x is the G.M. and y is the H.M. of two numbers. If x/y=5/4, then the numbers are in the ratio

A. 1:4. Let the numbers be a and b. Step 1: G.M. x=√(ab) and H.M. y=2ab/a+b Step 2: x/y=√(ab) (a+b)2ab=a+b2√(ab)=5/4 Step 3: Put a/b=t^2. Dividing top and bottom by b: t^2+12t=5/4 Step 4: 4t^2+4=10t 2t^2-5t+2=0 (2t-1)(t-2)=0 Step 5: t=2 or t=1/2, so a/b=4 or 1/4. The numbers are in the ratio 1:4. Check: for 1 and 4, G.M. =2 and H.M. =2× 4/5=8/5, so x/y=5/4.

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$x$ is the G.M. and $y$ is the H.M. of two numbers. If $\frac{x}{y} = \frac{5}{4}$ , then the numbers are in the ratio

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — $1 : 4$

Detailed Solution

Let the numbers be $a$ and $b$ .

G.M. $x = ab$ and H.M. $y = \frac{2 ab}{a + b}$

$\frac{x}{y} = \frac{ab ( a + b )}{2 ab} = \frac{a + b}{2 ab} = \frac{5}{4}$

Put $\frac{a}{b} = t^{2}$ . Dividing top and bottom by $b$ : $\frac{t ^{2} + 1}{2 t} = \frac{5}{4}$

$4 t^{2} + 4 = 10 t \Rightarrow 2 t^{2} - 5 t + 2 = 0 \Rightarrow (2 t - 1) (t - 2) = 0$

$t = 2$ or $t = \frac{1}{2}$ , so $\frac{a}{b} = 4$ or $\frac{1}{4}$ . The numbers are in the ratio $1 : 4$ .

✓ Check

For 1 and 4, G.M.

= 2

and H.M.

= \frac{2 \times 4}{5} = \frac{8}{5}

, so

\frac{x}{y} = \frac{5}{4}

Quick tip

$\frac{GM}{H M} = \frac{A M}{GM}$ ; numbers in ratio $1 : 4$ have $A M = 2.5, GM = 2$

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Arithmetic & Geometric Progression — Preparation

This question tests your understanding of Arithmetic & Geometric Progression, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

An arithmetic progression (AP) is a sequence where each term differs from the previous one by a constant amount, the common difference. A geometric progression (GP) is a sequence where each term is the previous one multiplied by a constant, the common ratio. This topic covers the nth term and sum of n terms of an AP and GP, the sum of an infinite GP, arithmetic and geometric means, and word problems. Key formulas are the AP nth term a + (n - 1)d, the AP sum n/2 [2a + (n - 1)d], the GP nth term ar^(n-1), and the infinite GP sum a/(1 - r) for |r| < 1. A useful trick is to choose symmetric terms, such as a - d, a, a + d for three terms of an AP, which simplifies many questions. Practise the AP and GP questions below, each with a step-by-step solution.

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$x$ is the G.M. and $y$ is the H.M. of two numbers. If $\frac{x}{y} = \frac{5}{4}$ , then the numbers are in the ratio