Hardest SAT Math Problems (updated for Digital SAT) - PrepMaven - Free Printable

Let me work through this problem step by step.

Given information:
1. Point (p, r) lies on the line y = x + b
2. Point (2p, 5r) lies on the line y = 2x + b
3. p ≠ 0
4. We need to find r/p

Step 1: Use the first point (p, r) on line y = x + b

Since (p, r) is on the line y = x + b, we can substitute:
- r = p + b

This means: b = r - p ... (Equation 1)

Step 2: Use the second point (2p, 5r) on line y = 2x + b

Since (2p, 5r) is on the line y = 2x + b, we can substitute:
- 5r = 2(2p) + b
- 5r = 4p + b

This means: b = 5r - 4p ... (Equation 2)

Step 3: Set the two expressions for b equal to each other

From Equation 1 and Equation 2:
r - p = 5r - 4p

Step 4: Solve for r/p

Let's rearrange:
r - p = 5r - 4p

Add 4p to both sides:
r + 3p = 5r

Subtract r from both sides:
3p = 4r

Divide both sides by p (since p ≠ 0):
3 = 4(r/p)

Divide both sides by 4:
r/p = 3/4

Let me verify this answer:
If r/p = 3/4, then r = 3p/4

From Equation 1: b = r - p = 3p/4 - p = 3p/4 - 4p/4 = -p/4

Check with Equation 2: b = 5r - 4p = 5(3p/4) - 4p = 15p/4 - 16p/4 = -p/4 ✓

Both equations give the same value for b, so our answer is correct!

Final Answer: B) 3/4