Three numbers are such that the second is 150% of the first, and the third is 80% of the second. What is the ratio of the first to the third?

Correct — A.
Take the first number as 100.

Second = 150% of 100 = 150
Third = 80% of 150 = 0.8 × 150 = 120

First : third = 100 : 120
Divide both by 20 = 5 : 6 → option (a)

Why the others are wrong

• (b)
  2:3 — 2 : 3 is first : second, 100 : 150. It stops one step early and never applies the 80% that turns the second number into the third.
• (c)
  5:4 — 5 : 4 is second : third, 150 : 120. The question asks for first to third, which is 100 : 120.
• (d)
  5:8 — 5 : 8 would need the third number to be 160 when the first is 100. The chain gives 0.8 × 150 = 120, not 160.

Chained percentages multiply. In the stem, "the second is 150% of the first" means second = 1.5 × first, and "the third is 80% of the second" means third = 0.8 × second.

So third = 0.8 × 1.5 × first = 1.2 × first. First : third is then 1 : 1.2, which clears to 5 : 6.

Taking the first number as 100 does the same work in whole numbers.

Key facts

• 150% of x = 1.5x = 3⁄2 of x
• 80% of x = 0.8x = 4⁄5 of x
• Successive "percent of" steps multiply: 0.8 × 1.5 = 1.2
• Scale a ratio to clear decimals: 1 : 1.2 = 10 : 12 = 5 : 6

Common traps

• Stopping at first : second (2 : 3) and never applying the 80%
• Writing second : third (5 : 4) when the question asks for first to third

No directly related past PYQ was found.