MP2 Section Four Notes on Percents



MP2 Section Four Notes on Percents and Probability

Fraction of a whole #

2/3 of 15 tuna cans – sample

*Think : Divide 15 into 3 equal sized groups. 5 in each group

10= in 2 groups bc…… xxxxx xxxxx xxxxx

2/3 of 15= 10

1. Take the WHOLE number and divide it by the denominator

2. that answer will give you how many are in each group

3. Then multiply that number by the numerator

Ex: 2/3 of 36

1. 36 divided by 3= 12

2. 12 x 2 = 24

OR

2/5 of 20

1. re write the fraction as 2/5 x 20/1 (20/1 is same as just the #20)

2. multiply across (after you cross reduce)

3. Solve and simplify the fraction

Understanding Percents

• percent means “per hundred” or “of 100”

• % is the percent sign

Fraction = 25/100

Decimal = .25

Percent = 25%

*When going from a decimal to a percent: multiply by 100 or move decimal two times to the right

Ex= .62 x 100= 62%

*When going from fraction to percent= make equivalent fractions with 100

Ex: 3/10 -- x10 on top and bottom -- 30/100 === 30%

4/25----x 4 = 16/100==== 16%

|Definition: |A percent is a ratio whose second term is 100. Percent means parts per hundred. The word comes from|

| |the Latin phrase per centum, which means per hundred. In mathematics, we use the symbol % for |

| |percent. |

Let's look at our comparison table again. This time the table includes percents.

|Comparing Shaded Boxes to Total Boxes |

|Grid |Ratio |Fraction |Percent |

|1 |96 to 100 |[pic] |96% |

|2 |  9 to 100 |[pic] |  9% |

|3 |77 to 100 |[pic] |77% |

|Writing Fractions | |

|as Percents | |

|[pic] |

|Problem: |Last marking period, Ms. Jones gave an A grade to 15 out of every 100 students and Mr. |[pic] |

| |McNeil gave an A grade to 3 out of every 20 students. What percent of each teacher's | |

| |students received an A? | |

|[pic] | |

|  |Solution | |

| | | |

| |Teacher | |

| |Ratio | |

| |Fraction | |

| |Percent | |

| | | |

| |Ms. Jones | |

| |15 to 100 | |

| |[pic] | |

| |15% | |

| | | |

| |Mr. McNeil | |

| |3 to 20 | |

| |[pic] | |

| |15% | |

| | | |

|[pic] |

|Solution: |Both teachers gave 15% of their students an A last marking period. |

In the problem above, the fraction for Ms. Jones was easily converted to a percent. This is because It is easy to convert a fraction to a percent when the denominator is 100. If a fraction does not have a denominator of 100, you can convert it to an equivalent fraction with a denominator of 100, and then write the equivalent fraction as a percent. This is what was done in the problem above for Mr. McNeil. Let's look at some problems in which we use equivalent fractions to help us convert a fraction to a percent.

|Example 1: |Write each fraction as a percent:    [pic] |

|[pic] |

|  |Solution |

| | |

| |Fraction |

| |Equivalent Fraction |

| |Percent |

| | |

| | |

| |[pic] |

| |[pic] |

| |50% |

| | |

| | |

| |[pic] |

| |[pic] |

| |90% |

| | |

| | |

| |[pic] |

| |[pic] |

| |80% |

| | |

| | |

|Example 2: |One team won 19 out of every 20 games played, and a second team won 7 out of every 8 games played. |

| |Which team has a higher percentage of wins? |

|[pic] |

|  |Solution |

| | |

| |Team |

| |Fraction |

| |Equivalent Fraction |

| |Percent |

| | |

| |1 |

| |[pic] |

| |[pic] |

| |95%     |

| | |

| |2 |

| |[pic] |

| |[pic] |

| |87.5% |

| | |

|[pic] |

|Solution: |The first team has a higher percentage of wins. |

In Examples 1 and 2, we used equivalent fractions to help us convert each fraction to a percent. Another way to do this is to convert each fraction to a decimal, and then convert each decimal to a percent. To convert a fraction to a decimal, divide its numerator by its denominator. Look at Example 3 below to see how this is done.

|Example 3: |Write each fraction as a percent:    [pic] |

|[pic] |

|  |Solution |

| | |

| |Fraction |

| |Decimal |

| |Percent |

| | |

| | |

| |[pic] |

| |[pic] |

| |87.5% |

| | |

| | |

| |[pic] |

| |[pic] |

| |95%     |

| | |

| | |

| |[pic] |

| |[pic] |

| | 1.5% |

| | |

| | |

Now that you are familiar with writing fractions as percents, do you see a pattern in the problem below?

|Problem: |If 165% equals [pic], and 16.5% equals [pic], then what fraction is equal to 1.65%? |

|[pic] |

|  |Solution |

| | |

| |Percent |

| |Fraction |

| | |

| | |

| | |

| |165% |

| |[pic] |

| | |

| | |

| | |

| |16.5% |

| |[pic] |

| | |

| | |

| | |

| |1.65% |

| |[pic] |

| | |

| | |

| | |

[pic]

|Summary: |To write a fraction as a percent, we can convert it to an equivalent fraction with a denominator of |

| |100. Another way to write a fraction as a percent is to divide its numerator by its denominator, then|

| |convert the resulting decimal to a percent. |

Converting a Fraction to a Percent

Do the following steps to convert a fraction to a percent:

For example: Convert 4/5 to a percent.

• Divide the numerator of the fraction by the denominator (e.g. 4 ÷ 5=0.80)

• Multiply by 100 (Move the decimal point two places to the right) (e.g. 0.80*100 = 80)

• Round the answer to the desired precision.

• Follow the answer with the % sign (e.g. 80%)

Converting a Percent to a Fraction

Do the following steps to convert a percent to a fraction:

For example: Convert 83% to a fraction.

• Remove the Percent sign

• Make a fraction with the percent as the numerator and 100 as the denominator (e.g. 83/100)

• Reduce the fraction if needed

Converting a Decimal to a Percent

Do the following steps to convert a decimal to a percent:

For example: Convert 0.83 to a percent.

• Multiply the decimal by 100 (e.g. 0.83 * 100 = 83)

• Add a percent sign after the answer (e.g. 83%)

Converting a Percent to a Decimal

How to convert a percent to a decimal:

For example: Convert 83% to a decimal.

Divide the percent by 100 (e.g. 83 ÷ 100 = 0.83)

Finding the Percent of a Number

To determine the percent of a number do the following steps:

• Multiply the number by the percent (e.g. 87 * 68 = 5916)

• Divide the answer by 100 (Move decimal point two places to the left) (e.g. 5916/100 = 59.16)

• Round to the desired precision (e.g. 59.16 rounded to the nearest whole number = 59)

Determining Percentage

Example: 68 is what percent of 87?

• Divide the first number by the second (e.g. 68 ÷ 87 = 0.7816)

• Multiply the answer by 100 (Move decimal point two places to the right) (e.g. 0.7816*100 = 78.16)

• Round to the desired precision (e.g. 78.16 rounded to the nearest whole number = 78)

• Follow the answer with the % sign (e.g. 68 is 78% of 87)

% to a fraction (example)

1. Take percent and place # over 100

2. Reduce

Fraction to a percent

1. First way you make equivalent to 100

2. second way= divide the numerator by the denominator

a. then multiply by 100

Percents of a number

Ex 60% 0f 80

Two ways8

First way

1. Place percent as a fraction over 100

2. Reduce fraction

3. Place whole number over 1

4. Multiply across to get answer and reduce till number over 1

5. “of “ means to multiply

6. Or you can cross reduce your fraction to make it smaller then multiply across

60/100 x 80/1

60/100= 3/5 x 80/1

3/5 x 80/1====the 80 and the 5 can both go into five. Therefore cross reduce them

5/5= 1 and 80/5 = 16

New fractions are : 3/1 x 16/1 = when multiplied across it b/cms 48/1 or the number 48.!

Second way

1. Take 100% of the percent number ..60/100 60 divided by 100 = .60

2. multiply that by the whole number

3. if any extra in the percent add it to the number

ex: 60% of 80

100% of 60% = .6

.6x 80= 48

% of dealing with MONEY!!

Regular and Sale price-transparency of notes

- also goes along with problem of the day worksheet

- teaching lesson

PROBABILITY

2 types : theoretical and experimental

1. use spinner

2. use dice for examples (1/6-16.7% is theory, but experiment is class results)

3. notes form transparency from math coach on probability

Notes from Harcourt!!!- on probability- both types and examples (

** probability lessons- YOU BET and TILES IN A BAG

Probability Concepts

About Probability

When we talk about probability we are assigning some measure of chance to an experiment. An experiment is an activity that has two or more clearly discernible results or outcomes. The collection of all outcomes is referred to as the sample space.

An event is any subset of the sample space. In performing an experiment, the probability is:

P = number of occurrences of the event

total number of trials

When all possible outcomes of a simple experiment are equally likely, the theoretical probability of an event is:

P = number of outcomes in the event

total number of possible outcomes

Example:

If you toss a coin, the theoretical probability of the coin landing with heads up is ½, because there is only one acceptable outcome (heads) out of two possible outcomes.

If you actually performed the experiment 100 times, you might find that the actual results were 45 heads and 55 tails. Experimental probability does not always match theoretical probability, which is based on logical analysis.

In an experiment, we are more confident that experimental probability approaches actual (theoretical) probability with a very large number of trials.

Implications for Instruction

There are several reasons why it is important to perform experiments and examine outcomes in the classroom. An experimental approach has these advantages:

1. It is significantly more intuitive and conceptual. Results begin to make sense and do not result from some abstract rule.

2. It eliminates guessing at probabilities and wondering, “Did I get it right?” Trying to determine the number of elements in a sample space can be difficult without some intuitive background knowledge.

3. It provides an experimental background for examining the theoretical model.

4. It helps students see that experimental probability approaches theoretical probability after a large number of trials.

5. It develops an appreciation for a simulation approach to solving problems. Many real world problems are actually solved by conducting experiments or simulations.

6. It is a lot more fun and interesting.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download