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Notes for January 13
Proportions!!!
SOLVING SIMPLE ONES
Word of the Day
Inanestupid; dumb;
pathetic
Today’s Objective
IWBAT solve algebraic proportions.
WARM-UP
Write each fraction in lowest terms (simplify).
1416
1.
972
3.
2464
2.
45120
4.
78
38
18
38
A ratio is a comparison of two quantities by division.
Ratios that make the same comparison are equivalent ratios.
In one rectangle, the ratio of shaded squares to unshaded squares is 7:5. In the other rectangle, the ratio is 28:20.
Both rectangles have equivalent shaded areas.
7:5 28:20
Example 1: Finding Equivalent Ratios
Find two ratios that are equivalent to each given ratio.
B.
1854
13
12848
A. =927
=9 • 227 • 2
=9 ÷ 927 ÷ 9
927
= Two ratios equivalent
to are and . 927
1854
13
Two ratios equivalent
to are and . 6424
12848
83
=64 • 224 • 2
6424
=
Multiply or divide the numerator and denominator by the same nonzero number.
83
=64 ÷ 824 ÷ 8
6424
=
Ratios that are equivalent are said to be proportional, or in proportion.
Equivalent ratios are identical when they are written in simplest form.
Simplify to tell whether the ratios form a proportion.
1215
B. and 2736
327
A. and 218
Since ,
the ratios are in
proportion.
19
= 19
19
=3 ÷ 327 ÷ 3
327
=
19
=2 ÷ 218 ÷ 2
218
=
45=
12 ÷ 315 ÷ 3
1215
=
34=
27 ÷ 936 ÷ 9
2736
=
Since ,
the ratios are not
in proportion.
45
34
Simplify to tell whether the ratios form a proportion.
1449
B. and 1636
Since ,
the ratios are in
proportion.
15
= 15
15
=3 ÷ 315 ÷ 3
315
=
15
=9 ÷ 945 ÷ 9
945
=
27
=14 ÷ 749 ÷ 7
1449
=
49=
16 ÷ 436 ÷ 4
1636
=
Since ,
the ratios are not
in proportion.
27
49
315
A. and 945
We can also use cross products to figure out whether two ratios are in proportion.
Tell whether the ratios are proportional.
410
615
Since the cross products are equal, the ratios are proportional.
60
=?
60 = 60
Find cross products.604
10615
Algebraic Proportions
Algebraic proportions are the same as regular proportions.
The cross-products must equal each other!
KEYPOINT
Solving Algebraic Proportions
To solve algebraic proportions, follow these steps:
1.) Cross-multiply
2.) Set the products equal to each other
3.) Solve for x
4.) Box your answer
Solving Algebraic Proportions
The most important thing to remember is to:
Solving Algebraic Proportions
Solve for x in the following proportion:
124
2 x
Solving Algebraic Proportions
Cross-multiply
124
2 x
2(12) = 24
4(x) = 4x
Solving Algebraic Proportions
Set the products equal to each other
4x = 24What am I
called?
Solving Algebraic Proportions
Solve for x 244 x
4
24
4
4x
6x
Solving Algebraic Proportions
Solve for x in the following proportion:
6
155
x
Solving Algebraic Proportions
Cross-multiply
6
155
x
5(-6) = -30
x(15) = 15x
Solving Algebraic Proportions
Set the products equal to each other
15x = -30
What am I called?
Solving Algebraic Proportions
Solve for x 3015 x
15
30
15
15 x
2x
Try some with your partner!
248
5 x
x
72
16
12
10025
2 x
36
91x
Try some on your own!
1
3x
6
3
x
1
2
x
2
12
3
2
4
5
x
Notes for January 14th
Proportions!!!
SOLVING COMPLEX
ONES
Let’s not make it too
hard to begin with. Let’s start
by just throwing a coefficient in front of
the x.
More Complex Algebraic Proportions
What happens when you see one of these?
10
8
5
2x
DO THE SAME THING!!!
More Complex Algebraic Proportions
Cross-multiply
10
8
5
2x
2x(10) = 20x8(5) = 40
More Complex Algebraic Proportions
Set the products equal to each other
20x = 40What am I
called?
Solve for x
4020 x
20
40
20
20x
2x
More Complex Algebraic Proportions
More Complex Algebraic Proportions
Solve the following proportion
x3
12
5
20
More Complex Algebraic Proportions
Cross-multiply
x3
12
5
20
20(3x) = 60x12(5) = 60
More Complex Algebraic Proportions
Set the products equal to each other
60x = 60What am I
called?
Solve for x
6060 x
60
60
60
60x
1x
More Complex Algebraic Proportions
Try some with your partner!
24
3
8
5 x
x3
72
16
12
100
2
25
2 x
36
9
4
4x
As a kicker, I have much expertise in
this manner …
LET’S KICK IT UP!!!
Even more complex algebraic proportions!
What happens when you see a proportion?
2
5
4
2
x
KEYPOINT!!! When solving proportions like
that, you must remember that each numerator and denominator are together – like a couple. You cannot separate them.
So in order to do this, you must use the
Distributive Property.
Steps for Solving Complex
Proportions1.) Cross-Multiply
2.) Set the products equal to each other
3.) Use the Distributive Property
4.) Solve for x
5.) Box your answer
Even more complex algebraic proportions
Cross-multiply
2
5
4
2
x
2(x – 2) = 2(x – 2)
-4(5) = -20
Even more complex algebraic proportions
Set the products each to each other
2(x – 2) = -20
Even more complex algebraic proportions
Use the Distributive Property and solve for x
20)2(2 x2042 x
420442 x162 x
2
16
2
2 x
8x
Even more complex algebraic proportions!
Solve the following proportion:
2
5
3
6
xx
Even more complex algebraic proportions
Cross-multiply
2
5
3
6
xx
-2(x + 6) = -2(x + 6)3(x - 5) = 3(x – 5)
Even more complex algebraic proportions
Set the products each to each other
-2(x + 6) = 3(x – 5)
Even more complex algebraic proportions
Use the Distributive Property and solve for x
5)– 3(x 6) 2(x -
153122 xx
15331232 xxxx15125 x
121512125 x
x 3
535 x
On Your Own!
2
x 3
4
6
PRACTICE!
It’ll be a Party in
Ms. Ryan’s Room!
2
4
3
2x
2
3x
4
6
Exit Ticket1. Are these two
ratios in proportion?j
A. YesB. NoC. Not sure
2. Solve for k: j
A. k = 40B. k = 4C. k = 5D. k = 8
3. Solve for x (simplify your answer): j
A. x = 12B. x = 4/7C. x = -21D. x = 12/21
4. Solve for b: j
A. b = 1.5B. b = 8.5C. b = -1.5D. B = -8.5
6
7
3
2 x
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