m is partly constant and partly varies as n. When n=6, m=11 and when n=20, m=32. Calculate the percentage change in m when n changes from 6 to 8.

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13. Percentage change in $m$

Step 1: Establish the relationship between $m$ and $n$.
Given that $m$ is partly constant and partly varies as $n$, we can write the relationship as:

$$m = k + cn$$

where $k$ is the constant part and $c$ is the constant of variation.

Step 2: Use the given conditions to find $k$ and $c$.
Condition 1: $m=11$ when $n=6$.

$$11 = k + 6c \quad (1)$$

Condition 2: $m=32$ when $n=20$.

$$32 = k + 20c \quad (2)$$

Subtract equation (1) from equation (2):

$$(32 - 11) = (k + 20c) - (k + 6c)$$ $$21 = 14c$$ $$c = \frac{21}{14} = \frac{3}{2} = 1.5$$

Substitute $c = 1.5$ into equation (1):

$$11 = k + 6(1.5)$$ $$11 = k + 9$$ $$k = 11 - 9$$ $$k = 2$$

So, the relationship is $m = 2 + 1.5n$.

Step 3: Calculate the initial value of $m$ when $n=6$.
From the given information, $m_1 = 11$.

Step 4: Calculate the new value of $m$ when $n=8$.

$$m_2 = 2 + 1.5(8)$$ $$m_2 = 2 + 12$$ $$m_2 = 14$$

Step 5: Calculate the percentage change in $m$.

$$\text{Percentage change} = \frac{m_2 - m_1}{m_1} \times 100\%$$ $$\text{Percentage change} = \frac{14 - 11}{11} \times 100\%$$ $$$$