Can someone help me with this problem?

[tonker]

1001^2 - 999^2 / 101^2 - 99^2 ?

a) 10
b) 20
c) 40
d) 80
e) 100

Answer is a. Can someone please explain it.

Thanks,
Vito

[beatthegmat]

What is the source of this question?

[tonker]

1001^2 - 999^2 / 101^2 - 99^2 ?

a) 10
b) 20
c) 40
d) 80
e) 100

Answer is a. Can someone please explain it.

Thanks,
Vito

This is from the GMAT Prep.

[rajs.kumar]

check this approach a ^ 2 - b ^ 2 = (a + b) (a - b)

[axefx]

Based on (a + b)^2 = a^2 + b^2 + 2*a*b
and (a - b)^2 = a^2 + b^2 - 2*a*b:

(1001^2 - 999^2) / (101^2 - 99^2)
= (1000 + 1)^2 - (1000 - 1)^2 / (100 + 1)^2 - (100 - 1)^2
= (1000^2 + 1^2 + 2*1*1000) - (1000^2 + 1^2 - 2*1*1000) / (100^2 + 1^2 + 2*1*100) - (100^2 + 1^2 - 2*1*100)
= 4000/400
= 10

[Stacey Koprince]

Alternatively, using Raj's suggestion (this is the shortcut):
1001^2 - 999^2 / 101^2 - 99^2

(1001 + 999)(1001 - 999) / (101 + 99)(101-99)
(2000)(2)
(200)(2)
cross off 2's, cross off two zero's each from top and bottom
20/2
10

[BTGmoderatorRO]

In solving this question, let us first recall the algebraic equation
a^2 - b^2 = (a-b)(a+b)
(1001^2 - 999^2) / (101^2 - 99^2)
solving the numerator first,
1001^2 - 999^2 = (1001-999)(1001+999)
=(2)(2000)
=4000
let solve the denominator
101^2 - 99^2 = (101-99)(101+99)
=2*200
=400
Solving mathematically using the division method,
numerator/denominator= 4000/400
=10

we got 10 as the answer which is option A

[Scott@TargetTestPrep]

1001^2 - 999^2 / 101^2 - 99^2 ?

a) 10
b) 20
c) 40
d) 80
e) 100

Answer is a. Can someone please explain it.

Thanks,
Vito

We can see that both the numerator and denominator are differences of two squares. Therefore, the numerator can be factored and simplified as:

(1001 - 999)(1001 + 999) = 2(2000) = 4000

Similarly, the denominator can be factored and simplified as:

(101 - 99)(101 + 99) = 2(200) = 400

Therefore, the quotient is 4000/400 = 10.

Answer: A