If A Equals the Sum of the Even Integers from 2 to 20, Inclusive GMAT Problem Solving

Question: If a equals the sum of the even integers from 2 to 20, inclusive, and b equals the sum of the odd integers from 1 to 19, inclusive, what is the value of a - b ?

1. 1
2. 10
3. 19
4. 20
5. 21

“If a equals the sum of the even integers from 2 to 20, inclusive”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2021”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Approach Solution 1:

Given:

• A equals the sum of the even integers from 2 to 20, inclusive
• B equals the sum of the odd integers from 1 to 19, inclusive

Find Out:

• The value of A - B

As per the formula, the sum of consecutive odd or even integers

= (no. of odd or even ints) * (first int + last int) / 2

Here A = sum of even ints from 2 to 20, inclusive
number of even ints = 10,
first int + last int = 2+20 = 22
A = 10*22 / 2
A = 110

B = sum of odd ints from 1 to 19, inclusive
number of odd ints = 10,
first int + last int = 1+19 = 20
B = 10*20 / 2
B = 100

Hence, we get A-B
A-B = 110 - 100
A-B = 10

Correct Answer: B

Approach Solution 2:

Given:

• A equals the sum of the even integers from 2 to 20, inclusive
• B equals the sum of the odd integers from 1 to 19, inclusive

Find Out:

• The value of A - B

As per the problem statement, A = 2+4+6 ... +20
As per the problem statement, B = 1+3+5 ...+19

We are looking to find the value of A-B
If we consider all the values,
We essentially get:
A-B = 2-1 + 4-3 + 6-5 + ... + 20-19

Hence, we get 1 every time.
The solution becomes
=> 1 + 1 + 1 .. + 1 (ten times)
=> 10

Correct Answer: B

Approach Solution 3:

# of integers: (highest multiple - lowest multiple)/increment + 1

average of evenly spaced sets: (highest + lowest)/2

Now:

even numbers:

(20-2)/2 + 1 = 10 even integers.
(20+2)/2 = 11 is the average of the even set.
sum = avg*(#of elements) = 11*10 = 110 = a

odd numbers:
(19-1)/2 + 1 = 10 odd integers.
(19+1)/2 = 10 is the average of the odd set.
sum = avg*(#of elements) = 10*10 = 100 = b

a-b = 110 - 100 = 10. 

Correct Answer: B

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