Proportions

Similarity & Proportions

Ratio, Fraction, Decimal, Percent, Rate, Proportion, Similar Polygons

Ratio

A ratio shows the relative amount of two or more values. Ratios can be shown in different ways:

• using the ":" to separate example values

• using the "/" to separate one value from the total

• as a decimal, after dividing one value by the total

• as a percentage, after dividing one value by the total

Example: if there is 1 boy and 3 girls you could write the ratio as:

1:3 (for every one boy there are 3 girls)

1/4 are boys and 3/4 are girls

0.25 are boys (by dividing 1 by 4)

25% are boys (0.25 as a percentage)

Multiple Select

In a classroom of 20 students, there are 12 boys and 8 girls. 9 students are wearing slippas, 4 students are wearing boots, 6 students wear sneakers and 1 is wearing yeezys. Which of the following ratios are correct?

• Ratio of Boys to Girls is 12:20

• Ratio of Girls to Students is 8:20

• 1/20 students wear yeezys

• 4 out of 20 students wear boots

• 9% of students wear slippas

Proportions

A Proportion is an equation (has equal sign in it, unlike ratios) which defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or ratios. If both sides are equivalent, then it is a true Proportion.

Multiple Choice

Are these two Fractions Equivalent...aka

Is this Proportion True?

    
 $\frac{1}{2}=\frac{3}{6}$  

• yes

• no

• i have no idea

Checking Proportions for Equivalency

Multiple Select

What is the Scale Factor between the sides of these triangles? Does it work between all pairs of corresponding sides?

• Scale Factor is 3, yes all sides multiply by the same scale factor

• Scale Factor is 1/3. You can get from the right to the left by multiplying all sides by this scale factor

Multiple Choice

Are the fractions equivalent?

• Yes, same scale factor

• No, different scale factors

Cross Products

Multiple Choice

Determine whether each pair of ratios forms a proportion

• Proportion

• Not a True Proportion

Multiple Choice

Determine whether each pair of ratios forms a proportion

 $\frac{10}{16}=\frac{15}{24}$  

• Proportion, cross products are equal

• Not a Proportion, cross products are not equal

From Fraction to Decimal

Multiple Select

For two given Ratios, how can you test if they are proportional?

• The cross products are equal.

• Two ratios are equivalent and could be reduced to the same fraction.

• When the numerator and denominator can be multiplied by the same number to get the other sides values.

• When two ratios have the same denominator

• When the decimal forms of both sides are equal

Fraction

Equivalent Fractions are always proportional.

Reduced Fractions are equivalent

Percents are fractions with a denominator of 100

Rates are fractions with a denominator of 1

Decimal

Percent

Per 100... Percents use the denominator of 100 and are equivalent and proportional to the given Ratio.

Reduce Percents to simplify as Fractions

Read Percents as Decimals

Working with Percents

Word Problems, no problem

The reason to revisit these concepts is to prepare you for Similar Polygons and then Trigonometry, both of which demonstrate similarity and use proportions. If you clearly define what is being compared, its easy to write the correct proportion and then solve the problem. You can also use units to guide you.

COnsider this problem......

In an animal shelter, the ratio of cats to dogs is 3 to 5. If there are a total of 25 dogs at the shelter, how many cats do they have?

In an animal shelter, the ratio of cats to dogs is 3 to 5. If there is a total of 40 animals at the Shelter, how many are dogs?

Multiple Choice

In an animal shelter, the ratio of cats to dogs is 3 to 5. If there are a total of 25 dogs at the shelter, how many cats do they have?

• 15

• 18

• 21

• 32

Multiple Choice

In an animal shelter, the ratio of cats to dogs is 3 to 5. If there are a total of 40 pets at the shelter, how many dogs do they have?

• 8

• 24

• 15

• 25

Multiple Choice

In an animal shelter, the ratio of cats to dogs is 3 to 5. What percent of the shelter is cats?

• 37.5%

• about half

• 62.5%

• 30%

Using Cross Products

an algebra shortcut...then some practice

Multiple Choice

Solve for x

• x = 19.6

• x = 5/98

• x = 101/5

Multiple Choice

The scale of a map says that 6 cm represents 15 km.

What distance on the map represents an actual distance of 6 kilometers?

• 3.6

• 2.5

• 2.4

• .42

Multiple Choice

A cake recipe calls for 1.5 cups of milk and 3 cups of flour. Seth made a mistake and used 5 cups of flour. How many cups of milk should he use to keep the proportions correct?

• 2.5 cups milk

• 1.75 cups milk

• 2 cups flour

• 2.25 cups milk

Multiple Choice

The currency in Argentina is the Peso. The exchange rate is approximately $3 = 1 Peso. At this rate, how many Pesos would you get if you exchanged $121.10?

• 20.2

• 363.3

• 40.4

Multiple Choice

Mary reduced the size of a painting to a width of 3.3 in. What is the new height if it was originally 32.5 in tall and 42.9 in wide?

• 13

• 4.356

• 0.76

• 2.5

Multiple Choice

You bought a tire for $120 that normally costs $150. By what percent was the tire discounted?

• 80%

• 20%

• 125%

• 12.5%

Rates

Rates take the ratio of two things and reduces the denominator to 1... This is called a unit Rate. Often rates are in time, but don't have to be.

Multiple Choice

If Justin can type 408 words in 4 minutes, how many words can he type per minute?

• 102 words per minute

• 100 words per minute

• 400 words per minute

• 412 words per minute

Multiple Choice

I paid $15.00 for 6 gallons of gasoline.

What was the price PER gallon?

• $0.40

• $9.00

• $2.50

• $0.90

Rate, Ratio, Slope

In this graph of Circumference v Diameter, the slope of the line is 3.15 which is close to the expected value of 3.14. You can clearly see that the relationship (C:D) is linear (constant). This shows us that the ratio of C:D stays the same no matter how big or small the circle measured.

For any proportional relationship, we will find that the

SLOPE IS THE RATIO... IS THE RATE.

SImilar Polygons

ALl of this so far has been to connect ideas you have already learned to Geometry. Now you are ready to consider where this all goes...Similarity. We saw these triangle before, we found them to be proportional and the scale factor is x3. Each side on the right is 3 times longer than the first. They are therefore SIMILAR POLYGONS.

Similar Polygons have Congruent corresponding angles and all sides proportional.

Multiple Choice

The two right triangles below are similar. What is x, the missing side length in triangle DEF?

• 3.75

• 9.75

• 10

• 12

Multiple Choice

Are these polygons similar?

• yes, all sides are proportional

• no, they are not proportional

• yes because corresponding angles are congruent

• yes because corresponding angles are congruent and corresponding sides are proportional.

Multiple Select

Which are expressions of the ratio and rate between y & x?

•  $\frac{y}{x}$  

• k

•  $\frac{y_2-y_1}{x_2-x_1}$