quantitative aptitude, maths applicable to Common Aptitude Test (CAT) Bank Competitive Exam UPSC Competitive Exams SSC Competitive Exams Defence Competitive Exams L.I.C/ G. I.C Competitive Exams Railway Competitive Exam University Grants Commission (UGC) Career Aptitude Test (IT Companies) and etc.

1. RATIOS
&
PROPORTIONS

2. RATIOS
For example:
There are eight girls and seven boys in a class.
Ratios are comparisons made between two sets
of numbers.
The ratio of girls to boys is 8 to 7.

3. 80 miles to 1 hour = 80mph
Ratios are used everyday. They are used for:
Miles per hour
The cost of items per pound, gallon, etc.
Hourly rate of pay

4. THERE ARE 3 WAYS TO WRITE RATIOS.
For example: There are 8 girls and 5 boys in my class.
What is the ratio of girls to boys?
1. Write the ratio using the word “to” between the two
numbers being compared.
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:
There are 13 boys and 17 girls in sixth grade.
Find the ratio of boys to the girls in sixth grade.
Directions: Write the ratio in three different ways.
13 to 17
13
17
13 : 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?
4 girls to 7 boys 4 girls : 7 boys
4 girls
7 boys
Hint: The question asks for girls to boys; therefore, girls
will be listed first in the ratio.

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 to 15 13:15
13
15

10. 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.
2. Amanda’s basketball team won 7 games and lost 5.
What is the ratio of games lost to games won?
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.
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?
Steps:
1. Read the word problem.
2. Set up the ratio.
40 : 4540 to 45 40
45

14. 3. Reduce the ratio if necessary.
Reduce means to break down a fraction or ratio into the
lowest form possible.
40 to 45
HINT: When having to reduce ratios, it is better to set up the ratio in the
vertical form. (Fraction Form)
=
40
45
Reduce = smaller number; operation will always be division.
40
45
Look at the numbers in the ratio. What ONE
number can you divide BOTH numbers by?
÷
÷
5
5
=
=
8
9
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
Factors of 45: 1, 3, 5, 9, 15, 45

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
52
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!)
÷
÷
26
26
=
=
1
2

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
÷÷ 3
3
=
=
5
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.
4
5
20
25
=
4 x 2520 x 5 =
If the cross products are equal, then it is a true proportion.
100 = 100
The cross products were equal, therefore 4 And 20 makes a true proportion.
5 25

20. Directions: Solve to see if each problem is a true proportion.
Guided Practice:
3 15
5 25=1.
2. 6
8
= 57
76
3. 7
12
= 37
60

21. Directions: Solve to see if each problem is a true proportion.
Guided Practice:
3 15
5 25=1. 2.
6
8
=
57
76
3. 7
12
= 37
60
15 x 5 = 3 x 25
75 = 75
true
57 x 8 = 6 x 76
456 = 456
true
7 x 60 = 37 x 12
420 = 444
false

22. SOLVING PROPORTIONS WITH VARIABLES
What is a variable?
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?
Look for the two sets of
ratios to make up a
proportion.
You have 52 miles in 4
hours. This is the first
ratio.
52 miles
4 hours
Next, the problem states “how
far can he travel in 7 hours.
The problem is missing the
miles. Therefore, the miles
becomes the variable.
n miles
7 hours
Set 1 Set 2
The proportion should be
set equal to each other.
52
4
=
n
7
HINT: The order of the ratio does matter!
A variable is any letter that takes place of a
missing number or information.

23. SOLVING THE PROPORTION:
3. Check the answer to see if it makes a true proportion.
Problem:
52 x 74 x n =
Which number is
connected to the variable? 4n = 364
Since the 4 is connected
to the variable, DIVIDE
both sides by the 4.
4
4 ÷ 4 = 1;
therefore you
are left with “n”
on one side.
When solving proportions, follow these rules:
1. Cross multiply.
2. Divide BOTH sides by the number connected to the variable.
52
4
=
n
7
n =
364 ÷ 4 = 91
91 miles
4

24. Check your answer!
52
4
=
91
7
52 x 7 = 91 x 4
364 = 364
If it comes out even, then the answer is correct.

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
3 miles
= $609.00
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
250 mi
x gal
600mi
=
250x 24000=
250x
250
24000
250
=
x = 96
40
250
x
600
=
40
250
96
600
=
2400024000
Check:
=

29. 2. If a 3 gallon jug of milk cost $9, how many 3
gallon jugs can be purchased for $45?
5 jugs of milk can be
purchased for $45
1
9
n
45
=
1
9
n
45
=
1x45 9n=
45
9
9n
9
=
5 = n
1
9
5
45
=
Check:
45 = 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
8
x_
12
=400 miles
8 hours
x miles
12 hours
=
400
8
x_
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
6
x_
15
=10 tulips
6 daffodils
x tulips
15 daffodils
=
10
6
x
15
=
6x 150=
6x
6
150
6
=
x 25=
x 25 tulips=
10
6
25
15
=
150 150=