SSC CGL Maths · Percentage, Profit and Loss

Successive Percentage Change

Each percentage change multiplies the current value by a factor, so several changes in a row multiply together instead of adding.

All Percentage, Profit and Loss notes

Why this matters

28 questions in the bank, 15 of them MODERATE: a price that rises then falls, money spent out of what is left, a value worked back to where it started, and price against consumption. One idea solves all of them, and it is the same idea the profit and discount pages lean on later, so learn it here first.

Concept 1 of 4: Successive changes as a product of factors

A rise of 20% turns 100 into 120, so it multiplies by 1.2. A fall of 15% multiplies by 0.85. The second change acts on the new value, not on the first one, so the factors multiply and the percentages never simply add.

Definition

A rise of r% multiplies by $1 + \frac{r}{100}$ ; a fall of r% multiplies by $1 - \frac{r}{100}$ .
Two changes of a% and b% (write a fall as a negative number) give a net change of $a + b + \frac{ab}{100}$ per cent.
The order does not matter: up 10% then down 20% ends where down 20% then up 10% ends.
An equal rise and fall of r% always leaves a loss of $\frac{r^2}{100}$ per cent.
The rule works for any product of two quantities: area (length times breadth, or side times side), a wage bill (workers times wage), a quotient (divide by the divisor's factor).
Fast route: for three or more changes, combine two with the formula, then combine the result with the next.

Net effect of two changes

\text{net}\% = a + b + \frac{ab}{100} \qquad \text{equal rise and fall: } -\frac{r^2}{100}\%

a, bthe two changes in per cent, a fall written as negative
rthe size of an equal rise and fall

Worked example

The price of a shirt rises by 20% and then falls by 10%. Find the net change.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 18 Sep 2024, Shift 3 · Q56Moderate

Example 1 · Percentage, Profit and Loss · Successive Percentage Change

An investor, invested ₹2,00,000 in shares. The shares rose by 18% on the first day, fell by 15% on the second day, and then fell by 3% on the third day. What is the net percentage increase or decrease in the value of the investor's shares? (Rounded up to one decimal place)

Adding the percentages

A rise of 30% followed by a fall of 30% is not 'no change'.

1.3 \times 0.7 = 0.91

, a 9% fall.

Dropping the sign in the product term

For a rise of 25% and a fall of 12%, b is

-12

, so the product term is

\frac{25 \times (-12)}{100} = -3

. The net change is

25 - 12 - 3 = 10\%

, not 16%.

Drill 11 more on successive changes as a product of factors

Concept 2 of 4: Spending a share of what is left

'18% of the rest' is 18% of whatever remains at that point, not 18% of the whole income. So each step keeps a share of the previous amount, and what is left at the end is the income times all the 'kept' factors.

Definition

After spending x% of an amount, $\left(1 - \frac{x}{100}\right)$ of it is left.
'y% of the remaining' multiplies what remains by $1 - \frac{y}{100}$ for the part still left.
Final amount $=$ total $\times$ all the kept factors. To find the total, divide the final amount by that product.
Shares that are all of the same total (30% rent and 20% food, both of the salary) add first. Only 'of the rest' shares multiply.

Amount left after spending out of the remainder

\text{left} = S\left(1 - \frac{x}{100}\right)\left(1 - \frac{y}{100}\right)\left(1 - \frac{z}{100}\right)

Worked example

Mohan spends 20% of his salary on rent and 25% of the rest on food. He saves the remaining ₹9,000. Find his salary.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 24 Sep 2024, Shift 2 · Q65Moderate

Example 2 · Percentage, Profit and Loss · Successive Percentage Change

Vipul spends 30% of his monthly income on rent, 60% of the remaining on groceries, and 40% of the remaining on others. If he saves ₹ 7056 in a month, then how much (in ₹) does he spend on groceries?

Adding shares of different amounts

20% of the income, then 50% of the rest, then 40% of the rest leaves

0.8 \times 0.5 \times 0.6 = 0.24

of the income. Adding 20, 50 and 40 gives 110%, which means nothing.

Drill 3 more on spending a share of what is left

Concept 3 of 4: Working back to the original value

If you know where a value ended and what changes it went through, divide by the combined factor to get back to the start. To undo a change, the reverse percentage is taken on the new value, so it is a different number.

Definition

Original $=$ final $\div$ (product of all the factors).
To undo a rise of r%, fall by $\frac{r}{100 + r} \times 100\%$ ; to undo a fall of r%, rise by $\frac{r}{100 - r} \times 100\%$ . This is the changing base from the Percentage Basics page.
A population or price growing by r% a year for n years is multiplied by $\left(1 + \frac{r}{100}\right)^n$ , so its value n years ago is today's value divided by that.
If a fall of x% and then a rise of x% leave the value k% lower, then $\frac{x^2}{100} = k$ .
When the rates are unknown, write the product of factors as an equation. With four options, putting each one in is often faster than solving.

Back to the start, and growth over years

\text{original} = \frac{\text{final}}{\left(1 + \frac{a}{100}\right)\left(1 + \frac{b}{100}\right)} \qquad P_n = P_0\left(1 + \frac{r}{100}\right)^n

Worked example

After a rise of 25% and then a rise of 20%, a price is ₹1,800. What was the original price?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 19 Sep 2024, Shift 2 · Q72Moderate

Example 3 · Percentage, Profit and Loss · Successive Percentage Change

The price of an item is increased twice successively. The final price is ₹2,295 and initial price was ₹1,000. The percentage increment the second time was twice the percentage increment the first time. By what percentage was the price increased the second time?

Undoing a change with the same percentage

A value that rose by 30% does not return by falling 30%. From 130, it must fall by 30 on a base of 130, which is about 23.08%.

Drill 6 more on working back to the original value

Concept 4 of 4: Price and consumption when spending is fixed

Spending is price times quantity. If the price goes up and the spending must stay the same, the quantity has to come down by just enough to cancel the price factor.

Definition

If the price rises by r%, consumption must fall by $\frac{r}{100 + r} \times 100\%$ to keep spending the same.
If the price falls by r%, the same money buys $\frac{r}{100 - r} \times 100\%$ more.
If price and consumption both change, new spending $=$ old spending $\times$ price factor $\times$ quantity factor.
'A fall of r% in price lets a buyer get Q kg more for ₹M': the money saved is r% of M, and it buys the Q kg at the new price. So new price $= \frac{r\% \text{ of } M}{Q}$ , and old price $=$ new price $\div \left(1 - \frac{r}{100}\right)$ .

Cut in consumption to keep spending fixed

\text{cut} = \frac{r}{100 + r} \times 100\% \qquad \text{spending} = \text{price} \times \text{quantity}

Worked example

The price of cooking oil rises by 25%. By what percentage should a family cut its use of oil so that its spending stays the same?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 26 Sep 2025, Shift 2 · Q62Easy

Example 4 · Percentage, Profit and Loss · Successive Percentage Change

A reduction of

20\%

in the price of sugar enables a person to buy 5 kg more for Rs. 200. What is the original price per kg?

Cutting consumption by the price rise

A price rise of 25% does not need a 25% cut. A 20% cut is enough, because

1.25 \times 0.8 = 1

Mixing up the new price and the old price

When a price cut lets you buy extra kilograms, the money saved buys them at the new, lower price. Divide that by the price factor to get the old price.

Drill 4 more on price and consumption when spending is fixed

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 (4)

Successive changes as a product of factors
Net effect of two changes
$\text{net}\% = a + b + \frac{ab}{100} \qquad \text{equal rise and fall: } -\frac{r^2}{100}\%$
Spending a share of what is left
Amount left after spending out of the remainder
$\text{left} = S\left(1 - \frac{x}{100}\right)\left(1 - \frac{y}{100}\right)\left(1 - \frac{z}{100}\right)$
Working back to the original value
Back to the start, and growth over years
$\text{original} = \frac{\text{final}}{\left(1 + \frac{a}{100}\right)\left(1 + \frac{b}{100}\right)} \qquad P_n = P_0\left(1 + \frac{r}{100}\right)^n$
Price and consumption when spending is fixed
Cut in consumption to keep spending fixed
$\text{cut} = \frac{r}{100 + r} \times 100\% \qquad \text{spending} = \text{price} \times \text{quantity}$

Watch out for (6)

Adding the percentages
→ Successive changes as a product of factors
Dropping the sign in the product term
→ Successive changes as a product of factors
Adding shares of different amounts
→ Spending a share of what is left
Undoing a change with the same percentage
→ Working back to the original value
Cutting consumption by the price rise
→ Price and consumption when spending is fixed
Mixing up the new price and the old price
→ Price and consumption when spending is fixed

Test yourself on Percentage, Profit and Loss

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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