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April 28, 2009

“Nobody can go back and start a new beginning, but anyone can start today and make a new ending.”

~Maria Robinson

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Final Exam

Tuesday, May 12

11am – 1pm

In our usual classroom

Cumulative (covers material from entire semester).

As always, you may use a calculator and/or manipulatives from your own pack.

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April 28, 2009

• Section 6.1 (finish)

• Exploration 6.3

• Section 6.2 – Percents

• Course evaluations

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6.1 (cont’d)

Ratios and Rates

• If a : b = c : d, then a/b = c/d.

• If a/b = c/d, then a : b = c : d.

Example:

• 35 boys : 50 girls = 7 boys : 10 girls

• 5 miles per gallon = 15 miles using 3 gallons

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6.1 (cont’d) Proportional word problems:

Start easy:• I can buy 3 candy bars for $2.00.• So, at this rate, 6 candy bars should cost…• 9 candy bars should cost…• 30 candy bars should cost…• 1 candy bar should cost… this is called a

unit rate.

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6.1 (cont’d)

Proportional word problems:

Here’s another.

• 7 small drinks cost as much as 5 large drinks. At this rate…

• How much should 14 small drinks cost?

• How much should 21 small drinks cost?

• How much should 15 large drinks cost?

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6.1 (cont’d)

Ratios are not the same as fractions

• The ratio of males to females is 3 : 2.

• That means 3/5 of the people are male, and 2/5 of the people are female. The ‘whole’ is the group of 5 people.

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6.1 (cont’d)

Ratios are not the same as fractions

• The mixture is 3 parts water and 1 part green dye.

• That means that 3/4 of the mixture is water and 1/4 of the mixture is green dye. The ‘whole’ is the mixture, which consists of 4 total parts.

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6.1 (cont’d)

Ratios are not the same as fractions

• A school’s enrollment increases by 25 students per year.

• This one can be expressed as a fraction – sort of: 25 students/1 year. There is no ‘whole’, though, since the two quantities being compared have different units!

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Exploration 6.3

Do Part 1.

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6.1 (cont’d) Reciprocal unit ratios

Suppose I tell you that 4 doodads can be exchanged for 3 thingies.

How much is one thingy worth? • 4 doodads/3 thingies means

1 1/3 doodads per thingy.How much is one doodad worth?• 3 thingies/4 doodads means

3/4 thingy per doodad.

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6.1 (cont’d)

To solve a proportion:

If a/b = c/d, then ad = bc.

Show that this is true:

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6.1 (cont’d)

To set up a proportion:

Ex: I was driving behind a slow truck at 25 mph for 90 minutes. How far did I travel?

• Set up equal rates: miles/minute

• 25 miles/60 minutes = x miles/90 minutes.

• Solve: 25 × 90 = 60 × x; x = 37.5 miles.

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6.1 (cont’d)

Strange looking problems:

Ex: I see that 1/4 of the balloons are blue, and there are 6 more red balloons than blue.

• Let x = number of blue balloons, and so x + 6 = number of red balloons.

• Also, the ratio of blue to red balloons is 1 : 3

• Proportion: x/(x + 6) = 1/3

• Alternate way to think about it. 2x + 6 = 4x

x x + 6

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6.1 (cont’d)

Which rates are (equivalent) proportional?Ex:

1. 6/10 mph

2. 1/0.6 mph

3. 2.1/3.5 mph

4. 31.5/52.5 mph

5. 240/400 mph

6. 18.42/30.7 mph

7. 60/100 mph

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6.2 – Percents

• Percent means per hundred.

• 50% means 50 per hundred, or 50/100.

• 95% means 95 per hundred, or 95/100.

• 2% means 2 per hundred, or 2/100.

• 315% means 315 per hundred, or 315/100.

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6.2 (cont’d)

Percents you should know:

1 = 100%

1/4 = 25%, 1/2 = 50%, 3/4 = 75%

1/5 = 20%, 2/5 = 40%, 3/5 = 60%, 4/5 = 80%

1/8 = 12.5%, 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5%

1/10 = 10%, 2/10 = 20%, etc…

%3.333/1 %6.663/2

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6.2 (cont’d)

Fractions, decimals, percents

• To write fractions as decimals or percents:

• a/b means a ÷ b. Divide, and write the answer to get the decimal. Then, multiply by 100 to get the percent.

Ex: 48/60 = 48 ÷ 60 = 0.8 = 80%

You try: 4/9, 4 3/20

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6.2 (cont’d)

Fractions, decimals, percents

• To write decimals as fractions or percents:

• Consider using expanded form, then combine fractions and simplify.

• To write a percent, multiply by 100.

Ex: 0.09 = 9/100 = 9%

You try: 7.007, 0.59

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6.2 (cont’d)

Fractions, decimals, percents

• To write a percent as a decimal, divide by 100.

• From there, you can convert the decimal to a fraction.

Ex: 591% = 5.91 = 5 91/100

You try: 3%, 62%, 0.4%

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6.2 (cont’d)

What happens if...

What if you have 3 2/5%

Rewrite 2/5 as 0.4.

So 3 2/5% = 3.4%

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HomeworkLink to online homework list:

http://math.arizona.edu/~varecka/302AhomeworkS09.htm

*Note: approximate grades are posted on D2L.