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A. 100 ml
B. 125 ml
C. 200 ml
D. 250 ml

20. P, Q, R, and [tex]S[/tex] are four 4-digit numbers, each having the digit 9 only once and in the place shown. None of the other digits are known.

What can be said with certainty about [tex]P, Q, R[/tex], and [tex]S[/tex]?
[tex]\[
\begin{array}{l}
P(9 \ldots \ldots) \quad Q( \ldots 9 \ldots \ldots) \\
R(\ldots \ldots 9 \ldots) \quad S(\ldots \ldots \ldots 9) \\
\end{array}
\][/tex]

A. [tex]S[/tex] is the smallest of the 4 numbers.
B. [tex]Q[/tex] is larger than [tex]R[/tex].
C. [tex]P[/tex] is the largest of the 4 numbers.
D. [tex]S[/tex] is the only odd number among the 4 numbers.



Answer :

GinnyAnswer
To determine what can be said with certainty about the numbers [tex]\( P, Q, R, \)[/tex] and [tex]\( S \)[/tex], let's analyze each of their configurations and certain properties step-by-step:

1. Configuration of each number:
- [tex]\( P \)[/tex] is a 4-digit number where 9 is in the thousand's place: [tex]\( P = 9xxx \)[/tex]
- [tex]\( Q \)[/tex] is a 4-digit number where 9 is in the hundred's place: [tex]\( Q = x9xx \)[/tex]
- [tex]\( R \)[/tex] is a 4-digit number where 9 is in the ten's place: [tex]\( R = xx9x \)[/tex]
- [tex]\( S \)[/tex] is a 4-digit number where 9 is in the unit's place: [tex]\( S = xxx9 \)[/tex]

2. Determining the smallest possible values:
- The smallest value for [tex]\( P \)[/tex] is [tex]\( 9000 \)[/tex].
- The smallest value for [tex]\( Q \)[/tex] is [tex]\( 2900 \)[/tex] (since [tex]\( 1900 \)[/tex] and below would put the 9 in a less significant place).
- The smallest value for [tex]\( R \)[/tex] is [tex]\( 2090 \)[/tex] (with 9 in the tens place).
- The smallest value for [tex]\( S \)[/tex] is [tex]\( 2009 \)[/tex] (as it ends with 9).

3. Determining the largest possible values:
- The largest value for [tex]\( P \)[/tex] is [tex]\( 9999 \)[/tex] (all remaining digits as 9).
- The largest value for [tex]\( Q \)[/tex] is [tex]\( 2999 \)[/tex].
- The largest value for [tex]\( R \)[/tex] is [tex]\( 2099 \)[/tex].
- The largest value for [tex]\( S \)[/tex] is [tex]\( 2999 \)[/tex].

4. Analysis for each statement:
- Statement A: [tex]\( S \)[/tex] is the smallest of the 4 numbers.
- The smallest possible value of [tex]\( S \)[/tex] is [tex]\( 2009 \)[/tex], which is smaller than the smallest possible values of [tex]\( P \)[/tex] ([tex]\( 9000 \)[/tex]), [tex]\( Q \)[/tex] ([tex]\( 2900 \)[/tex]), and [tex]\( R \)[/tex] ([tex]\( 2090 \)[/tex]). Thus, [tex]\( S \)[/tex] is indeed the smallest number.

- Statement C: [tex]\( P \)[/tex] is the largest of the 4 numbers.
- The largest value for [tex]\( P \)[/tex] is [tex]\( 9999 \)[/tex], which is greater than the largest values of [tex]\( Q \)[/tex] ([tex]\( 2999 \)[/tex]), [tex]\( R \)[/tex] ([tex]\( 2099 \)[/tex]), and [tex]\( S \)[/tex] ([tex]\( 2999 \)[/tex]). Hence, [tex]\( P \)[/tex] is the largest number.

- Statement B: [tex]\( Q \)[/tex] is larger than [tex]\( R \)[/tex].
- The smallest value of [tex]\( Q \)[/tex] is [tex]\( 2900 \)[/tex] and the smallest value of [tex]\( R \)[/tex] is [tex]\( 2090 \)[/tex]. Since [tex]\( 2900 \)[/tex] is greater than [tex]\( 2090 \)[/tex], [tex]\( Q \)[/tex] is indeed larger than [tex]\( R \)[/tex].

- Statement D: [tex]\( S \)[/tex] is the only odd number among the 4 numbers.
- [tex]\( S \)[/tex] ends in 9 and is therefore always an odd number. The other numbers [tex]\( P, Q, \)[/tex] and [tex]\( R \)[/tex] do not end in 9 (or any other odd digit with certainty in their smallest configurations), meaning [tex]\( S \)[/tex] is the only number guaranteed to be odd.

Based on this detailed analysis:

A. [tex]\( S \)[/tex] is the smallest of the 4 numbers. True
B. [tex]\( Q \)[/tex] is larger than [tex]\( R \)[/tex]. True
C. [tex]\( P \)[/tex] is the largest of the 4 numbers. True
D. [tex]\( S \)[/tex] is the only odd number among the 4 numbers. True

Therefore, all of the given statements about the numbers are correct.

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