"I cashed a check at the bank and spent $1.41 before I realized the teller had transposed the dollars with the cents. I now had exactly 5 times the value of the original check. What was the amount of the original check?"
Unsure how to set up an equation for a word problem.
Let x = the original money in the check.
x = 100a + b, cents.
After transposing dollar with cents, the amount of money became
100b + a, cents
100b + a - 141 = 5(100a + b) <==Attn: after spending 141 cents, the remaining part is five times the original.
Since a and b are integers, solving for b gives,
b = (499a + 141)/95
The integer solution is a = 6, b = 33. The next smallest integer solution is a = 101, b = 532, which doesn't make sense since b < 100.
To solve for the value of the check you must take into account that the teller transposed the dollars and cents. (e.g. gave you 41.56 when the check was supposed to be 56.41). This means that you must represent the dollars and cents as different variables within the same number (e.g. $d.c) How to do this? Well, $1 = 100 cents, so set everything equal to cents.
Therefore the correct check is (x*100) + y cents, where x is the dollar amount and y the cents.
The cashier gave you (y*100) + x cents, because the numbers were transposed.
With the information in the problem, you know that you spent 1.41 or 141 cents of that incorrect amount, and you are left with 5 times that correct amount. Plug everything into an equation:
100y + x - 141 = 5 (100x + y) incorrect amount - spent = 5 times correct amount
100y + x -141 + 141 = 500x + 5y + 141
100y + x = 500x + 5y + 141
95y = 499x + 141
y = (499x + 141)/95
There would be an infinite number of possibilities, except that you know some information about x and y. Because each at one point represents the cents part of a monetary figure, they can only be between 00 and 99. Also, y, the dollar amount of the payment received,is at least 5 times bigger than x. Also, y is a whole number so, looking at the equation, 499x + 141 has to be divisible by 95 evenly.
Just looking at units digits, only x values ending in 1 or 6 are eligible; otherwise 499x+141/95 doesn't yield an integer. And only 1, 6, 11, 16 qualify, because y is five times bigger than x but less than 99. So plug in each:
499(1) + 141/95 = 2.344355
499(6) + 141/95 = 33. You can plug in to the original equations to get x, and put x and y together to create the value of the original check. $6.33)
Let 'x' represent the amount of the original check, and let 'y' represent the amount the bank teller gave you.
We are given that you spent $1.41 from the amount the teller gave you (y), after which you realized that you now have 5 times the original amount of the check (x).
Working backwards, after you are given the money and spend $1.41, you have 5 time the amount of the original check. That is, you now have: 5x
Adding back the $1.41 you spent to what you now have yields the amount you were given by the teller. That is, the teller gave you: 5x + 1.41
With this, we arrive at the following equation for the amount the teller gave you:
y = 5x + 1.41
Solving for x in the above equation, we generate an equation for the amount of the original check:
y = 5x + 1.41
y - 1.41 = 5x + 1.41 - 1.41
y - 1.41 = 5x
(y - 1.41)/5 = (5x)/5
(y - 1.41)/5 = x