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RATIOS & PROPORTIONS
                                LESSON 1



                                                                      TENNESSEE
                                                                   ADULT EDUCATION




This curriculum was written with funding of the Tennessee Department of Labor and Workforce Development and may not be reproduced in any way without written permission. ©
RATIOS
Ratios are comparisons made between two sets
of numbers.


      For example:
There are eight girls and seven boys in a class.


          The ratio of girls to boys is 8 to 7.
Ratios are used everyday. They are used for:

Miles per hour
The cost of items per pound, gallon, etc.
Hourly rate of pay




                                       80 miles to 1 hour = 80mph
THERE ARE 3 WAYS TO WRITE RATIOS.
1. Write the ratio using the word “to” between the two
numbers being compared.

 For example: There are 8 girls and 5 boys in my class.
 What is the ratio of girls to boys?



             The ratio is: 8 girls to 5 boys

                               8 to 5
2. Write a ratio using a colon between the two
   numbers being compared.

For example: There are 3 apples and 4 oranges in the
   basket. What is the ratio of apples to oranges?



      The ratio is: 3 apples to 4 oranges.

                      3:4
3. Write a ratio as a fraction.


For example:
 Hunter and Brandon were playing basketball. Brandon
 scored 5 baskets and Hunter scored 6 baskets. What
 was the ratio of baskets Hunter scored to the baskets
 Brandon scored?



The ratio of baskets scored was:

                     6 baskets to 5 baskets
                                  6
                                  5
GUIDED PRACTICE:
Directions: Write the ratio in three different ways.
There are 13 boys and 17 girls in sixth grade.
Find the ratio of boys to the girls in sixth grade.



                                               13
       13 to 17            13 : 17
                                               17
RULES FOR SOLVING RATIO PROBLEMS.

  1.   When writing ratios, the numbers should be written in
       the order in which the problem asks for them.

 For example: There were 4 girls and 7 boys at the birthday
              party.
  What is the ratio of girls to boys?

Hint: The question asks for girls to boys; therefore, girls
will be listed first in the ratio.

                                                 4 girls
  4 girls to 7 boys       4 girls : 7 boys
                                                 7 boys
GUIDED PRACTICE:
Directions: Solve and write ratios in all three forms.

1. The Panthers played 15 games this season. They won 13
   games. What is the ratio of games won to games played?

 The questions asks for Games won to Games
 played.
                                        13
      13 to 15          13:15
                                        15
2. Amanda’s basketball team won 7 games and lost 5.
 What is the ratio of games lost to games won?
THE QUESTION ASKS FOR GAMES LOST TO GAMES
WON. THEREFORE, THE NUMBER OF GAMES LOST
SHOULD BE WRITTEN FIRST, AND THE GAMES WON
SHOULD BE WRITTEN SECOND.
    Games lost = 5 to Games won = 7


   5 to 7          5:7           5
                                 7
REDUCING RATIOS
Ratios can be reduced without changing their relationship.



                2 boys to 4 girls =




                 1 boy to 2 girls =
REDUCING RATIOS
Is this relationship the same?



                2 boys to 4 girls =




                 1 boy to 3 girls =
2. ALL RATIOS MUST BE WRITTEN IN
 LOWEST TERMS.
Steps:
   1. Read the word problem.
   2. Set up the ratio.
 For example:

 You scored 40 answers correct out of 45 problems on a
 test. Write the ratio of correct answers to total questions in
 lowest form.

 Step 1: Read the problem. What does it want to know?

              40 to 45    40 : 45     40
                                      45
3. Reduce the ratio if necessary.

  Reduce means to break down a fraction or ratio into the
  lowest form possible.
          Reduce = smaller number; operation will always be division.


HINT: When having to reduce ratios, it is better to set up the ratio in the
vertical form. (Fraction Form)

                                   40
               40 to 45      =
                                   45
 Look at the numbers in the ratio. What ONE
 number can you divide BOTH numbers by?

 Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
 Factors of 45: 1, 3, 5, 9, 15, 45                  40 ÷    5    =      8
                                                    45 ÷    5    =      9
Guided Practice:

Directions: Solve each problem. Remember to reduce.

 1. There are 26 black cards in a deck of playing cards. If there are 52 cards
 in a deck, what is the ratio of black cards to the deck of cards?

       Step 1: Read the problem. (What does it want to know?)

        Step 2: Set up the ratio.

                     26 black cards to 52 cards
        Step 3: Can the ratio be reduced? If so, set it up like a fraction.

                26 ÷ 26 =           1
                52 ÷ 26 =           2

                          What is the largest number that will go into
                          both the top number and the bottom number
                          evenly? (It can not be the number one!)
2. Kelsey has been reading Hunger Games for class. She read 15
chapters in 3 days. What is the ratio of chapters read to the
number of days she read?

                 15 chapters to 3 days



   15   ÷÷   3    =    5
    3        3    =
                       1



   Hint: When a one is on the bottom, it
   must remain there. If the one is dropped,
   there is no longer a ratio.
PROPORTIONS
Proportions are two ratios of equal value.



            1 girl                     4 girls
            4 boys                   16 boys


Are these ratios saying the same thing?
PROPORTIONS
Proportions are two ratios of equal value.



            1 girl                     5 girls
            4 boys                   16 boys


Are these ratios proportions?
DETERMINING TRUE PROPORTIONS:
 To determine a proportion true, cross multiply.
  If the cross products are equal, then it is a true proportion.


               4     =     20
               5           25

          20 x 5     =   4 x 25
           100       =     100


The cross products were equal, therefore 4 And 20 makes a true proportion.
                                        5    25
Guided Practice:

Directions: Solve to see if each problem is a true proportion.


1. 3 = 15
   5   25



2.   6 = 57
     8   76


3.     7 = 37
      12 60
Guided Practice:

Directions: Solve to see if each problem is a true proportion.

    3   15                    6          57
 1. 5 = 25               2.        =              3. 7      =     37
                              8          76          12           60



15 x 5 = 3 x 25         57 x 8 = 6 x 76
                                                7 x 60 = 37 x 12
                         456      =      456
  75 =      75                                   420 =      444

     true                         true
                                                    false
SOLVING PROPORTIONS WITH VARIABLES
 What is a variable?          A variable is any letter that takes place of a
                              missing number or information.

Eric rode his bicycle a total of 52 miles in 4 hours. Riding at
this same rate, how far can he travel in 7 hours?

                                                        Next, the problem states “how
                                                        far can he travel in 7 hours.
Look for the two sets of     You have 52 miles in 4
                                                        The problem is missing the
ratios to make up a          hours. This is the first
                                                        miles. Therefore, the miles
proportion.                  ratio.
                                                        becomes the variable.


    Set 1                     52 miles                            n miles
                                                    Set 2
                              4 hours                             7 hours

  The proportion should be                  52 = n
                                             4   7
  set equal to each other.

                                      HINT: The order of the ratio does matter!
SOLVING THE PROPORTION:
When solving proportions, follow these rules:

1. Cross multiply.
2. Divide BOTH sides by the number connected to the variable.
3. Check the answer to see if it makes a true proportion.

Problem:                               52        n
                                          =
                                       4         7

                                    4 x n = 52 x 7
 Which number is
 connected to the variable?           4n =      364         n = 91 miles
                                      4         4
Since the 4 is connected
to the variable, DIVIDE
both sides by the 4.
                              4 ÷ 4 = 1;        364 ÷ 4 =   91
                              therefore you
                              are left with “n”
                              on one side.
If it comes out even, then the answer is correct.

          Check your answer!

            52        91
                 =
             4        7

           52 x 7 = 91 x 4
            364 = 364
GUIDED PRACTICE
Directions: Solve each proportion.


1.For every dollar Julia spends on her Master
Card, she earns 3 frequent flyer miles with
American Airlines. If Julia spends $609 dollars on
her card, how many frequent flyer miles will she
earn?
GUIDED PRACTICE
Directions: Solve each proportion.
1. For every dollar Julia spends on her Master Card, she
earns 3 frequent flyer miles with American Airlines. If Julia
spends $609 dollars on her card, how many frequent flyer
miles will she earn?



Step 1: Set up the proportion.   $1.00    =   $609.00
                                  3 miles     d miles
Step 2: Cross multiply.
                                   1d   = 1827
Step 3: Divide
                                    1         1
Step 4: Check answer.              d    = 1827
1. Justin’s car uses 40 gallons of gas to drive 250
   miles. At this rate, approximately, how many
   gallons of gas will he need for a trip of 600 miles.

2. If 3 gallons of milk cost $9, how many jugs can
   you buy for $45?


3.   On Thursday, Karen drove 400 miles in 8 hours.
     At this same speed, how far can she drive in 12
     hours?
1. Justin’s car uses 40 gallons of gas to drive 250
   miles. At this rate, approximately, how many
   gallons of gas will he need for a trip of 600 miles.

 40 gal    x gal         40            x
        =                       =
 250 mi   600mi         250          600


 250x = 24000


  250x    = 24000                             Check:

  250        250
                                        40              96
                                                =
    x    =   96                        250             600
                                      24000     =      24000
2. If a 3 gallon jug of milk cost $9, how many 3
  gallon jugs can be purchased for $45?
                                  1   n
                                    =
                                  9 45

                                      Check:
 1= n
 9 45                             1             5
                              9        =
                                               45
1x45 = 9n
                              45       = 45
45 = 9n
 9      9
5=n

5 jugs of milk can be
purchased for $45
3.   On Thursday, Karen drove 400 miles in 8 hours. At this
     same speed, how far can she drive in 12 hours?


     400 miles    =      x miles
      8 hours           12 hours

     400           x_
            =
      8           12
     400           x_
           =
      8           12

      8x   =     4800

       x    =    600 miles
4.     Susie has two flower beds in which to plant tulips and
daffodils. She wants the proportion of tulips to daffodils to be
the same in each bed. Susie plants 10 tulips and 6 daffodils in
the first bed. How many tulips will she need for the second bed
if she plants 15 daffodils?
10 tulips    =     x tulips               10    x_
                                             =
 6 daffodils       15 daffodils            6   15


10            x                       x    = 25 tulips
       =
 6           15
6x     =    150                      10         25
                                        =
6x         150                        6         15
       =
 6          6                       150     =   150
 x   = 25

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1 - Ratios & Proportions

  • 1. RATIOS & PROPORTIONS LESSON 1 TENNESSEE ADULT EDUCATION This curriculum was written with funding of the Tennessee Department of Labor and Workforce Development and may not be reproduced in any way without written permission. ©
  • 2. RATIOS Ratios are comparisons made between two sets of numbers. For example: There are eight girls and seven boys in a class. The ratio of girls to boys is 8 to 7.
  • 3. Ratios are used everyday. They are used for: Miles per hour The cost of items per pound, gallon, etc. Hourly rate of pay 80 miles to 1 hour = 80mph
  • 4. THERE ARE 3 WAYS TO WRITE RATIOS. 1. Write the ratio using the word “to” between the two numbers being compared. For example: There are 8 girls and 5 boys in my class. What is the ratio of girls to boys? The ratio is: 8 girls to 5 boys 8 to 5
  • 5. 2. Write a ratio using a colon between the two numbers being compared. For example: There are 3 apples and 4 oranges in the basket. What is the ratio of apples to oranges? The ratio is: 3 apples to 4 oranges. 3:4
  • 6. 3. Write a ratio as a fraction. For example: Hunter and Brandon were playing basketball. Brandon scored 5 baskets and Hunter scored 6 baskets. What was the ratio of baskets Hunter scored to the baskets Brandon scored? The ratio of baskets scored was: 6 baskets to 5 baskets 6 5
  • 7. GUIDED PRACTICE: Directions: Write the ratio in three different ways. There are 13 boys and 17 girls in sixth grade. Find the ratio of boys to the girls in sixth grade. 13 13 to 17 13 : 17 17
  • 8. RULES FOR SOLVING RATIO PROBLEMS. 1. When writing ratios, the numbers should be written in the order in which the problem asks for them. For example: There were 4 girls and 7 boys at the birthday party. What is the ratio of girls to boys? Hint: The question asks for girls to boys; therefore, girls will be listed first in the ratio. 4 girls 4 girls to 7 boys 4 girls : 7 boys 7 boys
  • 9. GUIDED PRACTICE: Directions: Solve and write ratios in all three forms. 1. The Panthers played 15 games this season. They won 13 games. What is the ratio of games won to games played? The questions asks for Games won to Games played. 13 13 to 15 13:15 15
  • 10. 2. Amanda’s basketball team won 7 games and lost 5. What is the ratio of games lost to games won? THE QUESTION ASKS FOR GAMES LOST TO GAMES WON. THEREFORE, THE NUMBER OF GAMES LOST SHOULD BE WRITTEN FIRST, AND THE GAMES WON SHOULD BE WRITTEN SECOND. Games lost = 5 to Games won = 7 5 to 7 5:7 5 7
  • 11. REDUCING RATIOS Ratios can be reduced without changing their relationship. 2 boys to 4 girls = 1 boy to 2 girls =
  • 12. REDUCING RATIOS Is this relationship the same? 2 boys to 4 girls = 1 boy to 3 girls =
  • 13. 2. ALL RATIOS MUST BE WRITTEN IN LOWEST TERMS. Steps: 1. Read the word problem. 2. Set up the ratio. For example: You scored 40 answers correct out of 45 problems on a test. Write the ratio of correct answers to total questions in lowest form. Step 1: Read the problem. What does it want to know? 40 to 45 40 : 45 40 45
  • 14. 3. Reduce the ratio if necessary. Reduce means to break down a fraction or ratio into the lowest form possible. Reduce = smaller number; operation will always be division. HINT: When having to reduce ratios, it is better to set up the ratio in the vertical form. (Fraction Form) 40 40 to 45 = 45 Look at the numbers in the ratio. What ONE number can you divide BOTH numbers by? Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40 Factors of 45: 1, 3, 5, 9, 15, 45 40 ÷ 5 = 8 45 ÷ 5 = 9
  • 15. Guided Practice: Directions: Solve each problem. Remember to reduce. 1. There are 26 black cards in a deck of playing cards. If there are 52 cards in a deck, what is the ratio of black cards to the deck of cards? Step 1: Read the problem. (What does it want to know?) Step 2: Set up the ratio. 26 black cards to 52 cards Step 3: Can the ratio be reduced? If so, set it up like a fraction. 26 ÷ 26 = 1 52 ÷ 26 = 2 What is the largest number that will go into both the top number and the bottom number evenly? (It can not be the number one!)
  • 16. 2. Kelsey has been reading Hunger Games for class. She read 15 chapters in 3 days. What is the ratio of chapters read to the number of days she read? 15 chapters to 3 days 15 ÷÷ 3 = 5 3 3 = 1 Hint: When a one is on the bottom, it must remain there. If the one is dropped, there is no longer a ratio.
  • 17. PROPORTIONS Proportions are two ratios of equal value. 1 girl 4 girls 4 boys 16 boys Are these ratios saying the same thing?
  • 18. PROPORTIONS Proportions are two ratios of equal value. 1 girl 5 girls 4 boys 16 boys Are these ratios proportions?
  • 19. DETERMINING TRUE PROPORTIONS: To determine a proportion true, cross multiply. If the cross products are equal, then it is a true proportion. 4 = 20 5 25 20 x 5 = 4 x 25 100 = 100 The cross products were equal, therefore 4 And 20 makes a true proportion. 5 25
  • 20. Guided Practice: Directions: Solve to see if each problem is a true proportion. 1. 3 = 15 5 25 2. 6 = 57 8 76 3. 7 = 37 12 60
  • 21. Guided Practice: Directions: Solve to see if each problem is a true proportion. 3 15 6 57 1. 5 = 25 2. = 3. 7 = 37 8 76 12 60 15 x 5 = 3 x 25 57 x 8 = 6 x 76 7 x 60 = 37 x 12 456 = 456 75 = 75 420 = 444 true true false
  • 22. SOLVING PROPORTIONS WITH VARIABLES What is a variable? A variable is any letter that takes place of a missing number or information. Eric rode his bicycle a total of 52 miles in 4 hours. Riding at this same rate, how far can he travel in 7 hours? Next, the problem states “how far can he travel in 7 hours. Look for the two sets of You have 52 miles in 4 The problem is missing the ratios to make up a hours. This is the first miles. Therefore, the miles proportion. ratio. becomes the variable. Set 1 52 miles n miles Set 2 4 hours 7 hours The proportion should be 52 = n 4 7 set equal to each other. HINT: The order of the ratio does matter!
  • 23. SOLVING THE PROPORTION: When solving proportions, follow these rules: 1. Cross multiply. 2. Divide BOTH sides by the number connected to the variable. 3. Check the answer to see if it makes a true proportion. Problem: 52 n = 4 7 4 x n = 52 x 7 Which number is connected to the variable? 4n = 364 n = 91 miles 4 4 Since the 4 is connected to the variable, DIVIDE both sides by the 4. 4 ÷ 4 = 1; 364 ÷ 4 = 91 therefore you are left with “n” on one side.
  • 24. If it comes out even, then the answer is correct. Check your answer! 52 91 = 4 7 52 x 7 = 91 x 4 364 = 364
  • 25. GUIDED PRACTICE Directions: Solve each proportion. 1.For every dollar Julia spends on her Master Card, she earns 3 frequent flyer miles with American Airlines. If Julia spends $609 dollars on her card, how many frequent flyer miles will she earn?
  • 26. GUIDED PRACTICE Directions: Solve each proportion. 1. For every dollar Julia spends on her Master Card, she earns 3 frequent flyer miles with American Airlines. If Julia spends $609 dollars on her card, how many frequent flyer miles will she earn? Step 1: Set up the proportion. $1.00 = $609.00 3 miles d miles Step 2: Cross multiply. 1d = 1827 Step 3: Divide 1 1 Step 4: Check answer. d = 1827
  • 27. 1. Justin’s car uses 40 gallons of gas to drive 250 miles. At this rate, approximately, how many gallons of gas will he need for a trip of 600 miles. 2. If 3 gallons of milk cost $9, how many jugs can you buy for $45? 3. On Thursday, Karen drove 400 miles in 8 hours. At this same speed, how far can she drive in 12 hours?
  • 28. 1. Justin’s car uses 40 gallons of gas to drive 250 miles. At this rate, approximately, how many gallons of gas will he need for a trip of 600 miles. 40 gal x gal 40 x = = 250 mi 600mi 250 600 250x = 24000 250x = 24000 Check: 250 250 40 96 = x = 96 250 600 24000 = 24000
  • 29. 2. If a 3 gallon jug of milk cost $9, how many 3 gallon jugs can be purchased for $45? 1 n = 9 45 Check: 1= n 9 45 1 5 9 = 45 1x45 = 9n 45 = 45 45 = 9n 9 9 5=n 5 jugs of milk can be purchased for $45
  • 30. 3. On Thursday, Karen drove 400 miles in 8 hours. At this same speed, how far can she drive in 12 hours? 400 miles = x miles 8 hours 12 hours 400 x_ = 8 12 400 x_ = 8 12 8x = 4800 x = 600 miles
  • 31. 4. Susie has two flower beds in which to plant tulips and daffodils. She wants the proportion of tulips to daffodils to be the same in each bed. Susie plants 10 tulips and 6 daffodils in the first bed. How many tulips will she need for the second bed if she plants 15 daffodils? 10 tulips = x tulips 10 x_ = 6 daffodils 15 daffodils 6 15 10 x x = 25 tulips = 6 15 6x = 150 10 25 = 6x 150 6 15 = 6 6 150 = 150 x = 25