Solving Percentage Comparison Problems

This problem involves a series of percentage relationships between four quantities: A, B, C, and D. We are given the following information:

• A is 25% more than B.
• B is 40% less than C.
• C is 20% more than D.

Our goal is to find out what percentage A is less than D.

Step-by-Step Calculation

To solve this, we can assume a base value for one of the quantities and then calculate the others based on the given percentage relationships. It's often easiest to start with the quantity that is the base for the most relationships or is at the end of the chain of dependencies, which in this case is D.

Step 1: Assume a value for D.

Let's assume D = 100. This makes percentage calculations straightforward.

D = 100

Step 2: Calculate C based on D.

C is 20% more than D. This means C is D plus 20% of D.

C = D + 20\% \text{ of } D

C = D + \left(\frac{20}{100}\right) \times D

C = D(1 + 0.20)

C = 1.20 \times D

Substituting the value of D = 100:

C = 1.20 \times 100

C = 120

Step 3: Calculate B based on C.

B is 40% less than C. This means B is C minus 40% of C.

B = C - 40\% \text{ of } C

B = C - \left(\frac{40}{100}\right) \times C

B = C(1 - 0.40)

B = 0.60 \times C

Substituting the value of C = 120:

B = 0.60 \times 120

B = 72

Step 4: Calculate A based on B.

A is 25% more than B. This means A is B plus 25% of B.

A = B + 25\% \text{ of } B

A = B + \left(\frac{25}{100}\right) \times B

A = B(1 + 0.25)

A = 1.25 \times B

Substituting the value of B = 72:

A = 1.25 \times 72

A = \left(\frac{5}{4}\right) \times 72

A = 5 \times \left(\frac{72}{4}\right)

A = 5 \times 18

A = 90

Step 5: Compare A and D.

We have A = 90 and D = 100. We need to find the percentage A is less than D. Since A (90) is less than D (100), we calculate the difference and express it as a percentage of D (the base for comparison).

\text{Difference} = D - A

\text{Difference} = 100 - 90

\text{Difference} = 10

Step 6: Calculate the percentage A is less than D.

The percentage decrease is calculated with respect to the original value (D in this case).

\text{Percentage less} = \left(\frac{\text{Difference}}{D}\right) \times 100\%

\text{Percentage less} = \left(\frac{10}{100}\right) \times 100\%

\text{Percentage less} = 0.10 \times 100\%

\text{Percentage less} = 10\%

So, A is 10% less than D.

Summary of Values

Comparing A (90) and D (100), A is 10 less than D, which is $\frac{10}{100} \times 100\% = 10\%$ less than D.

Revision Table - Percentage Calculations

In our problem:

• C is 20% more than D: \(C = D \times (1 + 0.20)\)
• B is 40% less than C: \(B = C \times (1 - 0.40)\)
• A is 25% more than B: \(A = B \times (1 + 0.25)\)

Substituting these equations:

A = 1.25 \times B

A = 1.25 \times (0.60 \times C)

A = 1.25 \times 0.60 \times C

A = 0.75 \times C

A = 0.75 \times (1.20 \times D)

A = 0.75 \times 1.20 \times D

A = 0.90 \times D

So, A is 0.90 times D, which means A is 90% of D. If A is 90% of D, then A is (100 - 90)% = 10% less than D.

Additional Information - Percentage Changes

Percentage change problems often involve a base value. When we say 'X is P% more/less than Y', Y is the base. When we ask 'What percentage is X less than Y?', Y is the base for comparison in the denominator. When we ask 'What percentage is X more than Y?', Y is the base.

Consecutive percentage changes do not simply add up. For example, a 20% increase followed by a 20% decrease does not result in the original value. If you start with 100, a 20% increase gives 120. A 20% decrease on 120 is \(120 \times 0.20 = 24\), so the final value is \(120 - 24 = 96\), not 100.

In this problem, we chained the percentages: A depends on B, B on C, and C on D. This required calculating the effect of each percentage change sequentially or multiplying the multipliers as shown in the alternative method above.