Problem Solving

Hey,
Can anyone help me with the counting integers problem..
For questions such as:
How many integers are between 75 & 176?
how many even/odd number are between 39 & 789?
Find the sum of numbers between 45 and 87?
find the sum of even/odd number between 23 & 54?
and so on....

I am not getting the formula at all, Could someone please provide the explanation of formula and the problem, or a website which explains everything would be a life saver!!
Thank you!!

Hey Sapana,

Let me do one at a time.

Question # 1

How many integers are between A&B?
Answer: B-A-1
e.g. How many integers are between 75 & 176?
176-75-1 = 100
e.g. How many integers are between 5 & 10?
10-5-1 = 4 (6,7,8,9)

Question # 2

how many even/odd number are between A and B?
If A = Odd and B = Odd,
The answer: # of Odd numbers = (B-A-2)/2 # of Even numbers = (B-A)/2
If A = Odd and B = Even,
The answer: # of Odd numbers = (B-A-1)/2 # of Even numbers = (B-A-1)/2
If A = Even and B = Even,
The answer: # of Odd numbers = (B-A)/2 # of Even numbers = (B-A-2)/2
If A = Even and B = Odd,
The answer: # of Odd numbers = (B-A-1)/2 # of Even numbers = (B-A-1)/2

e.g. How many even/odd number are between 39 & 789?
# of Odd numbers = (B-A-2)/2 # of Even numbers = (B-A)/2
# of Odd numbers = (789-39-2)/2 = 374 # of Even numbers = (789-39)/2 = 375

p.s: A<B
p.p.s. To the best of my (limited) Knowledge !
p.p.p.s all question ask 'BETWEEN' values, so I have excluded the initial and the final numbers from the count

How many integers are between 75 & 176?

The number is always highest number - lowest number + 1 (All inclusive)
=> 176 - 75 + 1
102 numbers.
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how many even/odd number are between 39 & 789?

here the formula is : (highest number - lowest number)/2
If both beginning numbers are odd, the number of odd numbers between them will be one more than the even numbers.

The converse is true. If both ending and beginning numbers are even, the even numbers between them will be one more than the odd numbers.

=> so number of odd numbers is (789 - 39)/2
=> 375 + 1 = 376 Odd numbers (Since 789 and 39 are odd)
and even numbers will be 375.
--------------------------------------------------------
Find the sum of numbers between 45 and 87?

this is an ARITHEMATIC progression with common difference of 1.
and the formula is (n/2)[2a + (n - 1)d] where n is the count of numbers in the sequence and a is the first number.
so first count the numbers.
Applying the technique in the first answer. It would be 43.
so (43/2)[2*45 + 42(1)]
=>2838.
----------------------------------------------------------------------------
find the sum of even/odd number between 23 & 54?

first count the numbers.
Applying the technique in the second answer
for even numbers consider from 24 to 54. Then the number of even numbers is 16
for odd numbers consider from 23 to 53. Then the number of odd numbers is 16

Now apply the technique in 3rd answer.
for even numbers the a = 24 and d = 2 and n = 16
so sum is (16/2)[2*24 + 15*2]
sum of even numbers is 624

and similarly for odd numbers a = 23 and d = 2 and n = 16
so sum is (16/2)[2*23 + 15*2]
sum of even numbers is 608

Thank You so much.. Counting Problems are crystal clear for me now Thank You again for immediate responses.. Y'all are awesome!

No of integers between a and b:-

If inclusive set meaning a and included it is b-a+1

If exclusive meaning excluding a and b then b-a-1

If it is simply put between a and b then b-a

Odd or even or any arithmetic progression:-

If d is the common difference, meaning the difference between consecutive terms, then

[(last term of the AP)-(First term of AP)]/d+1 (inclusive) rest formulas are similar to the one discussed above

remember that the sum of an AP formula

Sn = (First Term + Last Term)*(No of terms inclusive)/2