Alex and Freddy collect stamps. The number of stamps in Alex’s collection is in a ratio of 3:5 to the number of stamps in Freddy’s collection. If Freddy gives Alex 10 stamps, the ratio will change to 7:9. How many stamps are in both collections together?
The Correct Answer and Explanation is:
Let the number of stamps Alex has be 3x, and the number of stamps Freddy has be 5x, since their collections are in the ratio 3:5.
Step 1: After the exchange
Freddy gives Alex 10 stamps.
- Alex’s new amount = 3x + 10
- Freddy’s new amount = 5x − 10
Now, their new ratio becomes 7:9.3x+105x−10=79\frac{3x + 10}{5x – 10} = \frac{7}{9}5x−103x+10=97
Step 2: Solve the proportion
Cross-multiply:9(3x+10)=7(5x−10)9(3x + 10) = 7(5x – 10)9(3x+10)=7(5x−10)27x+90=35x−7027x + 90 = 35x – 7027x+90=35x−70
Subtract 27x from both sides:90=8x−7090 = 8x – 7090=8x−70
Add 70 to both sides:160=8×160 = 8×160=8xx=20x = 20x=20
Step 3: Calculate number of stamps
- Alex originally had: 3x=3(20)=603x = 3(20) = 603x=3(20)=60
- Freddy originally had: 5x=5(20)=1005x = 5(20) = 1005x=5(20)=100
Total number of stamps = 60 + 100 = 160
Explanation
This problem is a classic example of a ratio and proportion word problem. We are told that Alex and Freddy collect stamps and their stamp counts are in a 3:5 ratio. This means if Alex has 3 parts, Freddy has 5 parts of some common quantity. Representing these quantities algebraically helps us keep track of the changes — so we let Alex have 3x and Freddy 5x stamps.
Next, Freddy gives Alex 10 stamps, altering both their totals. Now Alex has more, and Freddy has fewer. The new ratio is 7:9, which gives us a proportion we can solve. Writing the equation:3x+105x−10=79\frac{3x + 10}{5x – 10} = \frac{7}{9}5x−103x+10=97
we use cross multiplication, a standard technique in solving proportions. Multiplying across gives a linear equation, which we simplify to find x=20x = 20x=20.
With this value of x, we substitute back to find the actual number of stamps each person originally had: Alex had 60, and Freddy had 100. Adding both gives the total collection: 160 stamps.
This problem tests a student’s ability to model a situation with variables, understand how changes affect ratios, and apply algebraic methods to find a solution. It’s a practical example of how proportional reasoning is used to understand real-world exchanges.
