Station 1 There are red tiles and blue tiles in a box. The ratio of red tiles to blue tiles is 3:5. There are 12 more blue tiles than red tiles in the box. How many red tiles are in the box?
The Correct Answer and Explanation is:
Let the number of red tiles be RRR and the number of blue tiles be BBB.
We are given two pieces of information:
- The ratio of red tiles to blue tiles is 3:5, meaning: RB=35\frac{R}{B} = \frac{3}{5}BR=53 This can be rewritten as: R=35B(Equation 1)R = \frac{3}{5}B \quad \text{(Equation 1)}R=53B(Equation 1)
- There are 12 more blue tiles than red tiles, which means: B=R+12(Equation 2)B = R + 12 \quad \text{(Equation 2)}B=R+12(Equation 2)
Step 1: Solve the equations
From Equation 2, substitute R=B−12R = B – 12R=B−12 into Equation 1:B−12=35BB – 12 = \frac{3}{5}BB−12=53B
Step 2: Simplify the equation
To eliminate the fraction, multiply both sides of the equation by 5:5(B−12)=3B5(B – 12) = 3B5(B−12)=3B
Now expand the left side:5B−60=3B5B – 60 = 3B5B−60=3B
Step 3: Solve for BBB
Subtract 3B3B3B from both sides:2B−60=02B – 60 = 02B−60=0
Now add 60 to both sides:2B=602B = 602B=60
Finally, divide both sides by 2:B=30B = 30B=30
Step 4: Find the number of red tiles
Now that we know there are 30 blue tiles, substitute B=30B = 30B=30 into Equation 2:B=R+12B = R + 12B=R+1230=R+1230 = R + 1230=R+12
Subtract 12 from both sides:R=18R = 18R=18
Conclusion:
There are 18 red tiles in the box.
This problem used basic algebra to solve a system of equations. The ratio provided the first equation, and the difference in the number of tiles gave the second equation. By solving this system, we found that there are 18 red tiles and 30 blue tiles.
