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Lesson 30

PERCENT INCREASE OR DECREASE


In the last Lesson, we emphasized the equivalence of percents and fractions (ratios).  5 out of 20 is expressed as

     5 
20		 = 1
4		 = 25%

In this Lesson, we continue this emphasis with two types of word problems.


In this Lesson, we will answer the following:

  1. What are the two types of word problems in finding the Percent?

    Section 2

  2. In a banking problem, what is the Amount and what is the Base?

 1.   What are the two types of word problems in finding the Percent?
  1.  An "out of" problem:
                   The part is what percent of the whole?
  2.  A difference problem:
                   The difference is what percent of the original?

In the previous Lesson, we considered an out of problem.

Example 1.  An "out of" problem.    A team played 16 games and won 12.

a)  What fraction of their games did they win?

b)  What percent of their games did they win?

Solution.  First, the student should appreciate the similarity of those questions.

a)  The team won 12 out of 16 games:

12
16		 .

An "out of" problem will always give rise to a proper fraction.

4 is a common divisor of 12 and 16.  The fraction reduces to  3
4		 .
  b)  As for the percent of games they won,   3
4		  = 75%.  (Lesson 23.)

Example 2.  A difference problem.    Jimmy got a raise from $6.00 an hour to $8.00 an hour.  This was a raise of what percent?

Solution.  The previous example was an out of problem.  This one is a difference problem, because something has changed -- his salary.  In the previous example, nothing changed.

To find what percent of a raise, we must first find how much of a raise.  In this case, it was a raise of $2.00.  His original salary was $6.00.  The question is,

$2.00 is what percent of $6.00?

"The difference is what percent of the original?

Now, $2 is a third of $6. (Lesson 14.)  And a third is 33 1
3		 %. (Lesson 15,

Example 13.)

In fractional form:

Difference
 Original		  =  $2
$6		  =  1
3		  = 33 1
3		 % .
Jimmy's salary was increased by 33 1
3		 %.

"Difference problem" is shorthand for Percent increase or decrease.

In a difference problem, we must answer this question:

The difference is what percent of the original?

Example 3.   A cookbook was reduced from $24.50 to $17.95.  This was a reduction of what percent?

Solution.  This is a difference problem, because something has changed
-- the price of the book.

To find the difference with a calculator, press

2 4 . 5 - 1 7 . 9 5=

It is not necessary to enter the 0's on the right of a decimal.

See

 6.55  

The original price was $24.50.  Press

÷ 2 4 . 5 %

(Lesson 10.)  Displayed is

 26.73469  

On rounding off to one decimal place, this is approximately

26.7%.


With a simple calculator, the entire problem can be done at once by pressing

2 4 . 5 - 1 7 . 9 5 ÷ 2 4 . 5 %

"The difference divided by the original."

On some calculators, it is necessary to press = after each calculation; that is, before pressing ÷ .

Example 4.   Sarah earns $1800 a month and pays $430 for rent.  What percent of her income goes for rent?

Solution.  Is this a difference problem or an out of problem?  Do we subtract or not?

This is an out of problem because nothing has changed.  There has been no increase or decrease in her rent.  The problem means that Sarah pays $430 out of $1800 for rent.

With a calculator, press

4 3 0 ÷ 1 8 0 0 %

Displayed is

 23.888888  

On rounding off to one decimal place, this is approximately

23.9%.

In an "out of" problem, the smaller number is divided by the larger; the part by the whole .

Example 5.   Which of these is an out of problem, and which is a difference problem?

1.    Janet's salary is $400 a week, and she pays $53 in taxes.  What percent goes for taxes?
2.    Janet's salary increased from $400 a week to $440.  This was an increase of what percent?

Answer.   Problem 1 is an out of problem. For we could restate it:

Janet pays $53 out of $400 in taxes.

Problem 2 is a difference problem, because something has changed:  her salary. To find how much it has changed, we must subtract.

(If we tried to restate Problem 2 using the expression "out of," it would make no sense!)

Let us answer Problem 2:

Janet's salary, in going from $400 to $440, increased by $40.   The question is,

$40 is what percent of the original $400?

$40 is a tenth or 10% of $400.  Her salary increased by 10%.

$400 + $40 = +10%. $400 - $40 = -10%.

Now, if her salary had gone from $400 to $360, then it would have decreased by $40.  It would have decreased by the same 10%.

Summary
Every question that asks "What percent. . .?"
can be rephrased in this form:
      is what percent of      ?
In an out of problem, that question becomes:
The part is what percent of the whole?
Smaller number ÷ Larger
In a difference problem, that question is:
The difference is what percent of the original?
Difference ÷ Original

At this point, please "turn" the page and do some Problems.

or


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