All About Ratio

2) What is the ratio of videocassettes to the total number of items in the bag? There are 3 videocassettes, and 3 + 4 + 7 + 1 = 15 items total. 3 A ratio is a comparison of two or more quantities of the same The answer can be expressed as , 3 is to 15, or 3 :15. 15 kind. It is expressed in the form of a:b. We generally separate the two numbers in the ratio with a colon (:). Suppose we want to Comparing Ratios write the ratio of 8 and 12. 8 We can write this as 8:12 or as a fraction , and we read it as To compare ratios, write them as fractions. The ratios are equal if 12 eight is to twelve. they are equal when written as fractions.

Ratio

Example:

1:5 ↙

Are the ratios 3 is to 4 and 6:8 equal? 3 6 The ratios are equal as = . 4 8

↘

1 parts : 5 parts □ : □□□□□ 1 5 or 5 1

Examples: Are the ratios 7 : 1 and 4 : 81 equal?

Examples of Problems involving Ratios: Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange. 1) What is the ratio of books to marbles? Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer 7 would be . Two other ways of writing the ratio are 7 is to 4, 4 and 7:4.

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Example: Ratio Method

Ratios and Proportion A proportion is an equation with a ratio on each side. It is a statement that two ratios are equivalent. It is expressed as a = or a : b = c : d. b

A piece of string , 26 cm long, is cut in the ratio 4 : 9. What is the length of the longer piece? shorter piece

□□□□ 26 cm

In direct proportion , when one quantity increases, the other longer piece □□□□□□□□□ quantity increases in the same ratio. Likewise, when one quantity 4 + 9 = 13 parts → 26 cm decreases, the other quantity decreases in the same ratio. ( 26 I 13 ) x 9 = 18 cm

Examples  Equivalent ratio method Find the answer  18 : 27 = ____ : 3 = 8 : 12

9 parts →

The ratio of the length of A to the length of B is 4 : 9. If the length of A is 8 cm , find the length of B.

Therefore, the length of the longer piece is 18 cm.

A:B 4:9 x 2 ↙ 8

↘ x 2 : __

Therefore, the length of B is ______.

In indirect proportion, when one quantity increases , the other quantity decreases in the same ratio and vice versa. Example : A worker takes 4 hours to paint one room. How long will 4 workers take to paint the same room? How long will 4 workers take to paint 5 such rooms ? 1 worker : 1 room : 4 hours I 4æ

åx4

4 workers :1 room : _________ ↙ x 5

x 5 ↘

4 workers : 5 rooms : _________

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Method 2 è We can find the quantity for one unit before finding for the given units.

Rate Rate is used to expresse how long it takes to do something, such as traveling a certain distance. To walk 3 kilometers in one hour is to walk at the rate of 3 km/h.

30 min è 4 km One unit Ú 1 min è 30 ¸ 4 = 7.5 km 45 min Ú ( 45 ¸ 7.5 )km = 6 km

The fraction expressing Rate has units of distance as the numerator and units of time as the denominator. That is

3 km 1 hour 

=

6 km 2 hours

Converting rates

and so on.

We compare rates just as we compare ratios, by cross multiplying. When comparing rates, always check to see

Problems involving rates typically involve setting two ratios equal which units of measurement are being used. to each other and solving for an unknown quantity,

For instance, 3 kilometers per hour is very different from 3 meters

that is, solving a proportion. In other words rate can be used to

per hour! Look at the conversion below. 3 km/ hour = 3×1000 metres / hour =

compare any two quantities.

3000 m/ hour

As you can clearly see 3000 metres/hr is not the same as Example:

3 metres/hr.

Juan runs 4 km in 30 minutes. At that rate, how far could he run in 45 minutes? Method 1 è We can represent the data as fractions. 4 km 30 min

=

2 km 15 min

=

6 km 45 min

Therefore our answer is 6 km.

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Important:

Average Rate of Speed

One of the most useful tips in solving any math problem is to always write out the units when multiplying, dividing, or converting from one unit to another.

The average rate of speed for a trip is the total distance traveled divided by the total time taken for the trip. Remember this triangle when calculating speed, distance or time. Formulas :

Example: If Juan runs 4 km in 30 minutes, how many hours will it take him to run 1 km? Be careful not to confuse the units of measurement. While Juan's rate of speed is given in terms of minutes, the question is posed in terms of hours. Only one of these units may be used in setting up a proportion. Convert 30 minutes to hours. 4 km Ú 30 min = ½ hr Ô

÷

÷4

4

1 km The answer is

Ú

1 8

Ô hr

1 hr. 8

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Distance = Speed x Time Speed = Distance

÷

Time or

Time = Distance

÷

Speed or

 Distance  Time

 Distance  Speed 

On a vacation, Linda's family traveled 495 miles at 55 mph. How long did the trip take?

Example: A dog walks 8 km at 4 km per hour, then chases a rabbit for 2 km at 20 km per hour. What is the dog's average rate of speed for the distance he traveled? 4 km

Ò 1 hours

Ôx2 8 km

Ôx2 Ò 2 hours

20 km

Ò 1 hour

¯ ¸ 10

¯ ¸ 10

2 km

Ò 0.1 hours

Total Distance travelled  8 km + 2 km = 10 km Total time taken

 2 h + 0.1 h = 2.1 hours

Average Speed

 10 km ¸ 2.1 h » 4.761 km/h

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