TheCATExam
Sign in

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

Correct answer: 1/3

  • 1/3

  • B

    2/3

  • C

    3

  • D

    1

Solution

Easy

Take log⁡10\log_{10} of the common value 10001000. Each exponent equation gives 1x\tfrac1x and 1y\tfrac1y in terms of log⁡5.55\log 5.55 and log⁡0.555\log 0.555. Since 0.555=5.55/100.555=5.55/10, their logs differ by exactly 11, which makes the difference collapse.

1

Take logs. Apply log⁡10\log_{10} to each part of 5.55x=0.555y=10005.55^{x}=0.555^{y}=1000.

xlog⁡5.55=ylog⁡0.555=log⁡1000=3⇒ 1x=log⁡5.553,1y=log⁡0.5553\begin{aligned} &x\log 5.55=y\log 0.555=\log 1000=3\\ &\Rightarrow\ \frac{1}{x}=\frac{\log 5.55}{3},\qquad \frac{1}{y}=\frac{\log 0.555}{3} \end{aligned}
2

Relate the two logs. Since 0.555=5.55100.555=\dfrac{5.55}{10}.

log⁡0.555=log⁡5.55−log⁡10=log⁡5.55−1\begin{aligned} &\log 0.555=\log 5.55-\log 10=\log 5.55-1 \end{aligned}
3

Subtract. Use the two expressions from step 1 with step 2.

1x−1y=log⁡5.553−log⁡5.55−13⇒ =log⁡5.55−(log⁡5.55−1)3⇒ =13\begin{aligned} &\frac{1}{x}-\frac{1}{y}=\frac{\log 5.55}{3}-\frac{\log 5.55-1}{3}\\ &\Rightarrow\ =\frac{\log 5.55-(\log 5.55-1)}{3}\\ &\Rightarrow\ =\frac{1}{3} \end{aligned}
1x−1y=13\frac{1}{x}-\frac{1}{y}=\frac{1}{3}

Related Indices questions

See all Surds & Indices questions →
CAT 2019 Slot 1 QA Q32 — Indices solution | TheCATExam