Math Mammoth Ratios & Proportions & Problem Solving

Introduction to Ratios
A ratio is simply a comparison of some numbers or other quantities. When reading a ratio,
we use the word, “to.” We can use the colon (“:”) to indicate a ratio.
In this picture, there are four balls and three triangles.
So we say that the ratio of balls to triangles is 4:3 (read “four to three”).
Or we can say the ratio is “four balls to three triangles,”
or write: four balls : three triangles.
Here the ratio of hearts to stars is 5:3 (read “five to three”).
But the ratio of stars to hearts is 3:5.
So you see that the order in which you say things really does matter!
You can also compare one quantity to the whole group using a ratio.
The ratio of balls to all shapes is 4:7. The ratio of stars to all shapes is 3:8.
We can also compare three or more different things in a ratio.
The ratio of balls to hearts to triangles is 3:2:5 (read “three to two to five”).
The ratio of triangles to hearts is 5:2. The ratio of balls to all shapes is 3:10.
1. Write the ratios.
a. The ratio of boys to girls.
b. The ratio of boys to all children.
c. The ratio of girls to all children.
d. The ratio of circles to squares.
e. The ratio of squares to all shapes.
f. The ratio of squares to hearts to circles.
6 Math Mammoth Ratiios & Proportions & Problem Solving (Blue Series)

2. a. Draw a picture that represents the ratio
3 small balls : 8 bigger balls.
b. What is the ratio of bigger balls to all balls
3. The ratio of green shirts to blue shirts is 3:5.
a. What is the ratio of green shirts to all shirts
b. What is the ratio of blue shirts to all shirts
c. What is the ratio of blue shirts to green shirts
4. The picture on the right shows some men and some women.
a. What is the ratio of men to women
b. Suppose that each symbol represents 6 people:
How many men are there _______ How many women _______
How many people in all _______
We can use ratios even in the following situation:
The ratio of girls to boys in a gym class is 3:2,
and there are 25 students in all.
Since the ratio is 3:2, we draw a diagram with 3 blocks to represent the number of girls and 2
blocks to represent the number of boys. Thus we get five equal parts. Since there are 25 children
in all, each block in the diagram represents 5 of them.
So there are 15 girls and 10 boys.
Draw a diagram for each situation, and solve it.
5. The ratio of puppies to adult dogs in a kennel is 4:1.
(Draw a diagram with 4 “puppy” blocks and one “adult” block.)
If there are 80 dogs in total, how many dogs does each block represent
How many of the 80 dogs are adults
How many are puppies
6. Marsha has 147 marbles in her bag, some white and
some red. The ratio of white marbles to red ones is 3:4.
How many marbles are red
7. Anita and Shirley share a reward of $200 in a ratio of 3:5.
How much does Anita get And what about Shirley
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We can present the same situation with the same information using either a picture or a diagram, and
using either fractions or a ratio. Compare the examples below carefully.
Picture or Diagram As Fractions As a Ratio
3
of the shapes are triangles.
5
The ratio of triangles
2
5 of the shapes are circles. to circles is 3:2.
W W W W W B B B
——— 72 ———
5
of the 72 socks are white.
8
The ratio of white to blue
socks is 5:3. There are 72
3
8 of the 72 socks are blue. socks in all.
8. “Translate” between a picture or diagram representation, fraction language, and ratio language.
Picture or Diagram As Fractions As a Ratio
a.
of the shapes are squares.
of the shapes are circles.
The ratio of squares
to circles is 4:5.
b.
2
9 of the shapes are triangles. The ratio of triangles to
rectangles is ___ : ___ .
of the shapes are rectangles.
c.
of the ___ toy cars are Jack's.
of the ___ toy cars are Rick's.
The ratio of the number of
Jack's cars to the number of
Rick's cars is ___ : ___.
There are ____ cars in all.
d.
of the ___ eggs are white.
of the ___ eggs are brown.
The ratio of white eggs
to brown ones is 5:1.
There are 3,600 eggs in all.
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9. 5/11 of the 1,287 hospital patients are females.
a. What is the ratio of the female patients to all patients
b. What is the ratio of the female patients to the male patients
c. How many male patients are there
10. Michael has 30 coins. 2/10 of Michael's coins are nickels,
3/10 are dimes, and 5/10 are quarters.
a. What is the ratio of Michael's nickels to dimes to quarters
b. How many of each coin does he have
See if you can solve this problem where we know the ratio of girls to boys,
and the amount of girls, but not the total! The diagram will help.
The ratio of girls to boys in a cooking club was 5:2.
There were 20 girls.
How many boys were there in the club
How many members in all
11. A jar has black and white beans in a ratio of 5:4.
If there are 240 white beans, how many black beans are there
How many beans all in all
12. Peter's mom has made a rule that for each 2 library books
Peter reads, he may also read 3 comic books. If Peter reads
14 library books, how many comic books can he read
13. If Peter hauls home a stack of 45 books and comic books
according to mom's rule, how many of them are comic books
14. In a cookie mix, we find chocolate chip cookies, vanilla cookies,
and nut cookies in a ratio of 1:3:2. If a large sack contains
210 vanilla cookies, how many nut cookies are there
How many cookies in all
9 Math Mammoth Ratiios & Proportions & Problem Solving (Blue Series)

Ratios and Fractions
A ratio is simply a comparison of two numbers or other quantities.
In the picture on the right, there are three hearts and five stars. To compare the hearts to the stars,
we can say that the ratio of hearts to stars is 3:5 (read “three to five”).
The ratio of stars to hearts is therefore 5:3 (read “five to three”).
Notice that the order in which you mention the members of the ratio matters.
We can use fractions, too, to describe this same picture, like this:
3
8 of the shapes are hearts. 5
of the shapes are stars.
8
You can also compare one quantity to the whole group using a ratio.
The ratio of circles to all shapes is 2:9.
This is really similar to saying that 2 9
3
A theater has 110 plastic chairs in stacks.
5
We can present this same information, using
a ratio instead of a fraction, like this:
There are 110 chairs, some blue, some white.
The ratio of blue chairs to white chairs is 3:2.
of the shapes are circles.
of the chairs are blue, and the rest are white.
Notice carefully: The ratio does NOT mention that the chairs could be divided into five equal parts.
But if you draw a diagram of the situation, you can easily see that. (Also note that 3 + 2 = 5.)
1. Describe the images using ratios and fractions.
a.
The ratio of circles to pentagons is ____ : ____
The ratio of pentagons to all shapes is ____ : ____
of the shapes are pentagons.
b.
The ratio of diamonds to triangles is ____ : ____
The ratio of triangles to all shapes is ____ : ____
of the shapes are diamonds.
c. Mr. Hyde owns 1,200 acres of land. of it is forest, and the rest of
it is swampland. The ratio of the forest to swampland is ____ : ____.
Of the land, ______ acres are forest, and ______ acres are swampland.
10 Math Mammoth Ratiios & Proportions & Problem Solving (Blue Series)

2. “Translate” between a picture or diagram representation, “fraction language,” and “ratio language.”
Picture or Diagram As Fractions As a Ratio
a.
of the shapes are hearts.
of the shapes are diamonds.
The ratio of hearts
to diamonds is 1:5.
b.
3
4 of the shapes are circles. The ratio of circles to
triangles is ___ : ___ .
of the shapes are triangles.
c.
of the 112 songs are slow.
of the 112 songs are fast.
The ratio of the number of
slow songs to the number of
fast songs is ___ : ___.
There are ____ songs in all.
3. A store got a shipment of 155 calculators.
The ratio of basic calculators to scientific calculators was 4:1.
a. Draw a model to represent the situation.
b. What fractional part of the calculators were basic calculators
c. How many scientific calculators were there
4. There are 1,407 students in the local community college.
The ratio of male students to female students is 3:4.
a. Draw a model to represent the situation.
b. What fractional part of the students are male
c. How many male students are there
5. A jar contains 810 beans. Five-sixths of the beans
are white beans, and the rest are pinto beans.
a. Draw a model to represent the situation.
b. What is the ratio of white beans to pinto beans
c. How many white beans are there
11 Math Mammoth Ratiios & Proportions & Problem Solving (Blue Series)

6. Jack has a box filled with white and transparent marbles
in a ratio of 2:5. He has 38 white marbles.
a. Draw a model to represent the situation.
b. What fractional part of the marbles are white
c. How many marbles does he have in all
You can also have a ratio that compares
three (or more) different quantities.
In the picture, the ratio of triangles to hearts to stars is 3:2:4.
Using fractions, we say that
3
9 of the shapes are triangles, 2 9
of the shapes are hearts,
and 4 9
of the shapes are stars.
7. The ratio of circles to squares to triangles is 4:2:7.
What fractional part of the shapes are squares
8. If 4 11 of the shapes are hearts, 5 11
of them are circles, and the rest are diamonds, then:
a. What is the ratio of hearts to circles to diamonds
b. What is the ratio of diamonds to circles to hearts
9. A trail mix recipe contains raisins, nuts, and dried fruit (other than
raisins) in a ratio of 4:3:1. If you want to make 2 pounds of the mix,
then how many ounces of each ingredient do you need
10. Another trail mix recipe contains raisins, nuts, and dried fruit
in a ratio of 2:3:1. If you want to make 2 pounds of this recipe,
then about how many ounces of each ingredient do you need
11. A truck is carrying orange juice, peach juice, and pear juice
bottles in a ratio of 7:3:2. If there are 1,200 pear juice bottles,
then how many orange juice bottles are there
How many bottles of juice in total does the truck contain
12 Math Mammoth Ratiios & Proportions & Problem Solving (Blue Series)

Ratios
A ratio is a comparison of two numbers or quantities, using division. We can use a colon (“:”)
to indicate a ratio. We can also express the ratio using the word “to” or write it as a fraction.
The ratio of balls to triangles is 5:4.
The ratio of balls to triangles is 5 to 4.
The ratio of balls to triangles is 5 4 .
The two numbers in the ratio are called the first term and the second term of the ratio.
The order in which the quantities are mentioned does matter. In this picture, the
ratio of balls to triangles is 4:2, and the ratio of triangles to balls is 2:4.
We can also compare a part to the whole. The ratio of balls to all shapes is 4:6.
We can simplify ratios just like we simplify fractions. The ratio of balls
to triangles is 6 2 = 3 . We say that 6:2 and 3:1 are equivalent ratios.
1
The simplified ratio 3:1 means that for each three balls, there is one triangle.
When we simplify the ratio of hearts to stars to the lowest terms,
18
we get
12 = 3 . Or, 18:12 = 3:2. This means that for each
2
three hearts, there are two stars.
We could also simplify like this: 18:12 = 9:6. Those two ratios
are equivalent, but neither is simplified to the lowest terms.
1. Write the ratios. Also, simplify them to the lowest terms.
a. The ratio of hearts to stars is =
b. The ratio of hearts to stars is =
The ratio of stars to hearts is =
The ratio of hearts to all shapes is =
2. Write the ratios using the colon. Simplify the ratios if possible.
a. 15 to 20 b. 16 to 4
c. 25 to 10 d. 13 to 30
13 Math Mammoth Ratiios & Proportions & Problem Solving (Blue Series)

3. Write the equivalent ratios. Think about equivalent fractions.
a.
5
2 = 20 b. 3 : 4 = 9 : ____ c. 16 : 18 = ____ : 9 d. 5 1 = 4
e. 2 to 100 = 1 to ________ f. ____ to 40 = 3 to 5 g. 5 : ____ = 1 to 20
4. a. Draw a picture where there are a total of six balls,
and for each two balls, there are five squares.
b. Write the ratio of all balls to all squares,
and simplify this ratio to the lowest terms.
c. Write the ratio of all shapes to balls,
and simplify this to the lowest terms.
5. Look at the picture of the triangles and circles. If we drew more triangles and circles
in the same ratio, how many circles would there be …
a. … for 9 triangles
b. … for 15 triangles
c. … for 300 triangles
We can also form ratios using quantities with various units. For example, in the ratio
5 km
5 km : 8 km, both terms contain the unit “km”. We can then cancel that unit out:
8 km = 5 8 .
6. Write the ratios of the quantities using the fraction line. Simplify the ratios. In most of the problems,
you need to convert the one quantity so it has the same measuring unit as the other.
a. 2 kg and 400 g
b. 200 ml and 2 L
2 kg
400 g = 2,000 g
400 g = 2,000
400
= 5 1
c. 4 cups and 2 quarts d. 800 m and 1.4 km
e. 120 cm and 1.8 m f. 3 ft 4 in and 1 ft 4 in
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If the two terms in the ratio have different units, then the ratio is also called a rate.
For example, “5 miles to 40 minutes” is a rate. It is comparing the quantities “5 miles” and
“40 minutes,” perhaps for the purpose of giving us the speed at which a person is walking.
5 miles
We can also write this rate as 5 miles : 40 minutes or
or 5 miles per 40 minutes.
40 minutes
The word “per” signifies the same as the colon or the fraction line. This rate can also be
simplified:
5 miles
40 minutes
=
1 mile
8 minutes
. So the rate is also 1 mile per 8 minutes.
Example. Simplify the rate “15 pencils per 100 ¢”. Solution:
15 pencils
100 ¢
=
3 pencils
20 ¢
.
7. Fill in the missing numbers to form equivalent rates.
a.
2 cm
30 min
=
15 min
=
45 min
b.
$72
8 hr
=
1 hr
=
10 hr
c.
1/4 mi
10 min
=
1 hr
=
5 hr
d.
$84.40
8 hr
=
2 hr
=
10 hr
8. Express these rates in lowest terms.
a. $44 : 4 hr b. $30 : 8 kg
9. The rate of pencils to
dollars remains the same.
Fill in the table.
Pencils Dollars
1
c. 420 km : 8 hr d. 16 apples for $12
2
3 0.72
6
7
8
10. The rate of miles to gallons remains constant. Fill in the table.
Miles 345
Gallons 1 2 3 4 5 10 15 50
11. A car travels at a constant speed of 80 km/hour.
This means the rate of kilometers to hours
remains the same. Fill in the table.
Km 10 20 80 100 150 200 500
Minutes
15 Math Mammoth Ratiios & Proportions & Problem Solving (Blue Series)

Solving Problems Using Equivalent Ratios
Example. If Jake can ride his bike to a town that is 21 miles away in 45 minutes,
how far can he ride in 1 hour
Let’s form an equivalent rate. However, it’s not easy to
go from 45 minutes to 60 minutes (1 hour). Therefore,
let’s first figure the rate for 15 minutes, which is easy.
Why Because to get from 45 minutes to 15 minutes
you simply divide both terms of the rate by 3.
Then from 15 minutes, we can easily get to 60 minutes: Just multiply both terms by 4.
This is now easy to solve:
21 miles
45 minutes = 7 miles
15 minutes = 28 miles
60 minutes
÷ 3 × 4
21 miles
45 minutes = miles
15 minutes = miles
60 minutes
÷ 3 × 4
. He can ride 28 miles in 1 hour.
1. If Jake can ride 8 miles in 14 minutes, how long will he take to ride 36 miles
Use the equivalent rates below.
8 miles
14 minutes
=
4 miles
minutes =
36 miles
minutes
2. Write the equivalent rates.
a.
15 km
3 hr
=
1 hr
= 15 min
= 45 min
b.
$6
45 min
=
15 min
=
1 hr
=
1 hr 45 min
c.
3 in
8 ft
=
2 ft
=
12 ft
=
20 ft
d.
115 words
2 min
=
1 min
=
3 min
3. A car can go 50 miles on 2 gallons of gasoline.
a. How many gallons of gasoline would the car need for a trip of 60 miles
Use the equivalent rates below.
50 miles
2 gallons = 5 miles
gallons =
60 miles
gallons
b. How far can the car travel on 15 gallons of gasoline
(Hint: First consider the case with just 1 gallon of gasoline.)
16 Math Mammoth Ratiios & Proportions & Problem Solving (Blue Series)

Example. You get 20 erasers for $1.90.
How much would 22 erasers cost
We cannot easily find a number by which to multiply to get
from 20 to 22. It is possible, but we will solve this problem
in a different way.
We will first figure out the cost of just 2 erasers! To
find that, divide the price by 10. Then, from 2 erasers
to 22 erasers, the cost will increase 11-fold:
11 × $0.19 = $2.09. So 22 erasers would cost $2.09.
Another way to come up with the same answer is just to add $0.19
(the cost of 2 erasers) to $1.90 (the cost of 20 erasers). Yet another
way is to find the cost of 1 eraser first and multiply that cost by 22.
×
20 erasers 22 erasers
=
$1.90
×
÷ 10 × 11
20 erasers 2 erasers 22 erasers
=
=
$1.90 $ 0.19 $
÷ 10 × 11
4. On the average, Scott makes a basket nine times out of twelve shots when he is practicing.
How many baskets can he expect to make when he tries 200 shots
9 baskets
12 shots
=
5. You get 30 pencils for $4.50.
How much would 52 pencils cost
30 pencils
$4.50
=
6. A train travels at a constant speed of 80 miles per hour.
a. How far will it go in 140 minutes Use equivalent rates.
80 miles
1 hour
=
b. How long will it take for the train to travel 50 miles Use equivalent rates.
80 miles
1 hour
=
7. In a poll of 1,000 people, 640 said they liked blue.
a. Simplify this ratio to the lowest terms.
b. Assuming the same ratio holds true in another group of 125 people,
how many of those people can we expect to like blue
17 Math Mammoth Ratiios & Proportions & Problem Solving (Blue Series)