Joanie wants to mail a package that requires $1.30 in postage. If she has only 5-cent and 7-cent stamps, what is the smallest number of stamps she could use that would be totally exactly $1.30?
the smallest number of stamps is ____.
To find the smallest number of stamps Joanie could use, we need to consider the combinations of 5-cent and 7-cent stamps that add up to $1.30.
Let's start with the assumption that Joanie will use x 5-cent stamps and y 7-cent stamps. In this case, the equation representing the total value of the stamps is:
5x + 7y = 130 (since $1.30 is equivalent to 130 cents)
Now, to find the smallest number of stamps, we can use trial and error or, as we will do, use a straightforward approach:
1. Start with the assumption that Joanie will use 0 5-cent stamps.
- This would imply that the equation becomes: 7y = 130. However, there is no possible integer value of y that satisfies this equation. So, moving on to the next step.
2. Increase the number of 5-cent stamps by 1 and adjust the number of 7-cent stamps accordingly.
- If Joanie uses 1 5-cent stamp (x = 1), we can write the equation as: 5 + 7y = 130. Solving this equation, we find y = 18.57. However, since we are looking for a whole number of stamps, we need to increase x and adjust y.
3. Continue increasing the number of 5-cent stamps and adjusting the number of 7-cent stamps until we find a solution.
- By trying different values for x, we find that when x = 3 and y = 17, the equation 5x + 7y = 130 is satisfied. In this case, the smallest number of stamps Joanie would need is 3 (three 5-cent stamps) + 17 (seventeen 7-cent stamps) = 20 stamps.
Therefore, the smallest number of stamps Joanie could use to reach exactly $1.30 is 20 stamps.
number of 5's --- x
number of 7's ---- y
5x + 7y = 130
5x = 130-7y
x = (130-7y)/5
x must be a whole number, so
130-7y must be divisible by 5, only if 7y is a multiple of 5
to get the least number of stamps we should use as many of the 7's (the y) as we can
if y = 15 , the last positive value making 130-6y divisible by 5
x = 5
So she should use 5 fives and 15 sevens. total number = 20
check:
5(5) + 15(7) = 130
other choices:
y = 10, x = 12 , total = $130, number of stamps = 22
y = 20, x = -2 , not possible