Scott uses 45 -cent stamps and 8 -cent stamps to mail a gift card to a friend. If the postage is $2.07 , how many of each stamp did Scott use?
The Correct Answer and Explanation is:
To find how many 45-cent stamps and 8-cent stamps Scott used to make $2.07 in postage, we first convert the total cost into cents:2.07 dollars=207 cents2.07 \text{ dollars} = 207 \text{ cents}2.07 dollars=207 cents
Let:
- xxx = number of 45-cent stamps
- yyy = number of 8-cent stamps
The total postage equation becomes:45x+8y=20745x + 8y = 20745x+8y=207
This is a linear Diophantine equation, which means we are looking for integer solutions.
Step 1: Trial and Error to Find Integer Solution
Let’s test different values of xxx to find one where yyy is also a whole number.
Try x=3x = 3x=3:45(3)+8y=207⇒135+8y=207⇒8y=72⇒y=945(3) + 8y = 207 \Rightarrow 135 + 8y = 207 \Rightarrow 8y = 72 \Rightarrow y = 945(3)+8y=207⇒135+8y=207⇒8y=72⇒y=9
That works! So one solution is:
- 3 of the 45-cent stamps
- 9 of the 8-cent stamps
Let’s double-check:3(45)+9(8)=135+72=2073(45) + 9(8) = 135 + 72 = 2073(45)+9(8)=135+72=207
Final Answer:
Scott used 3 of the 45-cent stamps and 9 of the 8-cent stamps.
Explanation (300 words):
This problem involves solving a linear equation with two unknowns. The total postage required is $2.07. To make the math simpler, we convert dollars into cents, because the stamps are priced in cents. Thus, $2.07 becomes 207 cents.
We assign variables to represent how many stamps of each kind Scott used: let xxx be the number of 45-cent stamps, and yyy the number of 8-cent stamps. The total value of the stamps must equal 207 cents, which gives the equation 45x+8y=20745x + 8y = 20745x+8y=207.
Since we are working with whole items (stamps), the solution must be integers. We use substitution or trial-and-error to test small values of xxx and solve for yyy. We are looking for values that make both xxx and yyy integers.
Trying x=3x = 3x=3, we solve for yyy and find it equals 9. This means Scott used 3 stamps worth 45 cents and 9 stamps worth 8 cents.
We can verify this by calculating the total postage: 3×45=1353 \times 45 = 1353×45=135 cents and 9×8=729 \times 8 = 729×8=72 cents. Their sum is 135+72=207135 + 72 = 207135+72=207 cents, which is correct.
This method is useful for solving problems where combinations of fixed amounts must total a specific value. These types of problems appear in real-life budgeting, postage, and inventory scenarios.
