trish sends letters ans postcard to her friends.She needs 37 cents in postage for each letter and 21 cents for each postcard. She spends$3.11 in postage. How many letters and postcards did trisha send in all?
you want whole number solutions, ordered pairs, for the equation
37L + 21P = 311
This is an example of a Diophantine's equation.
Somehow I don't think you are studying that topic
by just doing simple trial and error, I got
L = 5, and P=6
check: 5x37 + 6x21 = 311
To solve this problem, let's use a system of equations. Let's assume Trish sent "x" letters and "y" postcards.
According to the information given, Trish needs 37 cents in postage for each letter, so the total cost of sending letters would be 37x cents.
Similarly, Trish needs 21 cents in postage for each postcard, so the total cost of sending postcards would be 21y cents.
We also know that Trish spent $3.11 in postage, which is equal to 311 cents.
Putting these equations together, we have two equations:
37x + 21y = 311 (Equation 1)
x + y = total number of letters and postcards (Equation 2)
To solve this system of equations, we can use substitution or elimination method. Let's use elimination:
Multiply Equation 2 by 21 to make the coefficients of "y" equal:
21x + 21y = 21(total number of letters and postcards)
Now subtract this equation from Equation 1:
37x + 21y = 311
-(21x + 21y = 21(total number of letters and postcards))
This simplifies to:
16x = 311 - 21(total number of letters and postcards)
We can now solve for "x":
16x = 311 - 21(total number of letters and postcards)
x = (311 - 21(total number of letters and postcards))/16
Since the number of letters and postcards cannot be fractional, we need to find a whole number solution. We can try different values for the total number of letters and postcards until we get a whole number for "x".
For example, if we try with the total number of letters and postcards = 1, then:
x = (311 - 21(1))/16 = (311 - 21)/16 = 290/16 which is not a whole number.
If we try with the total number of letters and postcards = 2, then:
x = (311 - 21(2))/16 = (311 - 42)/16 = 269/16 which is not a whole number.
Let's try again with a few more values:
If we try with the total number of letters and postcards = 3, then:
x = (311 - 21(3))/16 = (311 - 63)/16 = 248/16 = 15.5 which is not a whole number.
If we try with the total number of letters and postcards = 4, then:
x = (311 - 21(4))/16 = (311 - 84)/16 = 227/16 = 14.1875 which is not a whole number.
Continuing this process, we can try different values of the total number of letters and postcards until we find a whole number solution for "x".
Let's try with the total number of letters and postcards = 7:
x = (311 - 21(7))/16 = (311 - 147)/16 = 164/16 = 10.25 which is not a whole number.
As we continue this process, we find that there is no whole number solution for "x". Therefore, there is no possible solution for this problem with the given information.