Find the numbers whose GM is $5$ and AM is $7.5$:

The correct answer is Option b: $13.09$ and $1.91$. **Step 1: Set up equations.** Let the two numbers be $a$ and $b$. $$\frac{a+b}{2} = 7.5 \Rightarrow a + b = 15$$ $$\sqrt{ab} = 5 \Rightarrow ab = 25$$ **Step 2: Solve the quadratic.** $$(a-b)^2 = (a+b)^2 - 4ab = 225 - 100 = 125$$ $$|a-b| = \sqrt{125} = 5\sqrt{5} \approx 11.18$$ From $a + b = 15$ and $a - b = 11.18$: $$a = \frac{15 + 11.18}{2} = \frac{26.18}{2} \approx 13.09$$ $$b = \frac{15 - 11.18}{2} = \frac{3.82}{2} \approx 1.91$$ **Verification:** - AM = $(13.09 + 1.91)/2 = 15/2 = 7.5$ ✓ - GM = $\sqrt{13.09 \times 1.91} = \sqrt{25} = 5$ ✓ Hence, **Option B** is the correct answer.

Central Tendency & DispersionMTP June 24 Series IQuestion 3066 of 473

All Questions

Find the numbers whose GM is $\displaystyle 5$ and AM is $\displaystyle 7.5$ :

Options

\displaystyle 12

and

\displaystyle 13

\displaystyle 13.09

and

\displaystyle 1.91

\displaystyle 14

and

\displaystyle 11

\displaystyle 17

and

\displaystyle 19

For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option b — $\displaystyle 13.09$ and $\displaystyle 1.91$

All Options:

A $\displaystyle 12$ and $\displaystyle 13$
B $\displaystyle 13.09$ and $\displaystyle 1.91$
C $\displaystyle 14$ and $\displaystyle 11$
D $\displaystyle 17$ and $\displaystyle 19$

Detailed Solution & Explanation

**Step 1: Set up equations.** Let the two numbers be

\displaystyle a

and

\displaystyle b

\frac{a+b}{2} = 7.5 \Rightarrow a + b = 15

\sqrt{ab} = 5 \Rightarrow ab = 25

**Step 2: Solve the quadratic.**

(a-b)^2 = (a+b)^2 - 4ab = 225 - 100 = 125

|a-b| = \sqrt{125} = 5\sqrt{5} \approx 11.18

From

\displaystyle a + b = 15

and

\displaystyle a - b = 11.18

a = \frac{15 + 11.18}{2} = \frac{26.18}{2} \approx 13.09

b = \frac{15 - 11.18}{2} = \frac{3.82}{2} \approx 1.91

**Verification:** - AM =

\displaystyle (13.09 + 1.91)/2 = 15/2 = 7.5

✓ - GM =

\displaystyle \sqrt{13.09 \times 1.91} = \sqrt{25} = 5

✓ Hence, **Option B** is the correct answer.

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