## $$[y]$$ denotes the greatest integer less than or equal to $$y.$$ Is $$d < 1?$$

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### $$[y]$$ denotes the greatest integer less than or equal to $$y.$$ Is $$d < 1?$$

by Vincen » Fri Aug 06, 2021 7:09 am

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## Global Stats

$$[y]$$ denotes the greatest integer less than or equal to $$y.$$ Is $$d < 1?$$

(1) $$d = y - [y]$$
(2) $$[d]= 0$$

Source: Official Guide

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### Re: $$[y]$$ denotes the greatest integer less than or equal to $$y.$$ Is $$d < 1?$$

by [email protected] » Fri Aug 06, 2021 4:44 pm

00:00

A

B

C

D

E

## Global Stats

Vincen wrote:
Fri Aug 06, 2021 7:09 am
$$[y]$$ denotes the greatest integer less than or equal to $$y.$$ Is $$d < 1?$$

(1) $$d = y - [y]$$
(2) $$[d]= 0$$

Source: Official Guide

First, let's take a moment to get a good idea of what this strange notation means.
A few examples:
[5.1] = 5
 = 3
[8.9] = 8
[-1.4] = -2
[-13.6] = -14

IMPORTANT FACT #1: [y] < y
IMPORTANT FACT #2: The difference between y and [y] is always less than 1. In other words, y - [y] < 1

Target question: Is d < 1?

Statement 1: d = y - [y]
Take IMPORTANT FACT #1 from above: [y] < y
Subtract [y] from both sides to get: 0 < y - [y]
Now take IMPORTANT FACT #2 and add it to our inequality to get: 0 < y - [y] < 1
Statement 1 tells us that d = y - [y], so let's replace y - [y] with d to get: 0 < d < 1
PERFECT, we can now be certain that d < 1
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: [d] = 0
We're going to use IMPORTANT FACT #2 in our solution.
Take [d] = 0 and add d to both sides to get: [d] + d = d
Subtract [d] from both sides to get: d = d - [d]
From IMPORTANT FACT #2, we know that d - [d] < 1
Since we just showed that d = d - [d], we can be certain that d < 1
Since we can answer the target question with certainty, statement 2 is SUFFICIENT