Jorge's bank statement showed a balance that was $0.54 greater than wh

Key concept: a.bc \(= a + b(\frac{1}{10}) + c(\frac{1}{100})\)
For example: \(3.25 = 3 + 2(\frac{1}{10}) + 5(\frac{1}{100})=3+0.2+0.05=3.25\)

Jorge wrote a check for $x.yz BUT recorded it as $x.zy. His bank statement showed a balance that was $0.54 greater than what his records showed
This tells us that: $x.zy - $x.yz = 0.54
Now apply the above concept to write: \([x + z(\frac{1}{10}) + y(\frac{1}{100})]-[x + y(\frac{1}{10}) + z(\frac{1}{100})]= 0.54\)

Subtract \(x\) from both sides to get: \([z(\frac{1}{10}) + y(\frac{1}{100})]- [y(\frac{1}{10}) + z(\frac{1}{100})]= 0.54\)

Multiply both sides by \(100\) to get: \([10z + y]- [10y + z]= 54\)

Simplify to get: \(9z - 9y= 54\)

Divide both sides by \(9\) to get: \(z - y= 6\)

Rearrange to get: \(z = y + 6\)

Since y is a DIGIT, the possible solutions are
y = 0 and z = 6
y = 1 and z = 7
y = 2 and z = 8
y = 3 and z = 9
And that's it!

Since z can be 6, 7, 8 or 9, the correct answer is E

Cheers,
Brent

x.zy— x.yz = 0.54
—> zy —yz = 54
10z+ y—10y —z = 54
9z —9y = 54
z —y= 6
Each of x,y and z represents a digit from 0 through 9
—> 9 —3= 6
—> 8 —2 = 6
—> 7 —1 = 6
—> 6 —1 = 6
z could be 9, 8, 7 or 6

Answer ( E)

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Ok let's make it easy for beginners: straight to point

He has 0.54 extra---> reason: he recorded 0.54 less in his records i.e., x.zy instead of x.yz

>>>> (x*1+z*0.1+y*0.01) -(x*1+y*0.1+z*0.01) =0.54
multiply with hundred on both sides>>>>(10z+y)-(10y+z)=9(z-y)=54
>>>> z-y=6
>>>>z=6+y
There for z can be greater than or equal to 6(6 to 9 since they are digits as given in question).

Therefore answer is E.6

Hope this helps

THANKS

Given:
1. Jorge's bank statement showed a balance that was $0.54 greater than what his records showed.
2. He discovered that he had written a check for $x.yz and had recorded it as $x.zy, where each of x, y, and z represents a digit from 0 though 9.
Asked: Which of the following could be the value of z ?

x.zy - x.yz = .1(z-y) + .01(y-z) = .09(z-y) = $0.54
z-y = 6
(z,y) = {(9,3),(8,2),(7,1),(6,0)}
z could be 6,7,8 or 9

IMO E

Can someone please explain why its arranged so that x.zy - x.yz instead of x.yz - x.zy? I understand the solution but dont understand how it makes sense that his balance is .54 greater with records being greater than check instead of check being greater than records?

abalodi
rohansabat brings a logical point. Let's take it and try to solve.
Assume balance before the check was written = B
Balance as per Bank after the check was written, \(B_B\) = B - x.yz
Jorge's Balance with the error, \(B_J\) = B - x.zy

Now,
\(B_B - B_J\) = 0.54
B - x.yz - B + x.zy = 0.54 (basically B - x.zy < B - x.yz)
0.zy - 0.yz = 0.54
9(z - y) =54
z - y = 6
z = 9, 8, 7, 6

Only 6 is in the option.
Answer E.

rajpurohitchirag HTH.

(You may take example also:
B - 106 - B + 160 = 54 OR
B - 128 - B + 182 = 54 .....)
_________________

His Actual Bank Balance >>> What he recorded


This means that the Check he gave to someone must be written down in his records as GREATER THAN the Actual Check that he gave away ---- only in this scenario, logically, would he have more money in his Bank Account than he originally thought that he had


His Recorded Check = X.ZY > ACTUAL Check the Bank cashed = X.YZ


The Difference between: (Recorded Check) - (Actual Check) = $.54 Greater amount

X.ZY - X.YZ = .54

.ZY - .YZ = .54

----multiply Each Side by * (100) ------

ZY - YZ = 54

-----where Z and Y are Digits from 0, 1, 2 ......9-------

[ 10z + 1y ] - [ 10y + 1z ] = 54

10z - 1z + 1y - 10y = 54

9z - 9y = 54

9*(z - y) = 54

(z - y) = 6


The Difference between the Z-Digit and the Y-Digit must be Exactly = 6

therefore, the MIN Value the Z-Digit can take is = 6


only Answer Choice -E- is possible out of the Answer Choices provided


z = 6
y=0

X.ZY - X.YZ =

$10.60 - $10.06 = $.54 cents

-E-

Solution:

Since the bank statement has a balance that is greater than his records, the check he wrote must be less than what he had recorded. That is, x.yz < x.zy, and their difference is 0.54. Therefore, in cents, we can create the equation:

(100x + 10z + y) - (100x + 10y + z) = 54

9z - 9y = 54

z - y = 6

z = 6 + y

Since y is nonnegative, z must be at least 6. Therefore, E is the correct answer.

Answer: E
_________________

I have a doubt. Why are we subtracting x.yz from x.zy? I think that we should take the case of depositing because written a check could mean deposit too. If we take the case of withdrawal then only I think it should be x.zy - x.yz. this is ambiguous. Can someone please help? BrentGMATPrepNow @veritasprepkarishma TargetTestPrep Bunuel

When you "write a check" you are agreeing to transfer money from your bank account to somebody else's bank account (not the other way around).

10(z-y ) + y-z =54
=> 9*(z-y) = 54
=> z=y+6

It satisfies for y=0 , z=6
Therefore IMO E

There appears to be multiple right answers for this question.

For eg,

compare these two values: $1.82 VS $1.28, then answer is two (Option A)

Compare these two values: $1.60 VS $1.06, then answer is six (Option E)

In both cases, difference is $0.54, can someone correct me where I have gone wrong?

The larger number has Z in the tenths place in both your examples, which is correct.

Z in your first example is then 8, not 2, just as Z in your second example is 6.

Posted from my mobile device

Bunuel
Similar questions? That is: questions where you have to interpret \(a.bc \) as \(a+\frac{a}{10}+\frac{b}{100}\).

KarishmaB

The Bank showed $0.54 greater than the records kept by the consumer. So, if only it is $0.54 difference, then Z can also take the value of 3.
For example. 8.93(Bank statement)
Consumer's record: 8.39( Check)
The difference is $0.54

According to this, if bank shows $8.39 and the consumer recorded $8.93, then the bank will not clear the check. Further, It is said in the question that the Bank statement shows $0.54 more than his record. Then how come Z cannot take the value of 3?
_________________

x.yz and x.zy are not the amounts on bank statement vs consumer records. They are amounts of the check vs amount of the check on consumer records. So whatever his bank balance was, $x.yz, the smaller check amount, was deducted from it but the consumer recorded it as $x.zy (a larger amount) and hence expected a lower leftover bank balance. The larger amount was $0.54 more than the smaller amount.

Since x is the same for both, zy - yz = 54
10z + y - 10y - z = 54
9(z - y) = 54
z-y = 6

So z could be 9, 8, 7 or 6.

Answer (E)

This makes so much sense now because I solved it the other way around and got y-z= 6. This is the make it or break it for this question.

All you need to know is that, when you "write a check", it is taking money out of the bank, not storing a check in the bank

Hi,

A simple structure for these type of problems-

Let a and b are the digits of your number n, so that n = 10a + b.
The number after interchanging digits is 10b + a.
Let the difference because of interchanged digits be k
k = 9b - 9a or k/9= b - a

Change 10a or 10b to 100 or 1000 a or b depending if the misplaced digit is in the 100s place or 1000s place
Eg: 10a + b will become 100a + 10b | 10b + a will become 100b-10a and k = 9b - 9a will become k = 90b- 90a and so on

Solve for a and b. And there's the answer!