Let the positive number be denoted by 'k'. The problem describes a relationship between this number and a calculation involving a percentage of itself.

Setting Up the Equation for the Positive Number k

The question states that 'k' when multiplied by 30% of itself gives a certain result. Let's break this down:

• 30% of 'k' can be written as $\frac{30}{100} \times k = 0.3k$.
• 'k' multiplied by 30% of itself is $k \times (0.3k) = 0.3k^2$.

This result, $0.3k^2$, is equal to "a number which is 170% more than the number ‘k’". Let's understand what 170% more than 'k' means:

• 170% of 'k' is $\frac{170}{100} \times k = 1.7k$.
• 170% more than 'k' means adding 170% of 'k' to 'k'. So, it is $k + 1.7k = 2.7k$.

Now we can set up the equation based on the problem statement:

The number obtained by multiplying 'k' by 30% of itself ($0.3k^2$) is equal to 170% more than 'k' ($2.7k$).

So, the equation is:

\begin{equation*} 0.3k^2 = 2.7k \end{equation*}

Solving for the Positive Number k

We have the equation $0.3k^2 = 2.7k$. We are looking for a positive number 'k'.

Since 'k' is a positive number, $k \neq 0$. We can divide both sides of the equation by 'k':

\begin{align*} \frac{0.3k^2}{k} &= \frac{2.7k}{k} \\ 0.3k &= 2.7 \end{align*}

Now, to find 'k', divide both sides by 0.3:

\begin{align*} k &= \frac{2.7}{0.3} \\ k &= \frac{27}{3} \\ k &= 9 \end{align*}

The value of the positive number 'k' is 9.

Verifying the Solution

Let's check if k=9 satisfies the original condition:

• 30% of k = 30% of 9 = $0.3 \times 9 = 2.7$.
• k multiplied by 30% of itself = $9 \times 2.7 = 24.3$.
• 170% more than k = 170% more than 9 = $9 + (1.7 \times 9) = 9 + 15.3 = 24.3$.

Since $24.3 = 24.3$, the condition is satisfied. The positive number 'k' is indeed 9.

Revision Table: Positive Number and Percentage Problems

Reviewing key concepts related to this problem:

• Understanding percentages and converting them to decimals or fractions.
• Translating word problems into algebraic equations.
• Solving basic algebraic equations, including those with quadratic terms that can be simplified.
• Interpreting phrases like "X% of a number" and "X% more than a number".

Additional Information: Understanding Percentage Increase

When a quantity increases by a certain percentage, say P%, the new quantity is the original quantity plus P% of the original quantity. If the original quantity is Q, and the percentage increase is P%, the new quantity is:

\begin{equation*} \text{New Quantity} = Q + \left(\frac{P}{100} \times Q\right) = Q \left(1 + \frac{P}{100}\right) \end{equation*}

In our problem, the quantity is 'k', and it increases by 170%. So, the new value is:

\begin{equation*} k \left(1 + \frac{170}{100}\right) = k(1 + 1.7) = 2.7k \end{equation*}

This confirms our setup that 170% more than 'k' is equal to $2.7k$. This concept is crucial for solving many percentage-based word problems.

The positive number 'k' is 9.