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Percentage:

A commonly used relative quantity.

How large is one quantityrelative to

another quantity?

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How large is the compared quantity [cq]

relative to the reference quantity [rq]?

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Percentages simplify comparison of two or more groups of different sizes.

Remove the effects of the different sizes by placing them

on the same standard: “of one hundred”

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Compared quantity ═ Percentage

Reference quantity 100

Or

100 * cq/rq = percentage

Or

100 * proportion = percentage

General model

Steps in solving percentage problems: A general model

• In practice, steps often are skipped or done out of sequence in familiar or easy problems.

• The model is useful for showing the basic logic of solving even easy or familiar problems.

• The model is useful for coping with complex or confusing problems.

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Steps in solving percentage problems:

1. Figure out the type of problem

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Steps in solving percentage problems:

1. Figure out the type of problem.

2. Figure out which numbers go where.

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Steps in solving percentage problems:

1. Figure out the type of problem.

2. Figure out which numbers go where. Do necessary preliminary computations.

3. Solve using elementary algebra

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Steps in solving percentage problems:

1. Figure out the type of problem.2. Figure out which numbers go where. Do

necessary preliminary computations.3. Solve using elementary algebra

a. An equation is like a scale: whatever you do to one side you must do to the other.

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Steps in solving percentage problems:

1. Figure out the type of problem.2. Figure out which numbers go where.3. Solve using elementary algebra

a. Equation is like a scale: whatever you do to one side you must do to the other.

b. Isolate the unknown on one side of the equation.

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Steps in solving percentage problems:

1. Figure out the type of problem.2. Figure out which numbers go where.3. Solve using elementary algebra.4. Do the arithmetic.

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Types of problems:

1. Part-whole problems

2. “Percentage of” problems

3. Percentage difference problems

4. Percentage change problems

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Part-whole problems

The whole is the reference quantity; the part is the compared quantity.

For part-whole problems only, the percentage can never be more than 100%.

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1.a: Part-whole problems wherethe percentage is unknown (x)

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7 of 28 students in LSP 120 are juniors or seniors.

What percentage of the class is juniors or seniors?

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1.a: Part-whole problems wherethe percentage is unknown (x)

cq/rq = x/100

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1.a: Part/whole problems wherethe percentage is unknown (x)

cq/rq = x/100

100 * (cq/rq) = x

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1.b: Part-whole problems where the part is unknown (x)

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14% of 28 LSP 120 students are in colleges other than LA&S. How many non-LA&S students

are in the class?

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1.b: Part-whole problems where the part is unknown (x)

x/rq = p/100

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1.b: Part-whole problems where the part is unknown (x)

x/rq = p/100

x = p/100 * rq

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1.c.: Part-whole problems where the whole is unknown (x).

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16 students in an LSP 120 class are women. They make up

57% of the class. How many students are in the class?

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1.c.: Part-whole problems where the whole is unknown (x).

cq/x = p/100

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1.c.: Part/whole problems where the whole is unknown (x)

cq/x = p/100

cq=(p/100) * x

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1.c.: Part/whole problems where the whole is unknown (x).

cq/x = p/100

cq = (p/100) * x

cq/(p/100) = x

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A September, 2005, Gallup Poll found that 58% of a sample of 818 Americans disapproved of President Bush’s performance.

How many people in the poll gave Bush a disapproving

rating?

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Of a sample of Chicagoans, 780, or 25%, are smokers.

How many people were in the sample?

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“Percentage Of” problems

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“Percentage Of” problems

• These problems concern the direct comparison of two numbers where neither is a subset (part) of the other.

The number of Sox wins compared to number of Cubs wins.

The price of gas now compared to the price last year.

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“Percentage Of” problems

Characteristic phrasing:

• “X is what percent of Y?”

• “What percent of Y is X?”

• “In percentage terms, how large, small, (etc.) is X relative to Y?”

Does not include problems with “greater than/less than” language

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“Percentage Of” problems

100 * (x / y) = percentage

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Percentage difference problems

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Percentage difference problems

Concern the difference

between two numbers [n1, n2]

relative to one of the numbers [n2].

(The numbers represent the values of two cases on the same variable at the same

time.)

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Percentage difference problems

The difference [N1-N2]

is the compared quantity [cq]

The “than” number [N2]

is the reference quantity [rq].

Percentage difference problems

The standard verbal cues

to a percentage difference problem

are (1) the word percent or percentage and (2) words indicating relative size,

such as “more than” or “less than” or “greater than” or other synonymous

phrases.

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Percentage difference problems

(n1-n2)/rq = p/100

100*(n1-n2)/rq=p

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Percentage difference problems

In word problems,

the amount of the difference [cq]

often is not presented.

So it must be computed

from the information given in the problem.

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Percentage difference problems

Afghanistan has an area of 647,500 sq km. The area of Illinois is 150,007 sq. km. By what percentage is Afghanistan larger than Illinois?

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“percentage of” problems and percentage difference problems:

Rules of inter-conversion

• Convert percentage difference to “percentage of” by adding to 100. (Remember: If cq is less than rq, the percentage is negative, so addition is subtraction)

• Convert “percentage of” to percentage difference by subtracting 100

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Examples of converting between “Of” problems and “more than”

problems• “25% more than” is the same as “125% of”

• “17% less than” is the same as “83% of”

• “250% of” is the same as “150% more than”

• “60% of” is the same as “40% less than”

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Percentage change problems

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Percentage change problems

These are like percentage difference problems but involve comparing the same thing at two different times [“old” and “new”]. “New” is n1 and “old” is n2.

Compute the compared quantity by subtracting the value for the later time [old] from that for the earlier [new].

The number for the earlier time is the reference quantity.

Percentage change problems

(new-old)/old = p/100

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Percentage change problems

In 2000, the world population was 6.1 billion. In 1990, it was 5.3 billion.

What was the percentage rate of change in the world’s population between 1990 and 2000 than in 1990?

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Comparisons of types of problems

Percentage can exceed 100% for “percentage of” and percentage difference and “percentage change”

problems but never for “part-whole” problems

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Comparisons of types of problems

“part-whole” and “percentage of” problems normally require no calculations prior to

plugging in numbers;

percentage difference and percentage change problems do

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Special hint

• For percentage difference and percentage change problems:

• If the problem specifies the percentage and the compared quantity, but reference quantity is unknown

• Solve by converting 3to “percentage of” problem

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Example:

• Professor Muddle lost 15% of his weight on his new diet. He now weighs 180 pounds. How much did he weigh originally?

• To solve, you must restate the problem: His current weight (180 pounds) is 85 percent of his pre-diet weight.