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NY Regents Exam - Geometry: Help and Review10 chapters | 127 lessons

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Instructor:
*Norair Sarkissian*

Many values we come across on a regular basis change frequently. In this lesson we will look at examples when a quantity decreases in value, and how such decreases can be represented using percentages.

A brand new car depreciates in value immediately after its purchase. If a $17,000 car depreciates by 15%, what does that actually mean? Later in the lesson, we will answer this question. But first, let us examine a couple of key concepts, which may be familiar to us already: percentage and absolute change.

One way of looking at a percentage involves converting a ratio or fraction to a denominator of 100. We perform this conversion by multiplying the ratio by 100%.

For instance, if we have 23 students in a classroom and 10 of them are boys, the ratio of boys in the classroom is 10 out of 23. The percentage of boys in the classroom is determined by performing the multiplication

As for calculating a specific percentage of a given quantity, we first express the percentage either as a fraction or a decimal number, and then multiply it by the given number.

For example, to find 30% of 120, we first convert 30% into a decimal number. 30% actually means '30 out of a hundred', or '30 divided by 100'. Therefore, 30% of 120 is determined as follows:

If the value of a quantity changes over time, then **absolute change** is defined as the difference between the final and initial values of that quantity. Thus, if a number *A* changes to a new value *Z*, then the absolute change, *C*, is given by

If *C* is a positive number, it means the quantity has increased in value. If *C* is negative, the quantity has decreased in value. Finally, if *C* = 0, then *Initial Value* = *Final Value* and the quantity has remained constant.

For example, if the price of a laptop computer increases from $700 to $750, the absolute change is the difference, or $50. On the other hand, if the price of the computer drops from $700 to $650, the absolute change will be -$50, where the negative value indicates that the price has decreased.

Combining these two key concepts, the **percentage change** represents the absolute change *C* as a percentage of the initial value *A*. Thus, the percentage change in a quantity, p, is defined as

Please note that we use the absolute value of *A* (or |*A*|) in the expression, as it allows us to handle the case when the initial value *A* is negative. In this lesson, we will only consider problems where *A* is positive, and quite often we will discard the absolute value symbol for that reason.

When we are given the initial and final values, *A* and *Z* respectively, the formula we use for evaluating the percentage change directly is

If the quantity has decreased in value, i.e., if *Z* < *A*, then *p* will be a negative percentage, and *p* is called the percent decrease. The formula for percent decrease is the same as that of percentage change.

A gallon of gas was selling for $3.67 last month. This week, the price of gas has dropped to $3.57 a gallon. What is the percent decrease in price?

We are given the initial value *A* = 3.67 and the final value *Z* = 3.57. The percent decrease is

Therefore, the percent decrease from $3.67 to $3.57 is 2.72%.

Here's another example:

Cindy goes to her favorite shoe store and sees a sign that says that every item in the store is on sale, at a 20% discount. The shoes she had planned on buying had an original price of $120. How much money will she save if she buys the shoes on sale?

We need to determine the amount of (absolute) change in the price.

Using the percent decrease formula, we have

To solve for *C*, we multiply both sides by *A* and then divide both sides by 100%. Thus,

Now, we have *A* = 120 and *p* = -20%. Therefore,

Thus, Cindy will save $24 on this purchase. The negative sign indicates that the price of the shoes has decreased (since they are on sale).

Finally, here's a third example:

Returning to our example of a $17,000 car introduced at the beginning of the lesson, what is 15% of $17,000? Also, what will be the price of the car after the 15% depreciation?

In this problem, we are given *p* = -15%, *A* = 17,000. We are asked to solve for *Z*.

We can use the same approach we used in Example 2 above to solve for *C*, and then calculate the final value *Z*. The absolute change is

We can now evaluate the depreciated price of the car, using

Therefore, when a $17,000 car depreciates by 15%, its new price is $14,450.

The absolute change is the difference between the final value, *Z*, and the initial value, *A*, of a quantity. The absolute change is determined using the formula

The percentage change calculates the absolute change as a percentage of the initial value, *A*. It is evaluated using the formula

The percent decrease formula is identical to the percentage change formula given above, except that when we have a decrease, *p* is always a negative value.

The following are derived from the percentage decrease formula, and allow us to solve for other quantities in the formula.

1. When solving for *Z*, we can use the equation

2. When solving for *A*, assuming that *A* is positive, we can use the equation

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NY Regents Exam - Geometry: Help and Review10 chapters | 127 lessons

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