what is the smallest no. divisible by both 5/6 and 2 3/4?? my teacher said it'll be 2 7/24?

Question

criticize this please
i said the answer should be 55
please criticize me and my teacher
please tell me the correct ans.

Answer 1

We have to be clear about the definitions of "number" and "divisible" here. Usually when we speak about "divisible" we are talking about integers. Here, we clearly have fractions in the problem, so we might allow fractions in the answer.

Likewise, divisible usually means that the quotient of the numbers is an integer. If you would allow fractions here, all numbers are divisible by each other.

So you have two possible interpretations of the problem:
1) what is the smallest positive rational number x such that x / (5/6) and x / (2 3/4) are both integers; OR
2) what is the smallest positive integer x such that etc.

You answered version two correctly; he answered version one, but incorrectly.

The correct answers to both versions:
x / (5/6) must be an integer --> 6x must be a multiple of 5
x / (11/4) must be an integer --> 4x must be a multiple of 11

If 6x is a multiple of 5, then x must be a multiple of 5; similarly, x must be a multiple of 11. The answer is lcm(5,11) = 55.

However, if x can be a fraction, then its denominator must divide both factors 6 and 4, and therefore must divide the gcd of 6 and 4, which is 2. In other words, the denominator can be 2 at most. The correct answer would by 55/2 = 27 1/2.

Check: (27 1/2) / (5/6) = (55/2) * (6/5) = 330/10 = 33
... (27 1/2) / (2 3/4) = (55/2) * (4/11) = 220/22 = 10

His answer is WRONG:

(2 7/24) / (5/6) = (55/24) * (6/5) = 330/120 = 2 3/4 not an integer
(2 7/24) / (2 3/4) = (55/24) * (4/11) = 220/264 = 5/6 not an integer

Answer 2

Your teacher is right. 2 3/4=11/4,so the smallest number divisible by 5/6 and 11/4 would be 55/24, which is 2 7/24. You can check the answer. Multiply 5/6 X 11/4. The answer is 55/24.

Answer 3

Here you go: 5/6 and 2 3/4: 5/6 and 11/4.

Put both of these numbers in their least common denominator. In this case: 12. So 5/6 becomes 10/12 and 11/4 becomes 33/12.
10/12 & 33/12. The answer is 330/12. Or 27 1/2.
2 7/24 = 55/24. 11/4 does not go into 55/24 evenly, even though 5/6 does.

Answer 4

You are closer, you are on the right track, but 27 and a half is divisible by both. Convert both to 24ths. Then you see that it has to be a multiple of 20 and of 66, which give you 660/24. or 27.5

Take that, teacher!

Answer 5

Since 2and 7/24 is smaller than 2 3/4, it obviously can not be divided evenly by both numbers, so your teacher is incorrect.

Answer 6

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Answer 7

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