Solve the Puzzle: Find the Check Amount

• musicgold
In summary: This means that if we can find a pair of integers x and y that satisfy the equation, then we can solve for the whole number value of x using the equation.In summary, the man went into the bank and cashed a check for $100. He then spent 5 cents on his way home. He found that he had$200 left after spending the nickel.
musicgold

Homework Statement

I am trying to solve the following puzzle. The problem is that there are two unknowns and I could come up with only one equation.

A man went into a bank to cash a check. In handing over the money the cashier, by mistake, gave him dollars for cents and cents for dollars. He pocketed the money without examining it, and spent a nickel ( 5 cents) on his way home. He then found that he possessed exactly twice the amount of the check. He had no money in his pocket before going to the bank. What was the exact amount of that check?

Homework Equations

Let x - number of dollars and y - number of cents
check amount - ## 100x + y ##
paid amount - ##100y + x ##
amount left after spending a nickel - ## 100y + x - 5 ##

So the first equation we get is:
## 100y + x - 5 = 2 (100x + y) ##

After solving it, we get
## y = \frac { 199x + 5 } { 98} ##

The Attempt at a Solution

As there is not enough info for another equation, I tried the trial and error approach. To get a whole number the numerator has to be a multiple of 98. To have an even number in the numerator, x has to be an odd number.

I tried using a few values of x but didn't get an answer. I am not sure if I am headed in the right direction. Is there a better way of doing this?

Thanks.

Delta2
Have you tried to plot it and remember that both x and y must be integers?

musicgold said:

Homework Statement

I am trying to solve the following puzzle. The problem is that there are two unknowns and I could come up with only one equation.

A man went into a bank to cash a check. In handing over the money the cashier, by mistake, gave him dollars for cents and cents for dollars. He pocketed the money without examining it, and spent a nickel ( 5 cents) on his way home. He then found that he possessed exactly twice the amount of the check. He had no money in his pocket before going to the bank. What was the exact amount of that check?

Homework Equations

Let x - number of dollars and y - number of cents
check amount - ## 100x + y ##
paid amount - ##100y + x ##
amount left after spending a nickel - ## 100y + x - 5 ##

So the first equation we get is:
## 100y + x - 5 = 2 (100x + y) ##

After solving it, we get
## y = \frac { 199x + 5 } { 98} ##

The Attempt at a Solution

As there is not enough info for another equation, I tried the trial and error approach. To get a whole number the numerator has to be a multiple of 98. To have an even number in the numerator, x has to be an odd number.

I tried using a few values of x but didn't get an answer. I am not sure if I am headed in the right direction. Is there a better way of doing this?

Thanks.
You also have ## x = \frac{98 y - 5}{199}##.

Last edited:
jedishrfu
jedishrfu said:
Have you tried to plot it and remember that both x and y must be integers?
I tried plotting the equation using Winplot but it is hard to locate a pair that satisfies the equation.

musicgold said:
I tried plotting the equation using Winplot but it is hard to locate a pair that satisfies the equation.

##199## is a prime number. Does that help?

This is a linear diophantine equation because the x and y must be integers and they are related in a linear way.

https://en.wikipedia.org/wiki/Diophantine_equation

I did find a solution using a MATLAB clone (freemat.org) and you'll need to try solutions of x from 5 to 99 to find the corresponding y which can be done via google in the search bar. The MATLAB approach was faster because of its builtin vector calculating.

I'm not sure of any other way to solve it.

jedishrfu said:
This is a linear diophantine equation because the x and y must be integers and they are related in a linear way.

https://en.wikipedia.org/wiki/Diophantine_equation

I did find a solution using a MATLAB clone (freemat.org) and you'll need to try solutions of x from 5 to 99.

I'm not sure of any other way to solve it.

The fact that 199 is prime helps! But, I'm not sure how much for a pre-calc question.

This may be overkill for this problem but then again what is life for if not for learning new things:

https://en.wikipedia.org/wiki/Kuṭṭaka

Reduction of the problem
Aryabhata and other Indian writers had noted the following property of the linear Diophantine equations: "The linear Diophantine equation ax + by = c has a solution if and only if gcd(a, b) is a divisor of c." So the first stage in the pulverization process is to cancel out the common factor gcd(a, b) from a, b and c, and obtain an equation with smaller coefficients in which the coefficients of x and y are relatively prime.

For example, Bhaskara I observes: "The dividend and the divisor shall become prime to each other, on being divided by the residue of their mutual division. The operation of the pulveriser should be considered in relation to them."[1]

musicgold said:
I tried plotting the equation using Winplot but it is hard to locate a pair that satisfies the equation.

There is an elementary solution. Hint: consider the two cases where ##y < 50## and ##y \ge 50##.

Your equations are correct, but if you start again, there's a quicker way with this approach.

PS to keep things simple, assume he did have a nickel in change. I.e. ##x \ge 5##.

You can also quickly brute force it with Excel.

musicgold said:
## y = \frac { 199x + 5 } { 98} ##
.
If you rewrite that as ## y = 2x + \frac { 3x + 5 } { 98} ##, isn't a solution obvious?

musicgold, SammyS and TSny
haruspex said:
If you rewrite that as ## y = 2x + \frac { 3x + 5 } { 98} ##, isn't a solution obvious?
Oh...got it. Thanks.

\$31.63.

Jedishfru,

The solution (31, 63) doesn't seem to follow that GCD rule. Am I missing something?

musicgold said:
Jedishfru,

The solution (31, 63) doesn't seem to follow that GCD rule. Am I missing something?
Applying the GCD rule to the equation you ended up with in post #1, it says the gcd of 199 and 98 must be a factor of 5. Since 199 and 98 are coprime, the gcd is 1, so it is trivially true and not useful here.
Likewise, applying it to the answer, 31 and 63 are coprime, so, again, it just says 1 divides 5.

1. How do I solve the puzzle?

To solve the puzzle and find the check amount, you will need to examine all the clues provided and use deductive reasoning to eliminate possibilities until you are left with the correct answer.

2. What clues should I look for?

The puzzle will typically provide information such as the total amount paid, the number of people splitting the check, and the cost of each person's meal. Look for patterns and try to make logical deductions based on the information given.

3. Do I need any special knowledge to solve the puzzle?

No, the puzzle is designed to be solved using basic math and deductive reasoning skills. However, some knowledge of percentages and fractions may be helpful.

4. Is there a specific order in which I should solve the puzzle?

No, there is no specific order in which you should solve the puzzle. However, it may be helpful to start with the clues that provide the most information and work your way to the ones with less information.

5. What should I do if I get stuck?

If you get stuck, try stepping back and looking at the puzzle from a different perspective. You can also try solving the puzzle with a friend or colleague to bounce ideas off of each other. And don't be afraid to take a break and come back to the puzzle later with a fresh mind.

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