Bill buys x items at $3 each, y items at $5 each, and z items at 6 each. If x,y,and z are each multiple of 5 and the total price of the items costing $3 and $5 is 55, which of the following could be the total price of all x +y+z items?
3x + 5y = 55
the condition that must be satisfied is that x, y and z must all be multiples of 5.
first, we guess value of x (which must be a multiple of 5). And then we solve for corresponding value of y. Note that the value of y calculated must also be a multiply of 5. If it's not, guess another x value:
x: 5 , 10 , 15
y: 8 , 5 , 2
we see here that the only x and y value that satisfies the condition is
x = 10 and y = 5
now i think you forgot to type the choices.
To find the possible total price of all x + y + z items, let's start by understanding the given information.
1. Bill buys x items at $3 each.
2. Bill buys y items at $5 each.
3. Bill buys z items at $6 each.
4. x, y, and z are each multiples of 5.
5. The total price of items costing $3 and $5 is $55.
Let's break down the problem further:
The total price of items costing $3 can be calculated as 3x.
The total price of items costing $5 can be calculated as 5y.
According to the given information, the sum of these two prices is $55:
3x + 5y = 55 ---(Equation 1)
Now, let's consider the possible values of x and y.
Since x, y, and z are each multiples of 5, let's assign variables to them:
x = 5a
y = 5b
z = 5c
Substituting these values into Equation 1:
3(5a) + 5(5b) = 55
15a + 25b = 55
3a + 5b = 11 ---(Equation 2)
We can see that Equation 2 is a linear Diophantine equation. Now, to find the possible values of a and b, we can use the Extended Euclidean Algorithm or try different values until we find a solution that satisfies the equation.
By trying different values, we find that one possible solution is a = 1 and b = 2. Substituting these values back into the equations:
x = 5a = 5(1) = 5
y = 5b = 5(2) = 10
Now, let's find the possible total price of all x + y + z items:
Total price = 3x + 5y + 6z = 3(5) + 5(10) + 6z = 15 + 50 + 6z = 65 + 6z
The total price can be any value greater than or equal to 65, plus additional increments of 6 depending on the value of z.
Therefore, the possible total price of all x + y + z items could be 65, 71, 77, 83, 89, and so on.
(Note: This is just one possible solution. There may be other combinations of x, y, and z that satisfy the given conditions.)