Ratios and Rates. Goals Define and apply the concept of equivalent ratios to obtain the simplest form of a ratio. Big Idea The numerator and denominator

Ratios and Rates

Ratios and RatesGoals• Define and apply the concept of equivalent ratios to obtain the simplest form of a ratio.

Big Idea

The numerator and denominator in ratios in their simplest form do not have common multiples. Ratios that are equivalent to the ratio in the simplest form are obtained by multiplying numerator and denominator by the same value.

HW due Tuesday 12/7210.3, 210.10, 210.11, 210.12, 211.15, 211.17, 211.19, 211.23, 211.24, 211.30, 211.31, 211.35, 211.39, and 211.40


Notation

The ratio of variables a and b can be expressed as:

a

bor a ÷b or a:b

Examples:

3

5=3÷5=3:5

33

55=35 in simplest form


Notation: The ratio of variables a and b can be expressed as:

a

bor a ÷b or a:b

What is 36 to 12 in simplest form expressed as a fraction? expressed with a colon?

What is 0.2 to 8 in simplest form expressed as a fraction? expressed with a colon?

What is 72 to 1.2 in simplest form expressed as a fraction? expressed with a colon?

Do these:

Ratios and RatesWhat is 36 to 12 in simplest form expressed as a fraction?

expressed with a colon?What is 0.2 to 8 in simplest form expressed as a fraction?

expressed with a colon?What is 72 to 1.2 in simplest form expressed as a fraction?

expressed with a colon?What is 3c to 5c in simplest form expressed as a fraction?

expressed with a colon?What is 3/4 to 1/4 in simplest form expressed as a fraction?

expressed with a colon?What is 1.2 to 2.4 in simplest form expressed as a fraction?

expressed with a colon?What is 6 to 0.25 in simplest form expressed as a fraction?

expressed with a colon?

Do these:

Ratios and RatesWhat is 54 g to 90 g in simplest form expressed as a fraction?

What is 75 cm to 350 cm in simplest form expressed as a fraction?

What is 1 hr to 15 min in simplest form expressed as a fraction?

Do these:

Ratios and RatesGoals• Translate narrative descriptions into algebraic expressions involving proportions.

The perimeter of a rectangular garden is 30 feet, and the width is 5 feet. Find the ratio of the length of the rectangle to its width in simplest form.

Taya and Jed collect coins. The ratio of the number of coins in their collections, in some order, is 4 to 3. If Taya has 60 coins in her collection, how many coins could Jed have?


The ratio may be continued to describe multiple ratios

a : b : cA woodworker is fashioning a base for a trophy. He starts with a block of wood whose length is one-half its width and whose height is one-half its width. Write, in simplest form, the continued ration of length to width to height.

Ratios and RatesGoalApply the method of dimensional analysis for the conversion of units of measure.

Big Idea: Ratios are used to convert units.

HW237.4, 237.7, 237.12, 237.14

Example

10 millimeter = 1 centimeter so

1 centimeter

10 millimeter = 1

850 millimeters•

1 centimeter

10 millimeter=85 centimeters

Ratios and RatesDo these:

Change 1.5 meters to centimeters given that 1 meter is equal to 100 centimeters.

Change 2.5 miles to feet given that 1 mile is equal to 5280 feet.

Ratios and RatesGoal• Define and apply the concept of rate by examining the units of measure of the numerator and denominator.

Big Idea

A rate is a special kind of ratio in which the units of numerator and denominator are different.

Examples:

Speed is a rate … a bike travels 1 km in 5 minutes.

Price is a rate … 50 mp3 downloads costs $25

Density is a rate … there are 400 grams in a half liter

Ratios and RatesGoal• Define and apply the concept of unit rate.

Big Idea

In a unit rate the numerical coefficient in the denominator is 1.

Examples:

Speed is a rate … a bike travels 1 km in 5 minutes with a unit rate of 0.2 km/minute.

Price is a rate … 50 mp3 downloads costs $25 with a unit rate of 2 downloads/dollar

Density is a rate … there are 400 grams in a half liter with a unit rate of 800 grams/liter

Ratios and RatesGoal• Define and apply the concept of unit rate.

Big Idea

In a unit rate the numerical coefficient in the denominator is 1.

Do these, 213.11 and 213.16:

A commuter drove 48 miles to work in 1.5 hours. What is the unit speed?

Johanna can enter 920 words using a keyboard in 20 minutes. Al can enter 1290 words in 30 minutes. Who is fastest?


215.5 Two numbers have the ratio 7:5. Their difference is 12. What are the numbers?

215.8 The perimeter of a triangle is 48 centimeters. The lengths of the sides are in the ratio 3:4:5. Find the length of each side.

215.10 The sum of the measures of two angles is 90°. The ratio of the measures of the angles is 4:5. Find the measure of each angle.


215.17 The ratio of the number of boys in a school to the number of girls is 11 to 10. If there are 525 pupils in the school, how many of them are boys?

215.20 A chemist wishes to make 12.5 liters of an acid solution by using water and acid in the ratio 3:2. How many liters of each should she use?

Ratios and RatesGoalSimplify an equality of proportions by cross-multiplication of extremes and means.

Big Idea

Equivalent ratios can be expressed as a product of “means” and “extremes” which have a common denominator.

33

55=35 so 33•5=3•55 because

33•555•5

=3•555•55

Examples

HW220.2, and 2 of 220.3-8, 221.18, 221.21, 221.22, 221.23, and 221.27

Ratios and RatesGoalSimplify an equality of proportions by cross-multiplication of extremes and means.

Big Idea

Equivalent ratios can be expressed as a product of “means” and “extremes” which have a common denominator.

Consider 220.2

Mike said that if the same number is added to each term of a proportion, the resulting ratios for a proportion. Do you agree? If not, then give a counter example.

Ratios and RatesConsiderMike said that if the same number is added to each term of a proportion, the resulting ratios for a proportion. Do you agree? If not, then give a counter example.

a

b=

cd

Yes! If , then adding a number m to each ratio

a

b+ m=

a+ mbb

=cd+ m=

c+ mdd

The new proportion is

a + mbb

=c+ md

d

For example,

1

2=24

1

2+ 3=

1+ 3• 22

= 72 and

24+ 3=

2 + 3• 44

=144

Ratios and RatesConsider 220.3, 5, and 6

Do these ratios form a proportion?

3

4 and

30

40? 3• 40 =30•4Yes!

4

5 and

16

25?

2

5 and

5

2?

Ratios and RatesConsider

Rather than test the product of “means” and “extremes” you might see a common multiple, 220.8.

Do these ratios form a proportion?

3

4 and

30

40? 3• 40 =30•4Yes!

4

5 and

16

25?

2

5 and

5

2?

No!

2 • 2 ≠5•5No!

4 • 25≠5•16

36

30 and

18

15?

Ratios and RatesGoalSolve problems involving direct variation (proportionality).

Big Idea

You can use proportional thinking to solve problems.

Consider The denominator of a fraction exceeds the numerator by 3. If the numerator is decreased by 1 and the denominator is increased by 2 the fraction becomes 3:5. What is the original fraction?

Ratios and RatesConsider The denominator of a fraction exceeds the numerator by 3. If the numerator is decreased by 1 and the denominator is increased by 2 the fraction becomes 3:5. What is the original fraction?

Ratios and RatesConsider The denominator of a fraction exceeds the numerator by 3. If the numerator is decreased by 1 and the denominator is increased by 2 the fraction becomes 3:5. What is the original fraction?

x

x + 3; what is x if

x−1x+ 3+ 2

=35?

x −1x+ 3+ 2

=x−1x+5

if

x −1x+5

=35, then 5•(x−1) =3• (x+5)

5x −5=3x+15 so 2x=20 and x=10

Check:

10 −110 +5

=915

=35

Ratios and RatesConsider 220.29 The numerator of a fraction is 8 less than the denominator of a fraction. The value of the fraction is 3:5. Find the fraction.

HW:220.29, 220.30, 220.33, 220.35, and 220.36

Ratios and RatesConsider 220.29 The numerator of a fraction is 8 less than the denominator of a fraction. The value of the fraction is 3:5. Find the fraction.

x

x +8; what is x if

xx+8

=35?

if

x

x +8=35, then 5•x=3• (x+8)

5x =3x+ 24 so 2x=24 and x=12

Check:

12

20=3• 45• 4

=35

Ratios and RatesConsider 220.30 The denominator of a fraction exceeds the numerator of the fraction by 10. The value of the fraction is 5:12. Find the fraction.

HW:220.29, 220.30, 220.33, 220.35, and 220.36

Ratios and RatesConsider 220.33 What number must be added to both the numerator and denominator of the fraction 7:19 to make the resulting fraction equal to 3:4?

HW:220.29, 220.30, 220.33, 220.35, and 220.36

Ratios and RatesConsider 220.35 The numerator of a fraction is 7 less than the denominator. If 3 is added to the numerator and 3 is subtracted from the denominator, the resulting fraction is equal to 5:2. Find the original fraction.

HW:220.29, 220.30, 220.33, 220.35, and 220.36

Ratios and RatesConsider 220.36 Slim Johnson was usually the best free-throw shooter on his basketball team. Early in the season, however, he had made only 9 of 20 shots. By the end of the season, he had made all the additional shots he had taken, thereby ending with a season record of 3:4. How many additional shots had he taken?

HW:220.29, 220.30, 220.33, 220.35, and 220.36


Big Idea

You can use proportional thinking to solve problems in which variation is linear and constant.

Examples

Distance traveled at constant speed is proportional to time.The cost of items with constant unit price is proportional to the

number of items.The number of cups of flour added to a cake batter is

proportional to the number cakes to be baked.


Big Idea

A direct variation between two variables in when the variables form a proportion. The ratio of the variables is the constant of variation which is calculated from values of the variables.

Example

D=st and when D=35, t=5 so the constant of variation is 7.

HW:224.2, any 2 of 220.3 through 220.11


Big Idea

Ratios of values in data tables can be compared to determine if there is a direct variation. If so, the ratio of dependent and independent variables remains constant.

Example

Cause 1 2 3 4 5 6

Effect 3 6 9 12 15 18

Ratios and RatesBig Idea


Example

Cause 1 2 3 4 5 6

Effect 3 6 9 12 15 18

Ratio 3 3 3 3 3 3

Effect =3•Cause

The constant of variation is 3.



HW:224.14, 224.16

Cause 1 2 3 4 5 6

Effect 3 6 9 12 15 18

Ratio 3 3 3 3 3 3

Effect =3•Cause



Consider 224.14

x 4 5 6

y 6 8 10

ratio

Does x vary directly with y? If so what is the constant?

Does y vary directly with x? If so what is the constant?



Consider 224.16

x 2 3 4

y -6 -9 -12

ratio

Does x vary directly with y? If so what is the constant?

Does y vary directly with x? If so what is the constant?

Ratios and RatesGoalTranslate narrative descriptions into algebraic expressions involving proportions.

Solve problems involving direct variation (proportionality).

HW:225.21, 225.23, 225.25, 225.27, 225.32, 225.35, 225.39, 225.43, 225.45, 225.46, and 225.48


225.21

Is there a direct variation? If so what is the constant of variation?

R+T=80

225.23 e/i=20

225.24 bh=36


HW:225.23, 225.25, 225.27, 225.32, 225.35, 225.39, 225.43, 225.45, 225.46, and 225.48

225.25 C=7n is a formula for the cost of n articles that sell at the unit rate of $7 per item.

Is this a direct variation?

How will the cost of nine articles compare with the cost of three articles?

If n is doubled, what change takes place in C?


225.27 The variable d varies directly as t. If d = 520 when t=13, find d when t=9.

225.32 If four tickets to a show cost $17.60, what is the cost of seven such tickets?

225.35 There are about 60 calories in 30 grams of canned salmon. About how many calories are there in a 210-gram can?


225.39 The weight of 20 meters of copper wire is 0.9 kilograms. Find the weight of 170 meters of the same wire.

225.43 The scale on a map is given as 5 centimeters to 3.5 kilometers. How far apart are two towns if the distance between these two towns on the map is 8 centimeters?

225.45 A picture 3 and 1/4 inches long and 2 and 1/8 inches wide is to be enlarged so that its length will become 6 and 1/2 inches. What will be the width of the enlarged picture?


225.46 An 11-pound turkey costs $9.79. At this rate, find:

• the cost of a 14.4-pound turkey, rounded to the nearest cent• the cost of a 17.5-pound turkey, rounded to the nearest cent• the price per pound at which the turkeys are sold• the largest size turkey, to the nearest tenth of a pound, that can be bought for $20 or less.

225.48 If a family consumes q liters of milk in d days, represent the amount of milk consumed in h days.

Ratios and RatesGoalApply the concept of percent.Apply the concepts of percent of error, decrease, and increase.

Big Idea

A percent is the rate with a denominator of 100.

A percentage is product of a percent and a base.

ExampleA percentage of the cost of an item is due to a sales tax of 4%

base percent

Ratios and RatesGoalApply the concept of percent.

HW231.29, 30, 33, 34, 47, 48, 50, 52, and 53.

231.29 A test was passed by 90% of a class. If 27 students passed the test, how many students are the class?


HW231.29, 30, 33, 34, 47, 48, 50, 52, and 53.

231.30 Marie bought a dress that was marked $24. The sales tax is 8%. Find the percentage due to the sales tax. Find the total amount Marie had to pay.


HW231.29, 30, 33, 34, 47, 48, 50, 52, and 53.

231.33 The price of a new motorcycle that Mr. Klien bought was $5,430. Mr. Klein made payment of 15% of the price of the motorcycle and arranged to pay the rest in installments. How much was his down payment?


HW231.29, 30, 33, 34, 47, 48, 50, 52, and 53.

231.34 How much silver is in 75 kilograms of an alloy that is 8% silver?

Ratios and Rates231.47 At Relli’s Natural Goods, all items are being sold today at 30% off their regular prices. However, customers must still pay an 8% tax on items. Edie, a good-natured owner, allows each customer to choose one of two plans at this sale:

Plan 1: Deduct 30% of the cost of all items, then add 8% tax to the bill.

Plan 2: Add 8% tax to the cost of all items, then deduct 30% if this total.

Which plan save the customer more money? Why?


Big Idea

A percent of increase or decrease is the ratio of the change between the new amount and the original and the original amount.

Percent of increase or decrease =

original amount - new amount

original amount×100

Ratios and Rates231.48 In early March, Phil Kalb bought shares in two different companies.

Stock ABC rose 10% in value in March, then decreased 10% in April.

Stock XYZ fell 10% in value in March, then rose 10% in April.

What percent of its original price is each of these stocks now worth?

Ratios and Rates231.52 Isaiah answered 80% of the questions correctly on the math midterm, and 90% of the questions correctly on the math final. Can you conclude that he answered 85% of allthe questions correctly (the average of 80% and 90%)?

231.53 In January, Amy bought shares of stocks in two different companies. By the end of the year, shares of the first company had gone up by 12% while shares of the second company had gone up by 8%. Did Amy gain a total of 12%+8%=20% in her investments?


Big Idea

A percent error is the ratio of the difference between the value observed and the true value and the tru value multiplied by 100.

Percent of error =

observed amount - true amount

true amount×100

Ratios and Rates231.50 A carpenter measures the length of a board as 50.5 centimeters. The exact measure of the length was 50.1 centimeters. Find the percent of error in the carpenter’s measure to the nearest tenth of a percent.

Documents

Ratios and Rates. Goals Define and apply the concept of equivalent ratios to obtain the simplest form of a ratio. Big Idea The numerator and denominator