AFCAT Numerical Ability · Profit and Loss

Buying on Instalments, and Two Special Cases

An instalment plan costs more than the cash price; with interest, the cash price is what the payments are worth today. Two odd cases close the chapter: drying fruit and a broken stone.

All Profit and Loss notes

Why this matters

A small page with no AFCAT past-year question, yet five of its thirteen rows are HARD, all of them the cash price behind instalments with compound interest. The extra-paid rows are one addition; the cash-price rows need each payment divided by its own interest factor, and the marks go on dividing them all by the same one.

Concept 1 of 3: Extra paid on an instalment plan

An instalment plan's total is the down payment plus all the instalments. What it costs beyond the cash price is the extra, the price of paying later.

Definition

Total paid = down payment + number of instalments $\times$ each instalment.
Extra = total paid $-$ cash price; as a percentage, extra $\div$ cash price $\times 100$ .
Working back: cash price = total paid $-$ extra.

Extra paid on instalments

\text{extra} = \text{down payment} + n \times \text{instalment} - \text{cash price}

Worked example

A TV costs ₹42,000 cash, or ₹12,000 down and 18 monthly instalments of ₹1,900. Find the extra paid on the instalment plan.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Profit and Loss · Instalments and Special Cases

A generator is available for Rs.

60000

cash or Rs.

20000

down and

24

monthly installments of Rs.

2000

each. What is the total extra amount paid in installments?

Include the down payment

The total paid is the down payment plus every instalment. Comparing only the instalments with the cash price makes the plan look cheaper than cash.

Drill 4 more on extra paid on an instalment plan

Concept 2 of 3: Cash price from instalments with interest

A payment due a year from now is worth less today: at

r\%

compound interest it is worth the payment divided by

1 + \tfrac{r}{100}

. The cash price is the sum of what each instalment is worth today.

Definition

An instalment $x$ due after $k$ periods is worth $\dfrac{x}{\left(1 + \frac{r}{100}\right)^k}$ now.
Cash price = the sum of these, plus any down payment.
Half-yearly or quarterly instalments use the rate for that period: $20\%$ a year is $10\%$ a half-year and $5\%$ a quarter.
Two equal annual instalments: $\dfrac{x}{1 + r/100} + \dfrac{x}{(1 + r/100)^2}$ .

Cash price from n instalments

\text{cash price} = \sum_{k=1}^{n} \dfrac{x}{\left(1 + \frac{r}{100}\right)^k}

Worked example

A cooler is sold for two equal annual instalments of ₹4,410 each, at

5\%

compound interest. Find its cash price.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Profit and Loss · Instalments and Special Cases

A dining table is sold for

2

equal annual installments of Rs.

6000

each at

25\%

compound interest. What is the cash price?

Each payment has its own factor

The second annual instalment is divided by

\left(1 + \tfrac{r}{100}\right)^2

, not by

1 + \tfrac{r}{100}

again.

Use the rate per period

Quarterly instalments at

20\%

a year are discounted at

5\%

a quarter, over four quarters, not at

20\%

once.

Drill 5 more on cash price from instalments with interest

Concept 3 of 3: Drying fruit and a broken stone

Two kinds of question hide a fixed quantity. Drying removes only water, so the dry part does not change. When a price goes with the square of the weight, breaking an article loses value even though nothing is thrown away.

Definition

Find what does not change, or what the price depends on, and use it to link before and after. Each row of the table names it.

Situation	What stays fixed or sets the price	Worked case
Fresh fruit is dried	The dry pulp	Fruit with $75\%$ water has $25\%$ pulp, so 16 kg holds 4 kg of pulp. Dried to $60\%$ water, it weighs $4 \div 0.4 = 10$ kg. The water share changes; the pulp does not.
A stone priced by the square of its weight breaks	Price $= k \times \text{weight}^2$	Pieces $1 : 3$ of a stone weighing 4: value $4^2 = 16$ before, $1 + 9 = 10$ after, a loss of $\tfrac{6}{16} = 37.5\%$ .
The same stone breaks into equal halves	Price $= k \times \text{weight}^2$	Value $2^2 = 4$ before, $1 + 1 = 2$ after: a $50\%$ loss.

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The same idea in a practice question:

Practice questionModerate

Example 3 · Profit and Loss · Instalments and Special Cases

Fresh grapes contain 90% water and dried grapes contain 20% water. From 20 kg of fresh grapes, how many kg of dried grapes can be obtained?

Drying does not cut the weight by the water percentage

Going from

75\%

water to

60\%

water does not remove

15\%

of the weight: 16 kg becomes 10 kg. Hold the pulp fixed and divide.

Drill 1 more on drying fruit and a broken stone

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Extra paid on an instalment plan
Extra paid on instalments
$\text{extra} = \text{down payment} + n \times \text{instalment} - \text{cash price}$
Cash price from instalments with interest
Cash price from n instalments
$\text{cash price} = \sum_{k=1}^{n} \dfrac{x}{\left(1 + \frac{r}{100}\right)^k}$

Reference tables (1)

Drying fruit and a broken stone3 rows

Situation	What stays fixed or sets the price	Worked case
Fresh fruit is dried	The dry pulp	Fruit with $75\%$ water has $25\%$ pulp, so 16 kg holds 4 kg of pulp. Dried to $60\%$ water, it weighs $4 \div 0.4 = 10$ kg. The water share changes; the pulp does not.
A stone priced by the square of its weight breaks	Price $= k \times \text{weight}^2$	Pieces $1 : 3$ of a stone weighing 4: value $4^2 = 16$ before, $1 + 9 = 10$ after, a loss of $\tfrac{6}{16} = 37.5\%$ .
The same stone breaks into equal halves	Price $= k \times \text{weight}^2$	Value $2^2 = 4$ before, $1 + 1 = 2$ after: a $50\%$ loss.

Watch out for (4)

Include the down payment
→ Extra paid on an instalment plan
Each payment has its own factor
→ Cash price from instalments with interest
Use the rate per period
→ Cash price from instalments with interest
Drying does not cut the weight by the water percentage
→ Drying fruit and a broken stone

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