Writing a ProportionTo write a proportionproportion, you set one ratio equal to another ratio.

The only rule is this: Your left side and right side must be set up the Your left side and right side must be set up the

same way.same way.So, to see the Eiffel Tower was really 1064 feet tall, you measured

it’s shadow to be 798 feet long. Then, you held up a 4 foot broom, and the broom’s shadow was 3 feet long.

Write 2 different proportions that compare height Write 2 different proportions that compare height and shadow.and shadow.EF height

EF shadowbroom height

broom shadow

4 ft

3 ft

EF heightbroom height

EF shadowbroom shadow

1064 ft

798ft

798 ft

3 ft

1064 ft

4 ft

1.333...

1.333...

266 266

When you write a proportion, your left side and When you write a proportion, your left side and right side must be set up the same way.right side must be set up the same way.

Is it a Proportion?Dana earned a salary of $27,000/mo, and Teresa earned a

salary of $6,500/mo. Dana paid $10,530 in income taxes, but Teresa paid $1,625 in income taxes. Did they pay a

proportional amount of taxes?Step 1Step 1: Make a ratio (fraction) with Dana’s

Dana Teresa

=10,53027,000

1,6256,500

Step 2Step 2: Set that equal to Teresa’s

Step 3Step 3: Cross-multiply. Are the cross products equal?

When cross products are EQUAL, it IS a When cross products are EQUAL, it IS a proportionproportion

10,53027,000

1,6256,500

•68,445,000

43,875,000

When cross products are NOT equal, it is NOT a When cross products are NOT equal, it is NOT a proportionproportion

Since cross Since cross products are NOT products are NOT

equal, they do equal, they do NOT pay a NOT pay a

proportional proportional amount of taxes.amount of taxes.

Solving a ProportionDrew was bragging that he could bench press 245 newtons (N). To convert that weight to pounds (lb), use the chart to

the right:Step 1Step 1: Write your empty proportion.

Step 2Step 2: Set up your left side anyway you want. * I’d recommend starting with “ 245 N ”

Step 3Step 3: Find the correct conversion...

245● 1.88 ● x

Drew bench Drew bench pressed, about, 55 pressed, about, 55

pounds (lb).pounds (lb).

100 ounces (oz) = 6.25 pounds (lb)

8 newtons (N) = 1.8 pounds (lb)

25 ounces (oz) = 6.95 newtons (N)

... then write it as your right side ratio.

8 N

1.8 lb

Step 4Step 4: Cross-multiply, then set the cross-products equal to each

other

... since you’re trying to find weight in pounds, call that x.

x lb

•

=

8x = 441

Step 5Step 5: Solve the equation you made.

8 8

x = 55.125

Since newtons (N) is on

top on the left, then

put newtons

(N) on top on the right.

To solve a proportion, cross multiply, To solve a proportion, cross multiply, set the products equal to each other, set the products equal to each other,

then solve.then solve.

Writing & Solving ProportionsJack’s recipe makes 2.25 pounds (lb) of chili, and it calls for 1.5 tablespoons of lime juice. If he wanted to make 5.1 lb of chili,

how many tablespoons of lime juice should he use?

To write a proportion, each ratio must be set up the same way.

juice

Chili #1

=2.251.5

5.1

l chili

Now, find the cross products.

Jack should use 3.4 Jack should use 3.4 tablespoons of lime juice.tablespoons of lime juice.

Let’s set up both ratios as

Chili #2

•

2.25 •

l

= 5.1 • 1.5

2.25l= 7.6

52.25 2.25

l= 3.4

Simplify, then solve

Scale ModelsHarmony MS has the longest bus rides in the county. Here’s a map showing the route from Loudoun Heights (next to Harper’s Ferry) to Harmony. If their map length is 1 3 inches apart, then how far apart are they actually apart?

4

Step 1: Write an empty proportion.Step 2: On the left side, write the map scale map scale as a fraction.

mLet’s

change those

fractions to decimals.

0.25 • m = 1.75 • 2

0.25m = 3.50.25

Step 3: On the right side, fill in the distance given in the problem.

0.25

1.75

2

0.25 m = m = 1414

map scale map scale

Step 4: Call the missing distance, mm.

Step 5: Cross-multiply, then solve for mm.The bus ride would be, about, 14 miles.

Finding the Map ScaleThe longest drive in the contiguous U.S. is from Key West, FL to Seattle, WA. If the actual distance of the drive is 3,450 miles, what is the map scalemap scale?

Step 1Step 1: Write your empty proportion.

=

Step 2Step 2: Set your left side as

1

Step 3Step 3: Write the mapmap distance. as your numerator (top) …

in 3450 = x

x

Why 11 ? Since it’s your map scalemap scale, you get to pick the numerator.

Why x ? That’s the part you are trying to find.

Step 4Step 4: Write the actualactual distance. as your denominator (bottom)Step 5Step 5: Solve the proportion.

The The mapmap

scale isscale is

1 in1 in

mimi

Using a Map’s Scale Ratio

Set up your proportion. scale ratio = distance ratio

=Measure the distancedistance in cm.

cm

Now, measure the map’s scalescale ratioratio in cm...

... and fill it in your scalescale ratioratio.

cm

a

Solve. a a ==

The distance between Lisbon, Portugal and Vladivostok, Russia is The distance between Lisbon, Portugal and Vladivostok, Russia is miles. miles.

José and Rosa, from Lisbon, Portugal, are going to visit the grandparents, in Vladivostok, Russia,

for the holidays.. What will be the actual distance of their drive?

1.8 is1.8 is 124%124% of what number? of what number? (If needed, round to the (If needed, round to the nearest tenth.)nearest tenth.)

Using the Percent Proportion

Proportions compare a part/whole ratio to another part/whole ratio.

partwhole

partwhole

==

A percent proportion compares a ratio to a percent/100.

Which number is the “part”?

It’s 1.8What little word is next

to it?“is”

Which number is the “whole”?

I dunno.Let’s say x.

What little word is next to it?

“of”

Write and solve the proportion180 = 124x____ ____124 124

xx == 1.51.5

1.8 1.8 is 124% of is 124% of 1.51.5

=isof

p

100

=1.8

x

124

100

This is the

percentpercent proportionproportion..

What number is What number is 56.1%56.1% of 31of 31? ? (If needed, round to the nearest tenth.)

Using the Percent Proportion

Which number is the “is”?

(I dunno ?!) let’s call it xx

is

of__pp__100100

==

xx 17.417.4

Which number is the “of”?

it’s 31

Which number is the “percent”?

it’s 56.1

Solve for x

What percent What percent of 47of 47 is 41 is 41 ? ? (If needed, round your percent to the nearest tenth.)

Which number is the “is”?

41

is

of__pp__100100

==

pp 87.2%87.2%

Which number is the “of”?

Which number is the “percent”?

I dunno…let’s call it p

Solve for p

47

In 2009, Loudoun Valley High School had a capacity of 1350 In 2009, Loudoun Valley High School had a capacity of 1350 students, but 1497 students are enrolled. What percent of capacity students, but 1497 students are enrolled. What percent of capacity was Valley running?was Valley running?

Using the Percent Proportion

Which number is the “part”?

it’s actually 1497

xx 110.9110.9

Which number is the “whole”?

1350

Which number is the “percent”?

x

Solve for x

Harvard University accepted 4,983 freshman applications last year. Harvard University accepted 4,983 freshman applications last year. Only 7% of all students who applied were accepted. How many Only 7% of all students who applied were accepted. How many students applied?students applied?

Which number is the “part”? 4983

partwhole __pp__

100100==

xx 71,18671,186

Which number is the “whole”?

Which number is the “percent”?

it’s 7

Solve for x

(I dunno ?!) let’s call it xx

part

whole

__pp__100100

==

Valley was running at 110.9% Valley was running at 110.9% capacity.capacity.

71,186 students applied.71,186 students applied.

Are They Similar?Are They Similar?

Polygons are Polygons are similar similar when:when:

2. Corresponding sides are in 2. Corresponding sides are in proportionproportion

1.1. Corresponding angles are Corresponding angles are congruent.congruent.

*Put a pair of corresponding sides in a ratio.*Set them equal to another pair of corresponding sides.

*Cross-multiply – Are they equal?

PRPR

JLJL

QRQR

KLKL==??

J

L

K

252.88252.88 252.88252.88==http://www.mathopenref.com/similartriangles.html

21.21.8810.10.99

23.23.2211.11.66

* QPR KJL* PRQ JLK

The triangles

are SIMILAR

UU

WW

TT

Finding Missing Measures of Similar FiguresFinding Missing Measures of Similar Figures

VV2525

1515

2020

Finding Missing Measures of Similar FiguresFinding Missing Measures of Similar Figures

1212

1919xx

3030

Writing a Ratio

Write the ratio 3 feet (ft) to 8 inches (in) as a fraction in simplest form.

A ratio is a comparison of 2 values.

Compare these 2 values as a fraction.

3 ft

8 in Convert to the same unit.Hint: Always convert to the smaller unit.

=36 in

8 in

•12 =

Now, reduce

=9 in

2 in

9

2

There are other ways to write a ratio, also.

9 in

2 in

or

9 2

or 9:2

or 9 to 2

Finding a Unit RateIn 1987, Bo Jackson ran the 40 yd (yard) dash in 4.12 s (seconds).

In 1989, Deion Sanders ran the 60 yd dash in 6.68 s.Who ran at a faster rate?

To compare these runners, find each players yards second

sBoBo DeionDeion

40 yd

4.12 s

=9.708

≈ 9.7 yd s

60 yd

6.68 s

=8.982

≈ 9.0 yd s

Bo Jackson ran at the faster Bo Jackson ran at the faster rate.rate.Emerald’s 11 oz (ounce) can of almonds sells for $7.69.

Planters 5 oz bag of Trail Mix sells for $3.25. Which has the lower unit rate?

EmeraldEmerald’s’s

$7.69

11 oz=0.699≈$0.70

ozPlantersPlanters

$3.25

5 oz0.65 ≈$0.65

oz