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GMAT QUANTITATIVE REASONING
NUMBER PROPERTIES
PROBLEM SOLVING
Diagnostic Test
Question
A 2-digit positive integer ‘ab’ is written as ‘ba’, where a and btake values from 1 to 9, inclusive. The difference between ab andba is 36. Which of the following could be the value of ab?
I. 71 II. 62 III. 84
A. I only
B. II only
C. I and III only
D. II and III only
E. I, II and III
Method 1 – Best Method
Back substitute answer choices
Choice I: 71
The difference between ab and ba is 36.I. 71 II. 62 III. 84
· If ab = 71, ba = 17
Choice I: 71
The difference between ab and ba is 36.I. 71 II. 62 III. 84
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
Choice I: 71
The difference between ab and ba is 36.I. 71 II. 62 III. 84
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
Choice I: 71 Choice II: 62
The difference between ab and ba is 36.I. 71 II. 62 III. 84
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
· If ab = 62, ba = 26
Choice I: 71 Choice II: 62
The difference between ab and ba is 36.I. 71 II. 62 III. 84
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
· If ab = 62, ba = 26
ab – ba = 62 - 26 = 36
Choice I: 71 Choice II: 62
The difference between ab and ba is 36.I. 71 II. 62 III. 84
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
· If ab = 62, ba = 26
ab – ba = 62 - 26 = 36
Choice III: 84Choice I: 71 Choice II: 62
The difference between ab and ba is 36.I. 71 II. 62 III. 84
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
· If ab = 84, ba = 48· If ab = 62, ba = 26
ab – ba = 62 - 26 = 36
Choice III: 84Choice I: 71 Choice II: 62
The difference between ab and ba is 36.I. 71 II. 62 III. 84
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
· If ab = 84, ba = 48
ab – ba = 84 - 48 = 36
· If ab = 62, ba = 26
ab – ba = 62 - 26 = 36
Choice III: 84Choice I: 71 Choice II: 62
The difference between ab and ba is 36.I. 71 II. 62 III. 84
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
· If ab = 84, ba = 48
ab – ba = 84 - 48 = 36
· If ab = 62, ba = 26
ab – ba = 62 - 26 = 36
Choice III: 84Choice I: 71 Choice II: 62
The difference between ab and ba is 36.I. 71 II. 62 III. 84
Options II and III
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
· If ab = 84, ba = 48
ab – ba = 84 - 48 = 36
· If ab = 62, ba = 26
ab – ba = 62 - 26 = 36
Choice III: 84Choice I: 71 Choice II: 62
The difference between ab and ba is 36.I. 71 II. 62 III. 84
Correct Answer choice D.
Options II and III
· If ab = 71, ba = 17
ab – ba = 71- 17 = 54
· If ab = 84, ba = 48
ab – ba = 84 - 48 = 36
· If ab = 62, ba = 26
ab – ba = 62 - 26 = 36
Method 2
The theory behind the question
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
Question states ab – ba = 36·
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
Question states ab – ba = 36·
So, 9(a – b) = 36 or (a – b) = 4
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
Question states ab – ba = 36·
So, 9(a – b) = 36 or (a – b) = 4
i.e., the difference between the tens
and units place of the number is 4.
Check which choices match above
criterion
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
Question states ab – ba = 36·
So, 9(a – b) = 36 or (a – b) = 4
i.e., the difference between the tens
and units place of the number is 4.
Check which choices match above
criterion
Choice I: 71. (7 – 1) ≠ 4
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
Question states ab – ba = 36·
So, 9(a – b) = 36 or (a – b) = 4
i.e., the difference between the tens
and units place of the number is 4.
Check which choices match above
criterion
Choice I: 71. (7 – 1) ≠ 4
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
Question states ab – ba = 36·
So, 9(a – b) = 36 or (a – b) = 4
i.e., the difference between the tens
and units place of the number is 4.
Check which choices match above
criterion
Choice I: 71. (7 – 1) ≠ 4 Choice II: 62. (6 – 2) = 4
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
Question states ab – ba = 36·
So, 9(a – b) = 36 or (a – b) = 4
i.e., the difference between the tens
and units place of the number is 4.
Check which choices match above
criterion
Choice I: 71. (7 – 1) ≠ 4 Choice II: 62. (6 – 2) = 4
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
Question states ab – ba = 36·
So, 9(a – b) = 36 or (a – b) = 4
i.e., the difference between the tens
and units place of the number is 4.
Check which choices match above
criterion
Choice I: 71. (7 – 1) ≠ 4 Choice II: 62. (6 – 2) = 4 Choice III: 84. (8 – 4) = 4
The difference between ab and ba is 36.I. 71 II. 62 III. 84
01 Theoretical approach to solving the question
· Any two digit number, say 43 can be written as 4*10 + 3*1
So, ab can be written as a*10 + b*1 or 10a + b; a is
the tens digit and b the is units digit.
And, ba can be written as b*10 + a*1 or 10b + a; b
is the tens digit and a is the units digit.
ab – ba = 10a + b – (10b + a) = 9(a – b)
Question states ab – ba = 36·
So, 9(a – b) = 36 or (a – b) = 4
i.e., the difference between the tens
and units place of the number is 4.
Check which choices match above
criterion
Choice I: 71. (7 – 1) ≠ 4 Choice II: 62. (6 – 2) = 4 Choice III: 84. (8 – 4) = 4
Points to remember
Difference between ‘ab’ and ‘ba’; Sum of ‘ab’ and ‘ba’
For any two digit number ab, the difference between ab and ba(ab – ba) is always = 9(a – b)
Difference between ‘ab’ and ‘ba’; Sum of ‘ab’ and ‘ba’
For any two digit number ab, the difference between ab and ba(ab – ba) is always = 9(a – b)
Similarly, for any two digit number ab, the sum of ab and ba(ab + ba) is always = 11(a + b)
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