CAT 2019 Slot 1 — QA Question 32
IndicesEasy
If 5.55x = 0.555y = 1000, then the value of (1/x) − (1/y) is
Answer & solution
1/3
- B
2/3
- C
3
- D
1
Solution
Given, 5.55x = 0.555y = 1000 = k say.
⇒ 5.55x = k ⇒ 5.55 = k1/x ...(1)
⇒ 0.555y = k ⇒ 0.555 = k1/y ...(2)
⇒ 1000z = 103 = k ⇒ 10 = k1/3 ...(3)
⇒ 5.55 × 10 = 0.555
∴ k1/x × k1/3 = k1/y
⇒ k1/x + 1/3 = k1/y
⇒ 1/x + 1/3 = 1/y
⇒ 1/x - 1/y = 1/3
Alternately,
5.55x = 0.555y = 1000
Taking log to the base 10, we get;
log(5.55x) = log(0.555y) = log(1000) = log 103 = 3
∴ x log 5.55 = y log 0.555 = 3
So, (1/x) = (log 5.55)/3
(1/y) = (log 0.555)/3
Log 0.555 = log 5.55 × 10−1 = log 5.55 + log 10−1 = (log 5.55) − 1.
So, (1/y) = [(log 5.55) − 1]/3
∴ (1/x) − (1/y) = [(log 5.55)/3] − [{(log 5.55) − 1}/3] = 1/3.
Hence, option (a).