If 4x = 5y = 20z then z is equal to

Option d: xy/x+y. Solution: Let the given relation be equated to a constant k : 4x = 5y = 20z = k From this, we can express each base in terms of k : 4 = k1/x 5 = k1/y 20 = k1/z We know the mathematical relationship between the bases: 20 = 4 × 5 Substituting the expressions of the bases in terms of k into this relationship, we get: k1/z = k1/x × k1/y Using the law of indices ( am × an = am+n ): k1/z = k1/x + 1/y Since the bases on both sides are the same, we can equate their exponents: 1/z = 1/x + 1/y Simplify the right-hand side by taking the common denominator: 1/z = y + x/xy = x + y/xy Taking the reciprocal of both sides gives the value of z : z = xy/x + y Hence, Option D is the correct answer.

If $4^x = 5^y = 20^z$ then $z$ is equal to

The correct answer is Option d: $\frac{xy}{x+y}$. Let the given relation be equated to a constant $k$: $$4^x = 5^y = 20^z = k$$ From this, we can express each base in terms of $k$: $$4 = k^{1/x}$$ $$5 = k^{1/y}$$ $$20 = k^{1/z}$$ We know the mathematical relationship between the bases: $$20 = 4 \times 5$$ Substituting the expressions of the bases in terms of $k$ into this relationship, we get: $$k^{1/z} = k^{1/x} \times k^{1/y}$$ Using the law of indices ($a^m \times a^n = a^{m+n}$): $$k^{1/z} = k^{\frac{1}{x} + \frac{1}{y}}$$ Since the bases on both sides are the same, we can equate their exponents: $$\frac{1}{z} = \frac{1}{x} + \frac{1}{y}$$ Simplify the right-hand side by taking the common denominator: $$\frac{1}{z} = \frac{y + x}{xy} = \frac{x + y}{xy}$$ Taking the reciprocal of both sides gives the value of $z$: $$z = \frac{xy}{x + y}$$ Hence, **Option D** is the correct answer.

Ratio, Proportion, Indices, LogarithmsPYQ May 25Question 4003 of 220

All Questions

If $\displaystyle 4^x = 5^y = 20^z$ then $\displaystyle z$ is equal to

Options

\displaystyle xy

\displaystyle \frac{x + y}{xy}

\displaystyle \frac{1}{xy}

\displaystyle \frac{xy}{x+y}

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Correct Answer

✅ Option d — $\displaystyle \frac{xy}{x+y}$

All Options:

A $\displaystyle xy$
B $\displaystyle \frac{x + y}{xy}$
C $\displaystyle \frac{1}{xy}$
D $\displaystyle \frac{xy}{x+y}$

Detailed Solution & Explanation

Let the given relation be equated to a constant

\displaystyle k

4^x = 5^y = 20^z = k

From this, we can express each base in terms of

\displaystyle k

4 = k^{1/x}

5 = k^{1/y}

20 = k^{1/z}

We know the mathematical relationship between the bases:

20 = 4 \times 5

Substituting the expressions of the bases in terms of

\displaystyle k

into this relationship, we get:

k^{1/z} = k^{1/x} \times k^{1/y}

Using the law of indices (

\displaystyle a^m \times a^n = a^{m+n}

k^{1/z} = k^{\frac{1}{x} + \frac{1}{y}}

Since the bases on both sides are the same, we can equate their exponents:

\frac{1}{z} = \frac{1}{x} + \frac{1}{y}

Simplify the right-hand side by taking the common denominator:

\frac{1}{z} = \frac{y + x}{xy} = \frac{x + y}{xy}

Taking the reciprocal of both sides gives the value of

\displaystyle z

z = \frac{xy}{x + y}

Hence, **Option D** is the correct answer.

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