Lesson: Strange Operators

Comment on Strange Operators

If xθy=(x+1)/(y−1) for all y

If xθy=(x+1)/(y−1) for all y≠1, and aθb=1, then a=

A. b−2
B. b−1
C. b
D. b+1
E. b+2

I did (a+1)/(b-1) = 1
I don't understand what should I do to take out a

Let's take it from (a+1)/(b-1

Let's take it from (a+1)/(b-1) = 1
Multiply both sides of the equation by (b-1) to get: (a+1) = 1(b-1)
Expand right side: a + 1 = b - 1
Subtract 1 from both sides: a = b - 2

Hi Brent, for the below

Hi Brent, for the below question I am getting the different answer zero, here's why:
(√5 + √5)² - (√5 - √5)² = using the property a² - b² = (a+b)(a-b) I got, (√5 + √5) (√5 - √5) = (2√5)(0) = 0

If x ¤ y = (x + y)² - (x - y)². Then √5 ¤ √5 =

A. 0
B. 5
C. 10
D. 15
E. 20

You're using the property that says a² - b² = (a + b)(a - b)

So, in the expression, (√5 + √5)² - (√5 - √5)², what is a, and what is b?
Well, a = (√5 + √5) and b = (√5 - √5)

This means (a + b)(a - b) = [(√5 + √5) + (√5 - √5)][(√5 + √5) - (√5 - √5)]
= [(√5 + √5) + 0][(√5 + √5) - 0]
= (√5 + √5)(√5 + √5)
= (2√5)(2√5)
= 4√25
= 20

I have two different solutions posted here: https://gmatclub.com/forum/if-x-y-x-y-x-y-then-218476.html#p1727422

Does that help?

Cheers,
Brent

Hi Brent,

Hi Brent,

I was referring to your solution : https://gmatclub.com/forum/for-all-positive-integers-m-3m-when-m-is-odd-125937-20.html#p2048884

Am I right to infer that 81 = [27] or 81 = [162] and since [162] is not an option that is provided we may mark [27] as correct?

Thanks & Regards,
Abhirup

Great point!
[162] would also be correct.

Cheers,
Brent

Thanks Brent!!

Hi Brent,

Hi Brent,

Could you please explain this question a bit more (https://gmatclub.com/forum/for-any-real-number-x-the-operator-is-defined-as-x-x-1-x-213328.html). I'm not sure how the connection between the & equation with x and & equation with p were made. Thank you.

Hi Swatato,

Hi Swatato,

Here's the solution you're referring to: https://gmatclub.com/forum/for-any-real-number-x-the-operator-is-defined...

I spent the first part of the solution, showing that &(p + 1) = (p + 1)(-p) [upon simplification]
ASIDE: This step is crucial, because once we know that &(p + 1) and (p + 1)(-p) are EQUAL, we can later replace &(p + 1) with (p + 1)(-p).

Then I dealt with the given information that says: p + 1 = &(p + 1)

So, I took the part on the right side of the equation, &(p + 1), and replaced it with its equivalent value of (p + 1)(-p)

This results in the equation p + 1 = (p + 1)(-p), which we can now solve for p.

Does that help?

Cheers,
Brent

Hi Brent,

Hi Brent,

For question https://gmatclub.com/forum/an-operation-is-defined-by-the-equation-x-y-x-246192.html
Can we not factor the equation: x²/4 - xy - y² as (x/2 - y)²? as it will lead to the same solution when you solve the parenthesis.
Why did you first common out (1/4) and then solved the equation?

You're correct to say that x²/4 - xy - y² = (x/2 - y)².
However, I think it's safe to say that most students will not automatically see that factorization.
So, as an intermediate step, I factored out the 1/4 so that the resulting quadratic is easier to recognize as a special product.

Cheers,
Brent

Hi Brent,

Hi Brent,

question https://gmatclub.com/forum/if-denotes-the-least-integer-greater-than-or-equal-to-x-132223.html

"REPHRASED target question: Is -1 ≤ x < 0". Shouldn't it be -1 < x < 0. My understanding is that if x = -1 than [x]=-1. And if we allow x to be greater or equal to -1 and less than 0, then possible answers for [x] could be -1 or 0, hence we have to restrict it to only greater than -1 and less than 0. Am I missing something here?

Thanks.

Good catch!

Good catch!

You're partially right about how I should have rephrased the target question.

Notice that:
[2.1] = 3, since 3 is the smallest integer that's greater than or equal to 2.1.
[0] = 0, since 0 is the smallest integer that's greater than OR EQUAL TO 0
[1] = 1, since 1 is the smallest integer that's greater than or equal to 1.

The target question asks whether [x] = 0
This will occur if -1 < x ≤ 0

So, the REPHRASED target question should be "Is -1 < x ≤ 0"

I've edited my answer here: https://gmatclub.com/forum/if-denotes-the-least-integer-greater-than-or-...

Thanks again for the heads up!

Cheers,
Brent

Brent, hi.

Brent, hi.
Could you please explain one more time.
For any real number x, the operator & is defined as:

&(x) = x(1 − x)
If p + 1 = &(p + 1), then p = ?
How did we come to: &(p+1) = (p+1)[1 − (p+1)] = (p + 1)(-p) ?

GIVEN: &(x) = x(1 − x)

GIVEN: &(x) = x(1 − x)

So, the "&" symbol next to a number tells us to take that number and multiply it by that number subtracted from 1.

In other words, &number = number(1 - number)
So, &7 = 7(1 - 7)
And &88.3 = 88.3(1 - 88.3)
And &w = w(1 - w)
And &2k = 2k(1 - 2k)
And &j² = j²(1 - j²)
And &(p+1) = (p+1)[1 − (p+1)]

In all cases, we're taking the number (be it 7 or 88.3 or w or 2k or j²) and multiplying it by that number subtracted from 1.

Does that help?

Cheers.
Brent

Hi Brent,
I am struggling with this question. I am unable to understand what is [y] and how you could deduce the 2 facts from it.
Need help!

The first step is to figure

The first step is to figure out what the strange notation means.

GIVEN: [y] denotes the greatest integer less than or equal to y
A few examples:
[5.1] = 5
[3] = 3
[8.9] = 8
[-1.4] = -2
[-13.6] = -14

Once we have a better understanding of what the notation means, we can answer the question.
Here's my full solution: https://gmatclub.com/forum/denotes-the-greatest-integer-less-than-or-equ...

Please let me know if you have any questions about my solution.

Cheers,
Brent

Hi Brent,

Hi Brent,

I was not sure if this question goes here I hope it does...
100 students were selected to participate in a survey by the school in May 2013. The next month, another 100 students were randomly selected to participate in the same survey, out of which 2 had already participate in the survey the previous month. If the number of students did not change from May to June then what is the approximate number of students in the school?

A) 3,600
B) 2,400
C) 2,000
D) 4,700
E) 5,000

Thanks a lot for your help!

This is a not-so-great

This is a virtually identical version of the following question from the Official Guide: https://gmatclub.com/forum/in-a-certain-pond-50-fish-were-caught-tagged-...

The two questions are almost identical, except your question uses 100, whereas the official question uses 50.
Take a look at the solution in the linked question above, and let me know if you still need help.

Cheers, Brent