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Ratio Proportion Cross Rate

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					Ratio:

A ratio is a quotient of two terms.

                   a
a to b = a : b =
                   b

Examples:

1.) Write the ratio of 8 to 28 in simplest form.

                       8   2                       2
Answer: 8 : 28 =         =         final answer:     or 2 : 7 or 2 to 7
                       28 7                        7


Reduce both numerator and denominator. Divide both by 4.



2.) Write the ratio of 50 to 6 in simplest form.

                       50   25                       25
Answer: 50 : 6 =          =               final answer: or 25 : 3 or 25 to 3
                        6   3                         3
                                           *** Do NOT change a ratio to a mixed number.***


Reduce both numerator and denominator. Divide both by 2.



3.) Write the ratio of 12 to 4 in simplest form.

                       12 3                               3
Answer: 12 : 4 =          =               final answer:     or 3 : 1 or 3 to 1
                        4   1                             1


*** You must have the denominator of 1

Reduce both numerator and denominator. Divide both by 4.

    4.) Write the ratio of 8 to 16 in simplest form.

                           8   1                                   1
    Answer: 8 : 16 =         =            final answer: 1 : 2 or     or 1 to 2
                          16   2                                   2
5.) Write a ratio in reduced form for each of the following.
The Rocket’s baseball team has 20 wins and 8 losses.

                                                  20 5                            5
a.) Write the ratio of wins to losses. 20 : 8 =     =             final answer:     or 5 : 2 or 5 to 2
                                                  8   2                           2


Reduce both numerator and denominator. Divide both by 4.

b.) Write the ratio of losses to games played. There are 28 games played (losses + wins).

           8   2                                  2
8 : 28 =     =                  final answer:       or 2 : 7 or 2 to 7
           28 7                                   7


Reduce both numerator and denominator. Divide both by 4.


Rate:

A rate is a ratio of unlike quantities. For example: miles per hour (mph), miles per gallon (mpg),
and feet per second.

Examples:

1.) Write the rate for 30 telephone calls in 6 days in simplest form.

30 telephone calls 5 telephone calls
                                    or 5 telephone calls / day or 5 telephone calls per day
      6 days             1 day


Reduce both numerator and denominator. Divide both by 6.


*** The units must be included as part of the answer. *** The units in this example are
telephone calls and day.

2.) Write the rate for 250 calories for 10 ounces of drink in simplest form.

   250 calories       25 calories
                                   or 25 calories / ounce of drink or 25 calories per ounce
10 ounces of drink 1 ounce of drink


Reduce both numerator and denominator. Divide both by 10.
Proportion:

A proportion is when two equal ratios are set equal to each other.


           1 4                                           8 12
Example:     is a proportion.                               is a proportion.
           2 8                                          12 18



               8 12
Example: Is        a true proportion?
              12 15

To find out, you will cross multiply. If the cross products are equal, it is a true proportion. If the
cross products are not equal, it is not a true proportion.


                 8 12
                                  Cross multiply.
                12 15


                         ?
                  12 12  8 15
                   144  120

                                8 12
               Answer: No,          is not a true proportion.
                               12 15



               8 12
Example: Is        a true proportion?
              10 15

                 8 12
                  
                10 15


                                    8 12
               Answer: Yes,           is a true proportion.
                                   10 15
Solving Proportions:

To solve a proportion: Cross multiply then solve the algebraic equation.

Examples:

                                                           1
         8 x                                                   2
Solve:                                             Solve: 3 
         6 9                                               5 x

                                                     1
8 x                                                  32
              Cross Multiply.                                             Cross Multiply.
6 9                                                  5 x

This gives: 6x = 8 * 9
                                                                    1
Or             6x = 72                              This gives:         5*2
                                                                    3x
                                                      1
                                                    Or x  10
                                                      3
                          6 x 72
Divide both sides by 6:      
                           6   6
                                                                                1
                                                                               3 x 10
                                                                        1       1  1
                                                    Divide both sides by :
                                                                        3       3   3

                                                                1
Answer: x = 12                                       x  10 
                                                                3



                                                            3
                                                    x  10*  30
                                                            1
                                                    answer: x =30
Solving Word Problems using Proportions:

Example: Set up and solve a proportion for each of the following. Be sure to put the same units
across from each other.


1.) If a certain plant will grow 120 cm in 6 days, how many days will it take the plant to grow
140 cm?

                       120 cm 140 cm
Set up a proportion:           
                        6 days   x days

Solve: Cross Multiply. 120 x = 6 * 140

Or                             120 x = 840

                               120 x 840
Divide by 120 on both sides:        
                                120 120

                                      x=7                   Answer: 7 days



2.) Mark knows his car can travel 500 miles on 20 gallons of gasoline. How many gallons can he
expect to use if he travels 75 miles?

                       500 miles 75 miles
Set up a proportion:             
                       20 gallons x gallons

Solve: Cross Multiply. 500x = 75 * 20

Or                             500x = 1500

                               500 x 1500
Divide by 480 on both sides:        
                                500   500

                                      x=3                   Answer: 3 gallons
3.) In Gatlinburg, double fudge sells for $48.00 for 6 pounds. If Mary buys 5 pounds, what
should she expect to pay?

                           $48.00   x dollars
Set up a proportion:              
                          6 pounds 5 pounds

Solve: Cross Multiply. 6 x = 48 * 5

Or                              6 x = 240

                             6 x 240
Divide by 6 on both sides:      
                              6   6

                                       x = $40.00                   Answer: $40.00



Practice Ratio and Proportion: ( Answers on the next page).

I. Write each as a ratio or rate in simplest form.

1.) 9 to 24                            2.) 16 to 8


3.) 48 people in 8 cars                4.) 120 feet in 6 seconds


II. Solve each proportion.


      10 5                                            6 x
5.)                                            6.)    
       x 4                                            4 6


III. Solve each of the following by setting up a proportion. Show all steps.

7.) If 8 ounces of tomato sauce is used with every 6 ounces of meat, then how many ounces of
tomato sauce is needed for 9 ounces of meat.


Proportion: _____________________               Answer:____________
      8.) If a certain plant will grow 9 cm in 2 days, how many cm will the plant grow in 4 days?



Proportion: _____________________                 Answer: _____________


      9.) If a salesman makes 50 calls in 8 hours, how many calls can he make in 4 hours?



Proportion: _____________________                 Answer: _____________




Answers:

I. Write each as a ratio in the simplest form.

      3            2            6 people                                20 feet
1.)          2.)          3.)            or 6 people per car     4.)            or 20 feet per second
      8            1              1 car                                1 second

II. Solve each proportion.

5.) x = 8          6.) p = 9

      III.   Solve each of the following by setting up a proportion. Show all steps.

                   8s   x   8 x
7.) Proportion           or                            Answer x = 12
                   6m 9m    6 9

                   9cm    x cm     9 x
8.) Proportion                 or                            Answer x = 18
                    2d   4 days    2 4

                   50c x c    50 x
9.) Proportion            or                                 Answer x = 25
                   8h   4h    8 4

				
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Description: Ratio Proportion Cross Rate