Q.3 If
𝑝 ∶ 𝑞 = 1 ∶ 2
𝑞 ∶ 𝑟 = 4 ∶ 3
𝑟 ∶ 𝑠 = 4 ∶ 5
and 𝑢 is 50% more than 𝑠 , what is the ratio 𝑝 ∶ 𝑢?
(A) 2 ∶ 15
(B) 16 ∶ 15
(C) 1: 5
(D) 16: 45
Solution Overview
To solve the ratio problem where p:q = 1:2, q:r = 4:3, r:s = 4:5, and u is 50% more than s, first chain the ratios to express all variables relative to p. The correct ratio p:u is 16:45.
Step-by-Step Solution
Assign values using a common multiple for q from the first two ratios.
- From
p:q = 1:2, letp = 2kandq = 4k(makingqmatch the 4 inq:r). - For
q:r = 4:3,r = 3k. - For
r:s = 4:5, multiply by 3 to align:r = 12k,s = 15k. - Then
u = s + 0.5s = 1.5s = 1.5 × 15k = 22.5k. - Thus,
p:u = 2k:22.5k = 8:22.5 = 16:45after simplifying by dividing by 0.5.
Option Analysis
(A) 2:15: Too small; ignores chaining and underestimates
u‘s growth relative to p.(B) 16:15: Reverses the ratio;
u > s > p, so p:u cannot exceed 1:1.(C) 1:5: Approximate but imprecise; chaining yields exactly 16:45 (equivalent to 16/45 ≈ 0.355, not 0.2).
(D) 16:45: Correct, as derived from full proportion.
Chaining Technique
- Link p to q (1:2), scale q to match next ratio’s 4 (p=2k, q=4k).
- Extend to r=3k, then scale r for s (r=12k, s=15k).
- u=1.5×15k=22.5k yields p:u=2k:22.5k=16:45.
Common Pitfalls
- Forgetting to scale ratios properly leads to errors like option (A).
- Reversing growth (u>s) mistakes yield (B).
Practice similar ratio proportion problems for exam success.
