Math Revolution and GMAT Club Contest! If a, b, and c are the tenths

If a, b, and c are the tenths digit, the hundredths digit, and the thousandths digit of 0.abc, is 0.abc > 2/3?

Statement 1 - a + b > 14
Statement 2 - a + c > 15

ANSWER - OPTION D, both statements alone are sufficient

From question - Max value that a, b or c can have is 9. We have to check if 0.abc > 0.666...

a+b >14, min possible value of 'a' is 6 (by substituting b=9) making min value of term 0.69x
a+c >15, min possible value of 'a' is 7 (by substituting c=9) making min value of term 0.7X9

Thus both options independently can help us answer question

If a, b, and c are the tenths digit, the hundredths digit, and the thousandths digit of 0.abc, is 0.abc > 2/3?
the question really asks if a>6 and b>6 since 2/3 = ~0.66

(1) a + b > 14
to check whether a<=6, we need to maximize the value of b, which can only be 9, since 9 is the highest digit. in this case, a minimal value of a would be 6, because we need to satisfy the condition a+b>14.
in this case, we have 0.69c. Regardless of what C is, 0.abc is definitely greater than 2/3
note that we cannot put a<6, since a+b would not yield a number greater than 14, and since a and b can take values from 0 to 9, there is no such possibilities to make a<6.

A is sufficient, and we can clearly cross B, C, and E.

We are left with A and D, and the chances of getting the correct answer is 50% !

(2) a + c > 15
now, again trying to maximize one value and minimize the other one, so that to see whether our number is greater than 2/3.
suppose that a=7 and c=9 -> this is the smallest possible combination with the minimum value of a, but which satisfies the condition a+c>15.
we have 0.7b9. this number is definitely greater than 2/3.

we can cross A, and select confidently D as the correct answer!

If a, b, and c are the tenths digit, the hundredths digit, and the thousandths digit of 0.abc, is 0.abc > 2/3?

is 0.abc > 2/3 (0.666) ?
i.e. is abc > 666?

if a < 6 then NO
if a > 6 then YES
But if a = 6 then check the value of b which if equal to 6 then check the value of c.

(1) a + b > 14
if a + b = 15 then 6 + 9 = 15 OR 7 + 8 = 15 OR 8 + 7 = 15 OR 9 + 6 = 15, let us say abc = 69c
if the value of c is 0 to 9 and respectively abc = 690 to 699
in all cases abc > 666
Definite answer. Hence Sufficient.

(2) a + c > 15
if a + c = 16 then 7 + 9 = 16 OR 8 + 8 = 16 OR 9 + 7 = 16
in all cases a > 6, hence abc > 666.

Definite answer. Hence Sufficient.

Both (1) & (2) are individually sufficient. Hence "D" is the final answer!

My answer to this Question is D

If a, b, and c are the tenths digit, the hundredths digit, and the thousandths digit of 0.abc, is 0.abc > 2/3?

(1) \(a + b > 14\)
(2) \(a + c > 15\)

\(\frac{2}{3}= 0.666..\)
Is \(0.abc > 0.666\)?

1) \(a + b > 14\)
We know that these are digits => their sum cannot be a decimal. So take \(a + b = 15\)
a and b have to be single digits from 0 to 9. To form \(15\), possible values of a and b are -
\((9, 6)\) and\((8, 7)\)

Considering (9, 6), \(0.abc = 0.96c\) or\(0.69c\) -- Both are greater than \(0.666..\)
Considering (8, 7), \(0.abc = 0.87c\) or \(0.78c\) -- Both are greater than \(0.666..\)
Note - a + b cannot exceed 18, or else values of a and b will exceed 9 each. Any value of a + b> 15 will result in higher values of a and b, still resulting in \(0.abc > 0.666\)
Hence, Sufficient!

2) \(a + c > 15\)
Like (1), Lets take \(a + c = 16\)
To form \(16\), possible values of a and b are -
\((9, 7)\) and \((8, 8)\)
Considering (9, 7), \(0.abc = 0.9b7\) or \(0.7b9\). -- Both are greater than \(0.666..\)
Considering (8, 8), \(0.abc = 0.8b7\). -- Greater than \(0.666..\)
Note - a + c cannot exceed 18, or else values of a and c will exceed 9 each. Any value of a + c> 16 will result in higher values of a and b, still resulting in \(0.abc > 0.666\)
HHence Sufficient!

QUESTION #2:

If a, b, and c are the tenths digit, the hundredths digit, and the thousandths digit of 0.abc, is 0.abc > 2/3?

(1) a + b > 14
(2) a + c > 15

Solution:

0.abc = abc/1000
2/3 = 667/1000 (approximately)

(1) a+b > 14 --> min (a+b) = 15 --> min a = 6 --> min abc/1000 = 690/1000 > 667/1000 - Sufficient
(2) a+c > 15 --> min (a+c) = 16 --> min a = 7 --> min abc/1000 = 709/1000 > 667/1000 - Sufficient

Answer D

The answer is D

To simply the statement, is 0.abc>0.666666.. or in other words is any one of a or b or c >6. Even if one of the digits is great than 6, then 0.abc can turn out to be greater than 2/3.

1) a+b>14, in this case, the possible values of a/b are (9/6, 6/9, 8/7, 7/8, 9/8, 8/9, 9/9). In all cases either a or b is greater than 6. The lowest number is 0.69 which is still greater than 0.6666. so 0.abc > 2/3. SUFFICIENT

2) a+C>15, in this case, the possible values of a/c are (7/9, 9/7, 8/8, 9/8, 8/9, 9/9). In all cases a or c is greater than 6. This is clearly SUFFICIENT.

Hence answer is D

PS: (note that we cannot round of decimals, If you do round off then, 2/3 can be 0.7 & the answer will be B as option 1 will turn our insufficient. But this should not be the case as the question has clearly mentioned 2/3 and not the decimals itself which means 2/3=0.666666666 and not 0.7)

We have to check if 0.abc>0.667
Option A: If a+b>14 then taking a+b=15 we get max value of b can be 9 which makes a=6 thus Yes 0.69>0.67
A sufficient
Option B: If a+c> 15 then taking a+c=16 we get max value of c can be 9 which makes a=7 thus Yes 0.709>0.667
B sufficient
Thus each stmnt sufficient on a ta own
Ans: D

is 0.abc > 2/3 or 0.66

1. a+b>14, that means, 18>a+b>14, so the min digit value combinations could be 9+6, 8+7, 7+8, 6+9, 8+7 and 8+7. In any case 0.abc will be greater than 0.66 ... Sufficient
2. a+c>14, In all the cominations the minimum digit value of 'a' will be 7, So 0.7bc will be greater than 0.66. Sufficient

Ans: D

2/3=0.66666.....
Statement 1:
a+b>14
so, a>14-b or, a>5 [Highest value of b=9 is taken so that the least value of a can be derived]
Therefore, when least a=6 then b=9. [since value must be integer]
From here, we know that the number is at least 0.69 irrespective of C [Sufficient]
Statement 2:
a>15-c
Similarly, a>6 and hence sufficient.
Ans is D

2/3 = 0,667
We want to know if abc is greater than 667

1) a+b > 14: for A and B to be greater than 14, the minimum value for A would have to be 6, and B will have a maximum value of 9. So 0.69_ > 0.667. Statement 1 is sufficient.
2) a+c > 15: for A and C to be greater than 15, the minimum value for A has to be 7 and maximum c 9. So 0.7_9 is also greater than 0.667. Statement 2 is sufficient.
Answer D
_________________

We have
If a, b, and c are the tenths digit, the hundredths digit, and the thousandths digit of 0.abc, is 0.abc > 2/3?

(1) a + b > 14
(2) a + c > 15

for 0.abc > 2/3 to be true, the minimum value of 0.abc should be 0.667

1) a+b > 14
let's take a+b = 15, then for a=6 and b = 9 and assume c=0, we can conclude that 0.690 > 2/3. For all other possible values of a and b, 0.abc will be greater than 0.690. Sufficient.
2) a+c > 15
Let's take a+c = 16, the minimum value of 0.abc,assuming b = 0, will be 0.709. again we can conclude that 0.709>2/3. Sufficient.

Ans. D

we need to find if 0.abc> 0.666

1. a+b> 14 => a +b is atleast 15.
the least that a can be is 6 and in this case, b will be 9.
hence regardless of the value of c, 0.abc will be greater than .666

2. a+c> 15 => a + c is atleast 16.
the least that a can be is 7 and in this case, c will be 9.
hence regardless of the value of b, 0.abc will be greater than .666

Hence Ans.: D

Question says : 0.abc > 2/3 => 0.abc > 0.66

Case 1 : a+b>14 i.e minimum a+b=15,
consider smallest value of a=6, hence b=9 => 0.ab=0.69 > 0.66 True Sufficient

Case 2 : a+c>15 i.e minimum a+b=16,
consider smallest value of a=7, hence b=9 => 0.ab=0.79 > 0.66 True Sufficient

Sol. D. Both are sufficient separately.

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You know the importance of Kudos, be generous.

is 0.abc > 0.667?

St1: a + b > 14
Minimum value of a + b = 15;
Minimum value of a is possible when b is maximum --> b = 9 and a = 6 --> 0.abc = 0.69c > 0.667
Statement 1 alone is sufficient.

(2) a + c > 15
Minimum value of a + c = 16;
Minimum value of a is possible when c is maximum --> c = 9 and a = 7 --> 0.abc = 0.7b9 > 0.667
Statement 2 alone is sufficient.

Answer: D

My take is D.

(1) a+b>14 ; Say a+b = 15 ; then only possible combination is 0.96c or 0.69c. In any case its greater than 2/3= 0667 or 0.67. You can check any condition like a+b =18 etc. Hence sufficient.
(2) a+c> 15 ; Say a+c = 16; then only possible combination is 0.8b8 or 0.7b9 or 0.9b7. Hence sufficient.

Hope it helps!!!

D. Each statement alone is sufficient.

Is 0.abc>2/3?
2/3 is nothing but 0.6666

(i) Given a+b>14
The possible set of digits for a and b is (6,7,8,9); anything below 6 cannot add up to a sum greater than 14.
Now the minimum value for a is 6.
Hence the minimum value for 0.abc=0.69c, which will always be greater than 0.666
Therefore sufficient.

(ii) a+c>15
The possible set of digits for a and c is (7,8,9); anything below 7 cannot add up to a sum greater than 15.
The minimum value for a is 7.
And for 0.abc=0.7bc, which will always be greater than 0.666
Sufficient.

If a, b, and c are the tenths digit, the hundredths digit, and the thousandths digit of 0.abc, is 0.abc > 2/3?

(1) a + b > 14
(2) a + c > 15

st1) a+b is atleast 15
(7,8), (8,7), (9,6),(6,9)
so minimum is .69 which is >2/3

st2) a+c is minimum 16
(8,8), (9,7), (7,9)
So minimum is .79 > 2/3

Ans D

Ans: D (Either)
2/3 = 0.667. So the question simply wants us to find out if a, b, c > 6

1. a+b> 14.
a few ways of getting a 14 using single digit numbers only - 7+7, 6+8, 8+6, 5+9 ....
The smallest number a can take is 5 and largest that b can take is 9 to get 14. With this combination, 0.abc<2/3. with others it is >2/3 and hence sufficient.

2. c+a>15. Same logic as 1 and hence sufficient.

We need to prove if 0.abc>0.667 or not.

Statement 1- Since a+b>14, a should be minimum 6 (5+9=14). Sufficient.
Statement 2- Since a+c>15, a should be minimum 7 (6+9=15). Sufficient.

Answer is D.