Solving the Percentage Problem: Relating 'a' and 'b'

The question provides a relationship between two quantities, 'a' and 'b', using percentages. We are told that 20% of 'a' is equal to 'b'. Our goal is to express 'b% of 20' in terms of 'a'.

Understanding the Given Relationship: 20% of a = b

First, let's write the given relationship as a mathematical equation. Percentage means 'out of 100'. So, 20% can be written as $\frac{20}{100}$ or 0.20.

The statement "20% of a = b" translates to:

$\frac{20}{100} \times a = b$

This simplifies to:

$0.2a = b$

This equation tells us that 'b' is equal to 0.2 times 'a'.

Calculating b% of 20

Now, we need to find the value of 'b% of 20'. Similar to the first step, 'b%' means $\frac{b}{100}$.

So, "b% of 20" translates to:

$\frac{b}{100} \times 20$

Substituting the Value of 'b'

We know from the first part that $b = 0.2a$. We can substitute this expression for 'b' into the calculation for 'b% of 20'.

Substitute $b = 0.2a$ into $\frac{b}{100} \times 20$:

$\frac{0.2a}{100} \times 20$

Simplifying the Expression

Now, let's simplify this expression:

$\frac{0.2a \times 20}{100}$

Multiply 0.2a by 20:

$0.2a \times 20 = (0.2 \times 20) \times a = 4 \times a = 4a$

So the expression becomes:

$\frac{4a}{100}$

This fraction can be written as a decimal:

$\frac{4a}{100} = 0.04a$

Expressing the Result as a Percentage of 'a'

The result is $0.04a$. To express this as a percentage of 'a', we multiply the decimal by 100 and add the percentage sign.

$0.04a = (0.04 \times 100)\% \text{ of } a = 4\% \text{ of } a$

Comparing with Options

The result of 'b% of 20' is equal to '4% of a'. Let's compare this with the given options:

1. 5% of a
2. 30% of a
3. 4% of a
4. 20% of a

Our calculated result, 4% of a, matches option 3.

Step-by-Step Calculation Summary

• Given: $20\% \text{ of } a = b \implies 0.2a = b$
• Find: $b\% \text{ of } 20$
• Substitute b: $\frac{b}{100} \times 20 = \frac{0.2a}{100} \times 20$
• Simplify: $\frac{0.2a \times 20}{100} = \frac{4a}{100} = 0.04a$
• Convert to percentage: $0.04a = 4\% \text{ of } a$

Revision Table: Percentage Concepts

Additional Information: Working with Percentages and Variables

Problems involving percentages and variables, like the one we just solved, require converting the percentage statements into algebraic equations. This allows us to manipulate the expressions and solve for the unknown quantity or express one quantity in terms of another.

• Converting to Decimals: Often, the easiest way to work with percentages in calculations is to convert them to their decimal equivalents by dividing by 100. For example, 20% becomes 0.20, and 4% becomes 0.04.
• Algebraic Substitution: If you are given an equation (like $b = 0.2a$), you can substitute the expression for one variable into another expression. This was key in replacing 'b' with '0.2a' in the calculation of 'b% of 20'.
• Understanding 'of': In percentage problems, the word 'of' usually indicates multiplication. 'P% of X' means P/100 multiplied by X.

Mastering these basic conversions and algebraic techniques is crucial for solving a wide range of percentage-based problems.