In a certain two-digit integer, the ratio of the units digit

The possible integers are 32, 64 and 96.

1) 3-2 = 1, 6-4 = 2, 9-6 = 3. Only 96 satisfies the statement. Sufficient.

2) 3*2 = 6, 6*4 = 24, 9*6 = 54. Only 96 satisfies the statement. Sufficient.

Answer is hence D.

Infer - Ratio u/t = 2/3

S-1) t = u+3.
T = 3,4,5,6,7,8,9
U = 0,1,2,3,4,5,6
I missed to use the information given 2/3. If we see the options only 6/9 ratio gives u/t as 2/3. So sufficient.

S-2) Product = 54
T = 0,1,2,3,4,5,6,7,8,9
U = 0,1,2,3,4,5,6,7,8,9
For product = 54 and ration 2/3 only option is 6 & 9. Sufficient

the two digits be -9 and -6 in that case "B" option would yield 54 so the digits can be 9 & 6 as well as -9 & -6

Given:

units to tens = 2:3

Meaning it could be:

32
63
96

(It cannot be negative as the ratio given is positive.) Although we could say it's "-6 and -9" it does not make sense as these numbers together make up a two-digit integer.

Leaving us with the numbers above.

The first statement gives us that the digit in the tens place is 3 times larger than the one in the units place. As we've already worked out the three possibilities, we can rule out 32 and 63.

Statement 1 is sufficient.

Statement two gives us the product between the two numbers that make up the integer. We can simply find that the products are, respectively, 6, 18 and 54.

54 matches 96(9*6). Statement 2 is therefore sufficient.

Hence, D.

Why cant the two digits both be -9 and -6 ? I am still confused on this ...
Option A will fail , but B will hold ?
Can any one clarify this ....

U/T = 2/3, what is UT?

From 1, T = U + 3 => we know that 3U = 2T that means U is multiple of 2 and T is multiple of 3.
T => -9,-6,-3,3,6,9, negatives are not possible, when T=3 not possible, T=6 then U = 5 not possible, T=9 then U=6 and ratio is intact, UT = 96 in both cases. Sufficient.

From 2, UT=54 and U/T=2/3 => \sqrt{T} = \sqrt{9x9} => T = +-9 =>
if T = -9, U = -6 => Not possible ==> -9-6 ???
if T = +9, U = 6, =>UT = 96 yes


ANSWER: D

Question is asking what is UT and not what is U*T. Test maker confuse you with giving you information that stick to your mind, like in st2. Watch this trap!

U and T cannot be negative at all. So the UT can only be 96 in this case.

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