COMPARISON OF RATIOS

Example 1 :

Compare 2 : 3 and 5 : 3.

Solution :

2 : 3 = 2/3

5 : 3 = 5/3

The fractions 2/3 and 5/3 have the same denominator.

Compare the numerators.

2 < 5

So, 

2/3 < 5/3

2 : 3 < 5 : 3

Example 2 :

Compare 3 : 5 and 4 : 7.

Solution :

3 : 5 = 3/5

4 : 7 = 4/7

The fractions 3/5 and 4/7 do not have the same denominator.

Least common multiple of the denominators (5, 7) = 35.

Make the denominators of the fractions as 35 using multiplication.

3/5 = (3x7/5x7) = 21/35

4/7 = (4x5/7x5) = 20/35

The fractions 21/35 and 20/35 have the same denominator.

Compare the numerators.

21 > 20

So,

21/35 > 20/35

3/5 > 4/7

3 : 5 > 4 : 7

Example 3 :

Compare 3 : 4 and 5 : 6.

Solution :

3 : 4 = 3/4

5 : 6 = 5/6

The fractions 3/4 and 5/6 do not have the same denominator.

Least common multiple of the denominators (4, 6) = 12.

Make the denominators of the fractions as 35 using multiplication.

3/4 = (3x3/4x3) = 9/12

5/6 = (5x2/6x2) = 10/12

The fractions 9/12 and 10/12 have the same denominator.

Compare the numerators.

9 < 10

So,

9/12 < 10/12

3/4 < 5/6

3 : 4 < 5 : 6

Example 4 :

Compare 1.2 : 1.5 and 2 : 5.

Solution :

1.2 : 1.5 = 1.2/1.5

= (1.2x10)/(1.5x10)

= 12/15

= 4/5

2 : 5 = 2/5

The fractions 4/5 and 2/5 have the same denominator.

Compare the numerators.

4 > 2

So, 

4/5 < 2/5

1.2 :  1.5 > 2 : 5

Example 5 :

Compare 1 : 2.5 and 1.5 : 4.5.

Solution :

1 : 2.5 = 1/2.5

= (1x10)/(2.5x10)

= 10/25

= 2/5

1.5 : 4.5 = 1.5/4.5

= (1.5x10)/(4.5x10)

= 15/45

= 1/3

The fractions 2/5 and 1/3 do not have the same denominator.

Least common multiple of the denominators (5, 3) = 15.

Make the denominators of the fractions as 35 using multiplication.

2/5 = (2x3/5x3) = 6/15

1/3 = (1x3/3x5) = 3/15

The fractions 6/15 and 3/15 have the same denominator.

Compare the numerators.

6 > 3

So,

6/15 > 3/15

2/5 > 1/5

1 : 2.5 > 1.5 : 4.5

Example 6 :

Compare 2⅓ : 3⅓ and 3.6 : 4.8.

Solution :

2 ⅓ : 3 ⅓ = 7/3 : 10/3

= 7 : 10

= 7/10

3.6 : 4.8 = 3.6/4.8

= 36/48

= 3/4

The fractions 7/10 and 3/4 do not have the same denominator.

Least common multiple of the denominators (10, 4) = 20.

Make the denominators of the fractions as 20 using multiplication.

7/10 = (7x2/10x2) = 14/20

3/4 = (3x5/4x5) = 15/20

The fractions 14/20 and 15/20 have the same denominator.

Compare the numerators.

14 < 15

So,

14/20 < 15/20

7/10 < 3/4

2⅓ : 3⅓  < 3.6 : 4.8

Example 7 :

During a checkers game, there are 16 pieces left. The ratio of black to red is 3:5. How many black pieces are on the board? Explain how you found your answer.

Solution :

Ratio of black to red = 3 : 5

Number of black pieces = 3x

Number of red pieces = 5x

Number of pieces left = 16

3x + 5x = 16

8x = 16

x = 16/8

x = 2

Number of black pieces = 3(2) ==> 6

Number of red pieces = 5(2) ==> 10

Example 8 :

Your neighbor pays you $17 for every 2 hours you work. You work for 8 hours on Saturday. How much does your neighbor owe you?

Solution :

Ratio between number of hours to amount pays

= 2 : 17

When number of hours = 8, let x be the amount you are getting.

2 : 17 = 8 : x

2/17 = 8/x

2x = 8(17)

x = 4(17)

= $68

Example 9 :

You are downloading songs to your MP3 player. The ratio of pop songs to rock songs is 5 : 4. You download 40 pop songs. How many rock songs do you download?

Solution :

Let x be the number of rock songs.

Ratio between pop songs to rock songs = 5 : 4

5 : 4 = 40 : x

5/4 = 40/x

5x = 4(40)

x = 160/5

x = 32

So, number of rock songs is 32.

Example 10 :

A sports store donates basketballs and soccer balls to the boys and girls club. The ratio of basketballs to soccer balls is 7 : 6. The store donates 24 soccer balls. How many basketballs does the store donate?

Solution :

Ratio of basket balls to soccer balls = 7 : 6

Number of soccer ball = 24

Number of basket ball = x

7 : 6 = x : 24

7/6 = x/24

7(24) = 6x

x = 7(24)/6

x = 28

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