x, y are positive integers. When x is divided by y, the remainder is 6

x/y=6.12
(x-6)/y=6
subtracting,
6/y=.12
y=50
50*6.12=x=306
D

x, y are positive integers. When x is divided by y, the remainder is 6 and x/y=6.12, what is the value of x?

\(\frac{x}{y}\) = k + \(\frac{6}{y}\) = 6 + 0.12

so k = 6 & \(\frac{6}{y}\) = 0.12
so y = 50m ( m is a factor )

let's say m = 1 ; y = 50 ; x = 50 + 6 = 56 ; \(\frac{x}{y}\) = \(\frac{56}{50}\) <> 6.12
let's say m = 2 ; y = 50 ; x = 100 + 6 = 106 ; \(\frac{x}{y}\) = \(\frac{106}{50}\) <> 6.12
let's say m = 3 ; y = 50 ; x = 150 + 6 = 156 ; \(\frac{x}{y}\) = \(\frac{156}{50}\) <> 6.12
let's say m = 4 ; y = 50 ; x = 200 + 6 = 206 ; \(\frac{x}{y}\) = \(\frac{206}{50}\) <> 6.12
let's say m = 5 ; y = 50 ; x = 250 + 6 = 256 ; \(\frac{x}{y}\) = \(\frac{256}{50}\) <> 6.12

let's say m = 6 ; y = 50 ; x = 300 + 6 = 306 ; \(\frac{x}{y}\) = \(\frac{306}{50}\) = 6.12

since the remainder is 6 and the decimal part of the quotient is .12

0.12 * y = 6 {decimal part of quotient * divisor = remainder}

therefore y = 6/0.12 = 50
and x = 6y +6 = 6(50)+6 = 306

Option D

1. \(\frac{x}{y}\) = 6.12 = 6 + \(\frac{12}{100}\), or + \(\frac{3}{25}\)

2. \(\frac{x}{y}\) = a + \(\frac{6}{y}\), where latter = \(\frac{r}{y}\) from remainder formula

3. Set \(\frac{r}{y}\) equal to the decimal part of the decimal quotient from #1, and solve for y:

\(\frac{6}{y}\) = \(\frac{3}{25}\) --> y = 50

4. Rebuild the dividend x:

\(\frac{x}{y}\) = a + \(\frac{6}{y}\) --> x = (a)(y) + 6

From the quotient 6.12, we know a = 6 (the integer part of the quotient in decimal form = the integer quotient)

5. x = (6)(50) + 6 = 306, Answer D

An exceptionally clear explanation for problems like this one is from Mike McGarry, here:
https://magoosh.com/gmat/2012/gmat-quant-thoughts-on-remainders

We must remember that the remainder formula is:

x/y = Q + r/y

Substituting values into the equation, we have:

x/y = 6 + 12/100

x/y = 6 + 3/25

Thus, we see that 3/25 represents r/y, and since we are told that r = 6 we have:

3/25 = 6/y

3y = 6(25)

y = 2(25) = 50

Since y is 50, we see that x = 6.12y = 6.12(50) = 306.

Answer: D

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