.20)(0.80) = 0.96

Thus, after the increase and decrease, the final price is 96\% of the original price, which means it is a 4\% **decrease**.

**BIG IDEA #1**: when you go up by a percent, then down by the same percent, you do **not** wind up where you started: that’s the trap.

**BIG IDEA #2**: in this situation, as in any situation in which you have a series of percentage changes, simply multiply the respective multipliers.