51 ratio-proportion

Proportions

ProportionsTwo related quantities stated side by side is called a ratio.

ProportionsTwo related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4.

3 4

ProportionsTwo related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:

3 4

eggs cups of flour

ProportionsTwo related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:

3 4

eggs cups of flour

Proportions

This fraction is also the amount of per unit of the given ratio, in this case, ¾ egg / per cup of flour.

Two related quantities stated side by side is called a ratio. For example, if a recipe calls for 3 eggs and 4 cups of flour, then the ratio of eggs to flour is 3 to 4. We may write it using fractional notation as:

3 4

eggs cups of flour

Proportions

This fraction is also the amount of per unit of the given ratio, in this case, ¾ egg / per cup of flour.Two ratios that are equal are said to be in proportion.


3 4

eggs cups of flour

Proportions

This fraction is also the amount of per unit of the given ratio, in this case, ¾ egg / per cup of flour.Two ratios that are equal are said to be in proportion. Thus "3 to 4" is proportion to "6 to 8" since

3 4 = 6

8


3 4

eggs cups of flour

Proportions


3 4 = 6

8 Proportional equations are the simplest type of fractional equations.


3 4

eggs cups of flour

Proportions


3 4 = 6

8 Proportional equations are the simplest type of fractional equations. To solve proportional equations, we cross-multiply and change the proportions into regular equations.


A B

C D ,= If

Cross-Multiplication-RuleProportions

A B

C D ,= If then AD = BC.


A B



Example A. Solve for x.3 x

5 2 = a.

A B




5 2 = a. cross multiply

A B





6 = 5x

A B





6 = 5x

= x5 6

A B





6 = 5x

= x

2 3

(x + 2) (x – 5) = b.

5 6

A B





6 = 5x

= x5 6

2 3

(x + 2) (x – 5) = b. cross multiply

A B





6 = 5x

= x

2 3


2(x – 5) = 3(x + 2)

5 6

A B





6 = 5x

= x

2 3


2(x – 5) = 3(x + 2)2x – 10 = 3x + 6

5 6

–10 – 6 = 3x – 2x –16 = x

A B





6 = 5x

= x

2 3


2(x – 5) = 3(x + 2)2x – 10 = 3x + 6

5 6

2 (x + 1)

x= c. 3

–10 – 6 = 3x – 2x –16 = x

A B





6 = 5x

= x

2 3


2(x – 5) = 3(x + 2)2x – 10 = 3x + 6

5 6

2 (x + 1)

x= c.

x(x + 1) = 6

3

–10 – 6 = 3x – 2x –16 = x

A B





6 = 5x

= x

2 3


2(x – 5) = 3(x + 2)2x – 10 = 3x + 6

5 6

2 (x + 1)

x= c.

x(x + 1) = 6

3

x2 + x = 6 x2 + x – 6 = 0

–10 – 6 = 3x – 2x –16 = x

A B





6 = 5x

= x

2 3


2(x – 5) = 3(x + 2)2x – 10 = 3x + 6

5 6

2 (x + 1)

x= c.

x(x + 1) = 6

3

x2 + x = 6 x2 + x – 6 = 0 (x + 3) (x – 2) = 0 So the solutions are x = –3 and x = 2.

–10 – 6 = 3x – 2x –16 = x

ProportionsWhen setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions

ProportionsWhen setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.

Proportions

Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour?

When setting up a proportional equation for a word problem, the two quantities for the same type of measurement must occupy the same position in the fractions i.e. both must be in numerator, or both must be in the denominator.

Proportions

Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed.


Proportions

Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown.


Proportions

Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown.

number of eggscups of flour


Proportions

Example B. A recipe calls for 3 eggs and 4 cups of flour. How many eggs are needed if we have 10 cups of flour? Let x be the number of eggs needed. Write the two types of measurements, the number of eggs and the number of cups of flour, in a column as shown. Then write down the ratios.



Proportions



x 10


Proportions



3 4

x 10


Proportions



3 4

x 10

=

Set them equal, we get

x 10

3 4


Proportions



3 4

x 10

=


x 10

3 4 cross multiply

4x = 30


Proportions



3 4

x 10

=


x 10

3 4 cross multiply

4x = 30

x = 30 4


Proportions



3 4

x 10

=


x 10

3 4 cross multiply

4x = 30

x = 30 4 = 7½ We need 7½ eggs.


Example C. On a map, 4 inches corresponds to 21 miles in real distance. What is the real distance between two points if they are 14 inches apart on the map?

Proportions


Proportions

Let x be the number of real distance in miles and write the two types of measurements as


Proportions

Let x be the number of real distance in miles and write the two types of measurements asmilesinches


Proportions


x 14


Proportions


21 4

x 14


Proportions


21 4

x 14


21 4

x 14 =


Proportions


21 4

x 14


21 4

x 14 = cross multiply

4x = 294


Proportions


21 4

x 14


21 4


4x = 294

x =

294 4


Proportions


21 4

x 14


21 4


4x = 294

x =

294 4 = 73½

So 4 inches corresponds to 73½ miles.

ProportionsGeometric proportion gives us the sense of “similarity”, that is, objects that are of the same shape but different sizes.

Proportions

Given the following cats, which cat on the right is of the same shape as the one on the left?

A.

B.

Geometric proportion gives us the sense of “similarity”, that is, objects that are of the same shape but different sizes.

Proportions

Given the following cats, which cat on the right is of the same shape as the one on the left? The answer of course is A.


A.

B.

similar

not similar

Proportions

Given the following cats, which cat on the right is of the same shape as the one on the left? The answer of course is A. This sense of similarity is due to the fact that “all corresponding linear measurements are in proportion”.


A.

B.

similar

not similar

Proportions

Given the following cats, which cat on the right is of the same shape as the one on the left? The answer of course is A.

A.

B.

This sense of similarity is due to the fact that “all corresponding linear measurements are in proportion”.


similar

not similar

ProportionsFor example, given the following similar cats,

ProportionsFor example, given the following similar cats, let’s identify some corresponding points as shown.

Proportions

B.

For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known.

6

10 12

Proportions

B.

For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known.

6

10 12

3

Proportions

B.

For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion.

6

10 12

3

y z

Proportions

B.

For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion. Since the distances between the tips of the ears is 6 : 3 or 2 : 1 ratio,

6

10 12

3

y z

Proportions

B.

For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion. Since the distances between the tips of the ears is 6 : 3 or 2 : 1 ratio, we see that y =

6

10 12

3

y z

Proportions

B.

For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion. Since the distances between the tips of the ears is 6 : 3 or 2 : 1 ratio, we see that y = 10/5 = 2 and z = 12/2 = 6.

6

10 12

3

y z

Proportions

6

10 12

3

5 6

For example, given the following similar cats, let’s identify some corresponding points as shown. Suppose the following measurements are known. Then we may find the distances y and z by proportion. Since the distances between the tips of the ears is 6 : 3 or 2 : 1 ratio, we see that y = 10/5 = 2 and z = 12/2 = 6.

ProportionsDefinition. Two geometric objects are similar if all corresponding linear measurements are in proportion.

B.


B.

Hence if X, Y and Z are the measurements from the original cat as shown,

X

Y Z


B.

Hence if X, Y and Z are the measurements from the original cat as shown, and x, y and z are the corresponding measurements in the similar copy,

X

Y Z

x

y z


B.

Hence if X, Y and Z are the measurements from the original cat as shown, and x, y and z are the corresponding measurements in the similar copy, then it must be that

X

Y Z

x

y z

Xx = Y

y = Zz


B.


X

Y Z

x

y z

Xx = Y

y = Zz or that X

Y = xy etc…


B.


X

Y Z

x

y z

Xx = Y

y = Zz or that X

Y = xy etc…

This mathematical formulation of similarity is the basis for biometric security software such as facial recognition systems.

ProportionsThe reason that linear proportionality gives us the sense of similarity is due to visual geometry, that is, the visual images of an object at different distances form a “cone”.

ProportionsThe reason that linear proportionality gives us the sense of similarity is due to visual geometry, that is, the visual images of an object at different distances form a “cone”. One can easily demonstrate that in such projection, similar images of two different sizes must preserve the ratio of two corresponding distance measurements – as in the above definition of similarity.

ProportionsThe reason that linear proportionality gives us the sense of similarity is due to visual geometry, that is, the visual images of an object at different distances form a “cone”. One can easily demonstrate that in such projection, similar images of two different sizes must preserve the ratio of two corresponding distance measurements – as in the above definition of similarity.

Similar triangles

The Golden RatioProportions


An important ratio in arts and science is the golden ratio.


An important ratio in arts and science is the golden ratio.Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”,


An important ratio in arts and science is the golden ratio.Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1


Cut a line segment of length 1 into two, a long one L and a short one s such that the ratio of “1 to L” is the same as “L to s”, 1

L




L


1 L



L s


L s

1 L =



L s


L s

1 L =

The ratio1/L is the Golden Ratio.


L (1 – L)

1 L


L s


L s

1 L =

Since s = 1 – L, we’ve =



L (1 – L)

1 L

cross multiply


L s


L s

1 L =


1 – L

=

L2



L (1 – L)

1 L

cross multiply0 = L2 + L – 1


L s


L s

1 L =


1 – L

=

L2



L (1 – L)

1 L



L s


L s

1 L =


1 – L

=

L2


This is not factorable so by the quadratic formula

L =


L (1 – L)

1 L



L s


L s

1 L =


1 – L

=

L2

We’ll see later that the answer for L is about 0.618. http://en.wikipedia.org/wiki/Golden_ratiohttp://en.wikipedia.org/wiki/Golden_spiral


http://en.wikipedia.org/wiki/Golden_ratio

Ex. A. Solve for x.Proportions

3 2

2x + 1 3 =

21 x3 =1.

–2x/7 –5

1½ 3 =9.

2. 6 2=3. x + 1

–5 7 =4. –2x + 1 6 =5. x + 1 23x =6. 5x + 134x = 8

2/3 7/5

x 14 =7. 9

4/33 2x/5 =8.

10. – x + 2 2

–4 3 =11.

3 2x + 1

1 x – 2 =12. x – 5

22x – 3

3 =13.

3 2

x – 4 x – 1 =14. –3

52x + 1

3x =15.

3x + 2 2

2x + 1 3 =16. 3

22x + 1 x – 3 =17.

2 x

x + 1 3 =18. x

2x + 4

x =19.

1 2

Ex. B. (Solve each problem. It’s easier if fractional proportions are rewritten as proportions of integers.)

Proportions

20. If 4 cups of flour need 3 tsp of salt, then 10 cups flour need how much of salt?

Different cookie recipes are given, find the missing amounts.

21. If 5 cups of flour need 7 cups of water then 10 cups water need how much of flour?

23. If 2 ½ tsp of butter flour need 3/4 tsp of salt, then 6 tsp of salt need how much of flour.

22. If 5 cups of flour need 7 cups of water then 10 cups flour need how much of water?

24. If 1¼ inches equals 5 miles real distance, how many miles is 5 inches on the map?

For the given map scales below, find the missing amounts.

25. If 2½ inches equals 140 miles real distance, how many inches on the map correspond to the distance of 1,100 miles?

Proportions

A.

B.

C.

3

4

1

5

5

6

26. x

20 x

20

627.

x

3x + 4 2xx+2

15 x

x2x – 3

28.

29. 30. 31.

32–37. Find the surface areas of each cat above if A’s surface area is 2 ft2, B’s surfaces area is 7 ft2. and C’s is 280 ft2?

Ex. C. 26 – 31. Solve for x. Use the given A, B and C,.

38–43. Find the weight of each cat above if cat A is 20 lbs, cat B is 8 lb. and cat C is 350 lb?

51 ratio-proportion

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