A quantity Q is partly constant and partly varies inversely as p. Given that Q = 11 when p… — JAMB UTME

A quantity $Q$ is partly constant and partly varies inversely as $p$ . Given that $Q = 11$ when $p = 2$ , and $Q = 5$ when $p = 5$ , what is the value of $Q$ when $p = 10$ ?

A
$2$
$3$ Answer
C
$4$
D
$7$

Answer

The value of

Q

when

p = 10

3

The relationship for partial inverse variation is

Q = k_1 + \frac{k_2}{p}

. Substituting

(p=2, Q=11)

gives

2k_1 + k_2 = 22

, and substituting

(p=5, Q=5)

gives

5k_1 + k_2 = 25

. Solving these simultaneous equations gives

k_1 = 1

and

k_2 = 20

. Substituting

p = 10

into

Q = 1 + \frac{20}{p}

results in

Q = 1 + 2 = 3

Step-by-Step Solution

Set up the general formula for partial variation.

Q = k_1 + \frac{k_2}{p}

, where

k_1

and

k_2

are constants.

Partial variation consists of a constant part and a part that varies inversely with

p

Substitute the given values to form two simultaneous linear equations.

For

p = 2, Q = 11 \implies 11 = k_1 + \frac{k_2}{2} \implies 2k_1 + k_2 = 22

For

p = 5, Q = 5 \implies 5 = k_1 + \frac{k_2}{5} \implies 5k_1 + k_2 = 25

Substituting known data points provides equations to solve for the variation constants.

Solve the simultaneous equations for

k_1

and

k_2

Subtracting the first equation from the second gives

3k_1 = 3 \implies k_1 = 1

.
Substituting

k_1 = 1

into

2(1) + k_2 = 22

yields

k_2 = 20

Determining

k_1

and

k_2

establishes the explicit relationship between

Q

and

p

Calculate

Q

when

p = 10

Q = 1 + \frac{20}{10} = 1 + 2 = 3

Substitute

p = 10

into the established variation equation

Q = 1 + \frac{20}{p}

Key Concept

Partial Variation and Simultaneous Equations

Estimated Time:1m 30s

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