Percentages, Profit and Loss, Discounts and Interest Short Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Percentages, Profit and Loss, Discounts and Interest short notes for XAT: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: Profit & Optimization Toolkit

Quick Revision Checklist

Base check: is the percentage on CP or SP? Convert with $\frac{x}{100 - x} \times 100$ when it is on SP.
Mixtures: $Avg % = \frac{\sum ( Value _{i} \times % _{i} )}{\sum Value _{i}}$ — weight by cost value, never simple-average the rates.
Optimization: form $Π (x) = (P (x) - C) Q (x)$ ; the vertex sits at the midpoint of the roots.
Constraints: integer $x$ , $Q \geq 0$ , $P \geq C$ , and capacity caps can all move the answer off the vertex.
Non-linear value: breaking reduces value whenever $V \propto A^{n}$ with $n > 1$ ; loss $= 1 - \frac{p ^{n} + q ^{n}}{( p + q ) ^{n}}$ .
Discrete maxima: write the objective in terms of the free variables and check every boundary vertex.

Final tip: in advanced exams, if a profit question looks too simple, you are likely missing a base switch or a binding constraint. Re-read before you compute.

Summary: The Coupon Decision Toolkit

The Coupon Decision Toolkit

Read the triple: threshold

T

, benefit (

F

r

), cap

C

Eligibility:

B < T \Rightarrow

discount is zero.

Flat:

D = F

; effective rate

\frac{F}{B}

falls as

B

grows.

Percentage:

D = min (r B, C)

; flat-like beyond

B = \frac{C}{r}

Compare in rupees, never headlines; crossover

B^{*} = \frac{F}{r}

Per-transaction rule: splitting a bill can destroy eligibility.

Memory Hook

Threshold gates the deal, benefit sizes it, cap limits it, and the bill decides the winner.

Summary: The Interest & Percentage Toolkit

SI Formula:

S I = \frac{P \times R \times T}{100}

. Principal is the original amount.

Credit Purchases: Interest is charged only on the unpaid balance.

Reducing Balance: Interest for Year

N

is on the Principal remaining at the start of Year

N

Percentage Shifts:

New % = \frac{Old Part + Added}{Old Whole + Added}

Unit-Based Principal: Calculate total principal (Rate

\times

Area) before applying %.

Memory Hook

Interest is rent on the unpaid money. When the whole changes, update both the part and the whole in your equations.

Percentages, Profit and Loss, Discounts and Interest: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

In a revenue optimization model where Profit $Π (x)$ is a quadratic function of price change $x$ , given by $Π (x) = k (x - a) (b - x)$ with $a < b$ , at what value of $x$ does the maximum profit occur?

A.
$\frac{a + b}{2}$
B.
$\frac{b - a}{2}$
C.
$ab$
D.
$a + b$

Correct Answer:

Step-by-Step Solution

Key idea: This is a conceptual question about the properties of quadratic functions in optimization.

Step 1: Recognize the function form.

The profit function is given in intercept form: $Π (x) = k (x - a) (b - x)$ .

The roots of this quadratic equation (where Profit = 0) are $x = a$ and $x = b$ .

Step 2: Apply the vertex property.

For any parabola, the vertex (maximum or minimum point) lies exactly on the axis of symmetry.

The axis of symmetry is always at the midpoint of the roots.

Step 3: Calculate the midpoint.

$x_{v er t e x} = \frac{Root _{1} + Root _{2}}{2} = \frac{a + b}{2}$

Answer: $\frac{a + b}{2}$

Question 2 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

A food stall has a capacity constraint stating that "no more than 400 plates can be produced". If the stall currently plans to produce $P$ plates, which of the following expressions represents the maximum number of additional plates they can produce?

A.
$400 - P$
B.
$P - 400$
C.
$400 + P$
D.
$400 im es P$

Correct Answer:

Step-by-Step Solution

Key idea: This is a constraint translation question, converting a verbal limit into an algebraic expression for remaining capacity.

Step 1: The total capacity limit is 400 plates.

Step 2: The number of plates already planned is $P$ .

Step 3: The remaining capacity is the difference between the total limit and the planned amount.

Step 4: Remaining Capacity = $Total Limit - Planned Amount = 400 - P$ .

Step 5: This expression represents the maximum additional plates that can be produced without exceeding the constraint.

Answer: $400 - P$

More short notes in this unit

chapter

Percentages, Profit and Loss, Discounts and Interest Short Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Percentages, Profit and Loss, Discounts and Interest short notes for XAT: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved pra

A question from this chapter

Question 1

In a revenue optimization model where Profit Π (x) is a quadratic function of price change x, given by Π (x) = k (x - a) (b - x) with a < b, at what value of x does the maximum profit occur?

Question 2

A food stall has a capacity constraint stating that "no more than 400 plates can be produced". If the stall currently plans to produce P plates, which of the following expressions represents the maximum number of additional plates they can produce?

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