1 percentage: a commonly used relative quantity

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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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1. Figure out the type of problem.

2. Figure out which numbers go where.

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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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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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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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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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x/rq = p/100

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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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• 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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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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100 * (x / y) = percentage

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Percentage difference problems

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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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The difference [N1-N2]

is the compared quantity [cq]

The “than” number [N2]

is the reference quantity [rq].


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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(n1-n2)/rq = p/100

100*(n1-n2)/rq=p

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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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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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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.


(new-old)/old = p/100

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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.

1 percentage: a commonly used relative quantity

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