6th Grade - Ratios

 
     
 
     
 
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6th
Fractions, Decimals and Percentages
Ratios
Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a/b, a to b, a:b ).
Compare two quantities, where the quantities are known as terms. When “a” is term 1 and “b” is term 2, the quantities can be written in fraction form (a/b), word form (a to b), or using a colon (a:b). Ratios are used to compare; part to whole of the same set (ex. Number of dime out of total coins), part to part of different sets (ex. Number of dimes to number of quarters), two different quantities of the same variable (ex. Number of dimes in June to number of dimes in July), two different variables measured in the same units (ex. Can of soup to cans of water), and express rate (ex. Mph – compares distance to time). Equivalent ratios can be written by either; dividing or multiplying each term by the same number. To put a ratio in simplest form, the greatest common divisor (largest number that goes into both terms evenly with no remainders) must be divided by each term. It is only in simplest form when both terms no longer have any common divisors. For example, 4/16 can be simplified by dividing both numbers by 2. This would give the equivalent ratio 2/8. However, 2 and 8 are both also divisible by 2, so this is not yet simplest form. The simplest form of 4/16 is ¼ because both 4 and 16 have the greatest common divisor (a.k.a. GCF) of 4. (4⁄4=1 and 16⁄4=4, so the ratio in simplest form is ¼).
 

Sample Problems

(1)

There are 25 animals in the pet shop. Eight of the animals for sale are cats. What is the ratio of cats to total animals?

(8:25)

(2)

There are 12 cats and 8 dogs on Jackson Street. What is the ratio of cats to dogs? Is this ratio in simplest form? Write this ratio in simplest form.

(12:8 is not in simplest form, 3:2 is the simplest form of 12:8)

(3)

There are 8 dogs on Jackson Street and 12 on Jefferson Street. What is the ratio of dogs on Jackson Street to dogs on Jefferson Street?

(8:12)

(4)

Megan is making pitcher of lemonade. She needs one cups of lemon juice and 3 cups of water. What is the ratio of lemon juice to water? What would the equivalent ratio be if she wanted to make two pitchers?

(1:3, 2:6)

(5)

Juan can ride his bike 5 miles in one hour. What is the ratio of miles to hours?

(5:1)

Learning Tips

(1)

Review the three ways to write ratios. All three have the exact same meaning and representation. Remind students that whether the ratio is written as a fraction, ratio or with the colon that it is read by saying the first number followed by the word “to” and the last number. (ex. 4:6 is read “four to six”.

(2)

Each quantity in a ratio is called a term. The order in which the term is written is very important. For instance, if you are asked to find the ratio of cats to dogs, you must write the quantity of cats as the first term. (This would be the top term if written as a fraction). Pay close attention to the order of terms in word problems.

(3)

Have students create a word ratio before inputting terms. For example, cats:dogs, cats/dogs, or cats to dogs. They can then fill in the numbers in front of each word to ensure correct placement.

(4)

All ratios have equivalent ratios. These can have terms that are either smaller or larger than the original ratio. You find equivalent ratios the same way you find equivalent fractions. You can simplify a ratio (make the terms smaller) if there is a common divisor for both terms. For example, the ratio 4:2 can be simplified because both terms are can be divided by the divisor 2. Hence, the ratio 4:2 has the equivalent, simplified ratio of 2:1. Also, you can find equivalent ratios by multiplying both terms by the same number. For example, another ratio that is equivalent to 4:2 is 8:4. This is because

4x2=8 and 2x2=4.

(5)

A part can be zero. For example, If you’re asked to compare the number of pennies to quarters and there are no pennies and 10 quarters. The ratio would be written 0:10, 0/10 or 0 to 10.

(6)

Remember that all rates are ratios because you are comparing two quantities.

Extra Help Problems

(1)

Write the ratio of Xs to Os 3 ways. XXXXXXX OOOOOO

(7:6, 7/6, 7 to 6)


(2)

Write the ratio of Os to total 3 ways. XXXXXXX OOOOOO

(6:13, 6/13, 6 to 13)

(3)

Tom has 2 cats. Jose has 4 cats. Will has 2 dogs. What is the ratio of dogs to cats?

(2:6 or 1:3)


(4)

Tom has 2 cats. Jose has 4 cats. Will has 2 dogs. What is the ratio of Tom’s cats to Jose’s cats?

(2:4 or 1:2)


(5)

Tom has 2 cats. Jose has 4 cats. Will has 2 dogs. What is the ratio of dogs to total pets?

(2:8 or 1:4)

(6)

Hayley can drive 100 miles in two hours. Write her driving rate as a ratio. Find an equivalent ratio to show how many miles she can drive in one hour.

(100:2, 50:1)

(7)

Rogers Jr. High has 20 soccer balls, 45 dodge balls and 15 baseballs. What is the ratio of soccer balls to total balls? Can this ratio be written as 1/4 ? Explain.

(20:80, Yes, if both terms are divided by 4 you will get 1:4.)

(8)

Rogers Jr. High has 20 soccer balls, 45 dodge balls and 15 baseballs. What is the ratio of baseballs to dodge balls? Can this ratio be written 45:15? Explain.

(15:45, No, 45:15 gives the ratio of dodge balls to baseballs, not baseballs to dodge balls.)

(9)

Rogers Jr. High has 20 soccer balls, 45 dodge balls and 15 baseballs. What is the ratio of dodge balls to total balls? Find one equivalent ratio.

(45:80, sample equivalent ratio = 9/16)

(10)

Use the bolded ratio to find equivalent ratios and complete the table.

1

5

10

15



15

30


60

(1:3, 15:45, 20:60)


(11)

Use the bolded ratio to find equivalent ratios and complete the table.

1

2

3

4

5


12




(1:6, 3:24, 4:36, 5:48)

(12)

Complete the ratio table by finding equivalent ratios. Be sure to always start with the bolded ratio and pay attention to pattern changes.


1

4


20

48


12

36



(1:3, 12:36, 20:60, 48:144)

(13)

Complete the ratio table by finding equivalent ratios. Be sure to always start with the bolded ratio and pay attention to pattern changes. Write two of your own in the last boxes.

1


9




6

18



(1:2, 3:6, sample ratios: 36:72, 45:90)

(14)

Complete the ratio table by finding equivalent ratios. Be sure to always start with the bolded ratio and pay attention to pattern changes. Write two of your own in the last boxes.

11

33




4


20



(33:12, 55:20, Sample ratios: 66:24, 44:16)

(15)

Write the ratio in simplest form.

100: 50

(2:1)

(16)

Write the ratio in simplest form.

111 to 111

(1 to 1)

(17)

Write the ratio in simplest form.

21: 35

(3:5)

(18)

Write the ratio in simplest form.

13/39

(1/3)

(19)

Write the ratio in simplest form.

25 to 250

(1 to 10)

(20)

Tell whether or not the two ratios below are equivalent and explain how you know.

25:75 1/3

(Yes, they have a GCF of 3.)

(21)

Tell whether or not the two ratios below are equivalent and explain how you know.

50 to 100 100:200

(Yes, they have a GCF of 2.)

(22)

Tell whether or not the three ratios below are equivalent and explain how you know.

5/8 25/40 1/ 4

(5/8 and 25/40 are because of a GCF of 5, but ¼ is not because there are no common factors for ¼ and the other two ratios.)


(23)

Tell whether or not the three ratios below are equivalent and explain how you know.

22/11 2/1 66/33

(Yes, they have common factors: 11 and 3.)

(24)

The table below shows the favorite ice cream flavors of 6th grade students at Skylark School. Use the table to find the ratio of students that prefer mint-n-chip to the students that prefer cookie dough. Write the ratio in simplest form.

Ice Cream

Chocolate Chunk

Mint-N-Chip

Cookie Dough

# of students

15

10

45

(10:45, 2:9)

(25)

The table below shows the music type of 6th grade students at Smith School. Use the table to find the ratio of students that prefer hip hop to all students interviewed. Write the ratio in simplest form.

Music Type

Alternative

R & B

Hip Hop

# of students

18

15

11

(11:44, 1:4)

 

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