A cashier has 26 bills consisting of three times as many ones as fives, one less ten than fives, and the rest twenties. If the value is $120, find how many of each she has.
:{ how do I know I can trust your answer?
u+f+t = 26
u = 3f
t = f-1
u+5f+10t+20x = 120
Before we try to solve that mess, we know u is a multiple of 5 (because the total is a multiple of 5, and all the other bills are worth multiples of 5). It is also a multiple of 3, so let's say there are 15 ones and 5 fives.
That gives t=6, so there are 6 tens.
Adding up all the values, we see that 15+25+60+20x = 120
That gives us 1 twenty.
You have a typo. It should have said there were one less fives than tens.
He's got that person icon next to his name, it means he's highly qualified.
you can also trust the answer, because it fits all the conditions.
That's how you can trust it. It works when checked.
When you do a problem, don't you check your answer to see whether it's right?
To solve this problem, we can set up a system of equations based on the given information.
Let's assume the cashier has:
- x number of ones,
- y number of fives,
- z number of tens, and
- w number of twenties.
We can then translate the information provided into equations:
Equation 1: x + y + z + w = 26 (The total number of bills is 26)
Equation 2: x = 3y (Three times as many ones as fives)
Equation 3: z = y - 1 (One less ten than fives)
Equation 4: x + 5y + 10z + 20w = 120 (The total value of the bills is $120)
To solve this system of equations, we will do the following:
Step 1: Substitute the expressions from Equation 2 and Equation 3 into Equation 1:
3y + y - 1 + y + w = 26
5y + w - 1 = 26
Step 2: Rearrange Equation 4 by substituting the expressions from Equation 2 and Equation 3:
3y + 5y - 1 + 10(y - 1) + 20w = 120
18y + 20w - 11 = 120
Step 3: Now, we have a system of two equations with two variables:
5y + w - 1 = 26 ...(equation A)
18y + 20w - 11 = 120 ...(equation B)
Step 4: Solve this system of equations to find the values of y and w. Subtract equation A from equation B to eliminate w:
18y + 20w - 5y - w = 120 - 26
13y + 19w = 94 ...(equation C)
Step 5: Now, let's solve equations A and C simultaneously. We can do this by substituting w from equation A into equation C:
13y + 19(26 - 5y - 1) = 94
13y + 19(25 - 5y) = 94
13y + 475 - 95y = 94
-82y = -381
y = 4.646
Since y represents the number of fives, it cannot be a decimal. Therefore, there must be an error in our calculations or assumptions. Please recheck the problem statement or confirm if I have made any mistakes.