Percentages represent a part of a whole, with the whole being represented as [latex]100\%[/latex]. They are a special kind of fraction that are used to indicate proportions out of a hundred.
Conversion of percentages to decimals or fractions is straightforward. To convert a percent to a decimal, you simply divide the percentage by [latex]100[/latex], which effectively means moving the decimal point two places to the left. To convert a percent to a fraction, place the percentage over [latex]100[/latex] to form a fraction and then simplify if possible. For instance, [latex]45\%[/latex] becomes [latex]0.45[/latex] as a decimal and [latex]\frac{45}{100}[/latex] as a fraction, which simplifies to [latex]\frac{9}{20}[/latex].
This is [latex]60.75\%[/latex]. Notice that the percent can be found from the equivalent decimal by moving the decimal point two places to the right.
In the news, you hear “tuition is expected to increase by [latex]7\%[/latex] next year.” If tuition this year was [latex]$1200[/latex] per quarter, what will it be next year?
The tuition next year will be the current tuition plus an additional [latex]7\%[/latex], so it will be [latex]107\%[/latex] of this year’s tuition: [latex]$1200(1.07) = $1284[/latex]. Alternatively, we could have first calculated [latex]7\%[/latex] of [latex]$1200[/latex]: [latex]$1200(0.07) = $84[/latex]. Notice this is not the expected tuition for next year (we could only wish).
Instead, this is the expected increase, so to calculate the expected tuition, we’ll need to add this change to the previous year’s tuition: [latex]$1200 + $84 = $1284[/latex].
Rates
The Main Idea
Rates are a specific kind of ratio, used to compare quantities of different kinds, such as miles per hour or price per pound. A unit rate is a rate that is simplified so that it has a denominator of one, making it easier to compare different rates directly. For instance, if you can drive [latex]180[/latex] miles on [latex]10[/latex] gallons of gas, the unit rate would be [latex]18[/latex] miles per gallon.
Proportions are equations that show two ratios or rates as being equivalent. For example, if you know you can drive [latex]180[/latex] miles on [latex]10[/latex] gallons of gas, and you want to know how far you can drive on [latex]15[/latex] gallons, you would set up a proportion: [latex]\frac{180}{10} = \frac{x}{15}[/latex], and solve for [latex]x[/latex]. The proportion equation allows us to solve problems by finding missing quantities in equivalent ratios or rates.
Solve the proportion [latex]\displaystyle\frac{5}{3}=\frac{x}{6}[/latex] for the unknown value [latex]x[/latex].
This proportion is asking us to find a fraction with denominator [latex]6[/latex] that is equivalent to the fraction[latex]\displaystyle\frac{5}{3}[/latex]. We can solve this by multiplying both sides of the equation by [latex]6[/latex], giving [latex]\displaystyle{x}=\frac{5}{3}\cdot6=10[/latex].
A map scale indicates that [latex]\frac{1}{2}[/latex] inch on the map corresponds with [latex]3[/latex] real miles. How many miles apart are two cities that are [latex]\displaystyle{2}\frac{1}{4}[/latex] inches apart on the map?
We can set up a proportion by setting equal two [latex]\displaystyle\frac{\text{map inches}}{\text{real miles}}[/latex] rates, and introducing a variable, [latex]x[/latex], to represent the unknown quantity—the mile distance between the cities.
Many proportion problems can also be solved using dimensional analysis, the process of multiplying a quantity by rates to change the units.
Suppose you’re tiling the floor of a [latex]10[/latex] ft by [latex]10[/latex] ft room, and find that [latex]100[/latex] tiles will be needed. How many tiles will be needed to tile the floor of a [latex]20[/latex] ft by [latex]20[/latex] ft room?
In this case, while the width the room has doubled, the area has quadrupled. Since the number of tiles needed corresponds with the area of the floor, not the width, [latex]400[/latex] tiles will be needed. We could find this using a proportion based on the areas of the rooms:[latex]\displaystyle\frac{100\text{ tiles}}{100\text{ft}^2}=\frac{n\text{ tiles}}{400\text{ft}^2}[/latex]
Absolute and Relative Change
The Main Idea
Absolute change and relative change are two ways of quantifying the difference between two values.
Absolute change refers to the simple difference between the initial value and the final value. For example, if a stock price goes from [latex]$10[/latex] to [latex]$15[/latex], the absolute change is [latex]$5[/latex].
Relative change, on the other hand, expresses the absolute change as a percentage of the original value. This provides a sense of the scale or significance of the change in relation to the starting point. In the above example, the relative change would be [latex]50\%[/latex], as the stock price increased by half of its original value. This is calculated by dividing the absolute change ([latex]$5[/latex]) by the initial value ([latex]$10[/latex]), and then multiplying the result by [latex]100[/latex] to get a percentage.
Suppose a stock drops in value by [latex]60\%[/latex] one week, then increases in value the next week by [latex]75\%[/latex]. Is the value higher or lower than where it started?
To answer this question, suppose the value started at [latex]$100[/latex]. After one week, the value dropped by [latex]60\%: $100 – $100(0.60) = $100 – $60 = $40[/latex]. In the next week, notice that base of the percent has changed to the new value, [latex]$40[/latex]. Computing the [latex]75\%[/latex] increase: [latex]$40 + $40(0.75) = $40 + $30 = $70[/latex].
In the end, the stock is still [latex]$30[/latex] lower, or [latex]\displaystyle\frac{\$30}{100} = 30\%[/latex] lower, valued than it started. A video walk-through of this example can be seen below: