If p : q = $\frac{2}{3}$ : $\frac{5}{6}$ and q : r = $\frac{3}{4}$ : $\frac{1}{2}$, find p : q : r

Step 1: Simplify p : q

p : q = 2/3 : 5/6. Multiply both sides by 6: 4 : 5.

Step 2: Simplify q : r

q : r = 3/4 : 1/2. Multiply both sides by 4: 3 : 2.

Step 3: Make q the same in both ratios

• p : q = 4 : 5
• q : r = 3 : 2

The LCM of 5 and 3 is 15.

• p : q = 4 : 5 = 12 : 15 (multiply by 3)
• q : r = 3 : 2 = 15 : 10 (multiply by 5)

Step 4: Combine

p : q : r = 12 : 15 : 10

Check: p/q = 12/15 = 4/5, which matches 4 : 5. q/r = 15/10 = 3/2, which matches 3 : 2.

Tip: To combine two ratios that share a middle term, take the LCM of that middle term's two values and scale each ratio to match it. Then simply line up the terms. Clearing fractions first (multiplying by the LCD) makes this much easier.