If s and t are positive integers such that s/t = 64.12 which

If s and t are positive integers such that s/t = 64.12, which of the following could be the remainder when s is divided by t ?

(A) 2
(B) 4
(C) 8
(D) 20
(E) 45

When we say \(\frac{s}{t} = 64.12 = 64 \frac{12}{100} = 64 \frac{3}{25}\)
We get \(\frac{s}{t} = 64 \frac{3}{25}\)
What does the mixed fraction of the right hand side signify? It signifies that s > t and when s is divided by t, we get 64 as quotient and 3 as remainder if t is 25.

e.g. \(\frac{13}{5} = 2 \frac{3}{5}\). Here, remainder is 3
\(\frac{130}{50} = 2 \frac{30}{50}\). Here remainder is 30.
\(\frac{26}{10} = 2 \frac{6}{10}\). Here remainder is 6.
So in our question, the remainder will be 3 or any multiple of 3. 45 is the only multiple of 3.

We can rewrite the given equality as \(s = 64t + 0.12t\). The divider is t, the quotient is 64.
The remainder is \(0.12t\) (it is less than t) and it is an integer, being equal to \(s-64t\).
Since \(0.12t=\frac{3}{25} t\), it follows that t should be a multiple of 25, so \(t=25n\), for some positive integer n.
Therefore, the remainder is \(3n\), or a multiple of 3. The only answer that is a multiple of 3 is 45.

Answer: E

\(\frac{S}{t} = 64 + .12\)
\(S = 64t + .12t\)

The remainder is equal to .12t.

R = .12t
R/.12 = t

We have to look for R where R/.12 is an integer.

A)2/.12 = 200/12 is not an integer
B)4/.12 = 400/12 = 100/3 is not an integer
C) 8/.12 = 800/12 = 400/6 = 200/3 is not an integer
D) also not
E) 45/12 = 4500/12 = 1500/4 = 15*25 is an integer

Answer: E
C)

We know that \(\frac{s}{t} \)= 64.12, and we need to find the remainder when s is divided by t

s/t = 64.12,
=> s = t*64.12
=> s = 64t + t*.12
So, when s is divided by t then we will get t*.12 as reminder (as t*.12 will be less than t)
Now t is an integer and .12 is 12*.01 which means it is 3*something
So, only answer choices which are multiple of 3 are contenders.

Only possibility is E!
Hope it helps!

Watch the following video to learn the Basics of Remainders

Answer is E

S/T = 64.12 or you can write it as 6412/100 or 1603/25

So, If we divide 1603 by 25 we will get remainder of 3.

From the five options, only 45 is divisible by 3 So, The answer should be 45.

out of the answer explanation, but can we also say that t must be a multiple of 25 using the same fraction?

Yes, t must be a multiple of 25: s/t = 1603/25 = 3206/50 = 6412/100 = ... = 64.12 --> the remainders 3, 6, 12, ..., respectively.

I did not understand why r must be a multiple of 3? From answer choice E 45 is 15 times 3 (a multiple of 3) does that mean t will be 15 times 25?

Because \(\frac{3}{25}\)cannot be simplified further (say factorised further)

\(\frac{3}{25}= \frac{3*15}{25*15}= \frac{45}{25*15}\) ... That is possible

Answered this question wrong but when i finished reading karishma's blog https://www.veritasprep.com/blog/2011/05/quarter-wit-quarter-wisdom-knocking-off-the-remaining-remainders/ . I am confident about these type of questions.


s/t=64.12

when we divide "s" by "t" then 64.12, Here .12 is a Remainder which we are representing in quotient. so to find the possible remainder...

0.12--12/100 -- 3/25 , so 25 or multiple of 25 has to be a "t" & 3 or multiple of 3 has to be remainder.

Answer E.

If the question had been which of the following cannot be the remainder. Then can we use the propertry that Remainder must be divisibe by 25 as well???

After expressing s/t = 64.12 as:

s = 64t + 0.12t

In case a student faces doubts about how to draw inferences about remainder r from the above equation, here is an alternate line of thought:

It is clear that the remainder r will come from the term 0.12t

So, we can write: r = 0.12t

=> t = \(\frac{r}{0.12}\) = \(\frac{100r}{12}\) = \(\frac{25r}{3}\)

So, t = \(\frac{25r}{3}\)

But, the question statement gives us a constraint on t: that t is a positive integer.

This means, \(\frac{25r}{3}\) is a positive integer.

This is only possible when r is a multiple of 3.

As I said, an alternate route to the same deduction.

Hope this was useful for you!

Best Regards

Japinder

Dear Ankush

Good to see you here on GC!

Here's the answer to your question:

It is not necessary for the remainder to be divisible by 25.

Let's look at this in terms of constraints:

Constraint 1: The remainder is always a non-negative integer.

From the equation

\(\frac{r}{t}=\frac{3}{25}\), we get that

r=\(\frac{3t}{25}\)

From constraint 1, we see that

\(\frac{3t}{25}\) must be a non-negative integer.

This means either t = 0 or t is a multiple of 25.

But t cannot be equal to 0 because then the expression \(\frac{s}{t}\) becomes undefined

This means, t is a multiple of 25.

Constraint 2: The question states that t is a positive integer.

As explained in the post I made just above, this means \(\frac{25r}{3}\) is a positive integer, which leads you to the inference that r is a multiple of 3.

So, the bottom-line is that the only 2 inferences that we can conclusively draw from the given information is that:

i) t is a multiple of 25 and
ii) r is a multiple of 3

Hope this helped!

Thanks and Best Regards

Japinder

Decimal part x Divisor = Remainder

i.e. o.12*t = remainder
i.e. Remainder = (12/100)*t = (3/25)*t

Since the Result has to be multiple of 3 so Option E is the only choice that fits the requirement

Answer: Option E

This problem will be best solved using the remainder formula. Let’s first state the remainder formula:

When positive integer x is divided by positive integer y, if integer Q is the quotient and r is the remainder, then x/y = Q + r/y.

In this problem we are given the following:

s/t = 64.12

We can simplify this to read as the remainder formula:

s/t = 64 + 0.12

s/t = 64 + 0.12

s/t = 64 + 12/100

s/t = 64 + 3/25

Because Q is always an integer, we see that Q must be 64, and thus the remainder r/y must be 3/25. We can now equate r/y to 3/25 and determine a possible value for r.

r/y = 3/25

Note that some equivalent values for r/y could be 6/50 or 9/75 or 12/100, and so forth. Note that in all cases, the value of r is a multiple of 3.

Of the answer choices, the only multiple of 3 is 45, so that is a possible value of r.

Answer: E

any Similar questions?

Check this one: https://gmatclub.com/forum/if-s-and-t-a ... fl=similar