SSC CGL Maths · Linear Equations

Word Problems and Applications

Name the unknowns, turn each sentence into an equation, solve the system, and answer exactly what the question asks for.

All Linear Equations notes

Why this matters

Twenty-three questions in the bank, thirteen EASY and ten MODERATE. Eight need one unknown, eight need two, five swap two quantities and two count coins. Four of the swap questions come from the 2025 shifts and share one pattern: an order delivered with its quantities swapped, so the bill rises by a percentage. Writing the equations is the real work here; solving them is the first page of this chapter.

Concept 1 of 4: Turning a word problem into one equation in one unknown

Many word problems look as if they have two or three unknowns but really have one. Name one quantity x, write every other quantity in terms of x, and the story becomes a single equation.

Definition

Call the quantity the question asks for, or the one everything else is described by, $x$ .
Translate phrase by phrase: "a quarter of the number" is $\frac{x}{4}$ , "decreased by 3" is $-\,3$ , "exceeds AB by 4" is $AB + 4$ , "3 less than twice AB" is $2AB - 3$ .
Two kinds, one total: if 15 items are of two kinds, call one kind $x$ and the other $15 - x$ . One unknown, not two.
Steady change: a quantity that starts at $a$ and changes by $d$ each step is $a + dt$ after $t$ steps. Two such quantities are equal when $a_1 + d_1t = a_2 + d_2t$ . From two known values, $d$ is the change divided by the number of steps between them.
Seating: with $c$ seats, "one person has no seat" means $c + 1$ people; "two to a seat leaves one seat empty" means $2(c - 1)$ people.
Fast route for two kinds: pretend every item is the cheaper kind; each swap to the dearer kind adds the price difference. The number of swaps is the shortfall divided by that difference. Safe only when there are exactly two kinds.

Steady change

\text{value after } t \text{ steps} = a + dt \qquad d = \frac{\text{later value} - \text{earlier value}}{\text{steps between them}}

athe starting value
dthe change in each step (negative for a fall)
tthe number of steps

Worked example

A purse holds 20 notes, each worth ₹10 or ₹5, and ₹130 in all. How many ₹10 notes are there?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 26 Sep 2024, Shift 1 · Q63Easy

Example 1 · Linear Equations · Word Problems and Applications

A number whose fifth part increased by 5 is equal to its third part decreased by 7. Find half of the number.

Using two letters where one will do

When the total count is given, the second kind is the total minus the first. Two letters with one equation cannot be solved; one letter with one equation can.

Miscounting the steps between two values

From the value after year 6 to the value after year 15 there are 9 yearly steps, not 15 and not 10. Divide the change by the difference of the two step numbers.

Answering x instead of the question

Questions often ask for something built from x: half the number, the other kind of item, the smallest side. Read the last line again before picking an option, since x itself is usually one of the wrong options.

Drill 7 more on turning a word problem into one equation in one unknown

Concept 2 of 4: Two unknowns from two statements

When the story gives two separate facts about two quantities, each fact is one equation. Two equations, two unknowns, then the elimination you already know.

Definition

Name both unknowns, write one equation for each statement, and solve by elimination.
Transfers: "if A gives 1 to B, they are equal" means $A - 1 = B + 1$ , so $A - B = 2$ . A transfer of $k$ takes $k$ from the giver and adds $k$ to the receiver.
Rates per day: when totals are earned over different numbers of days, divide each total by its days first, then compare.
Percentages of counts: 18% of $y$ is $0.18y$ ; multiply the whole equation to clear the decimals.
Fast route, scaling: if the asked combination is a fixed fraction of a given one (6 pens and 4 pencils is half of 12 pens and 8 pencils), scale the given price. No solving at all. It is safe only when every item scales by the same factor.
Fast route, combine: if the question asks for a total such as $x + y$ , adding or subtracting the equations often gives it directly.

Equal after a transfer of k

A - k = B + k \;\Rightarrow\; A - B = 2k

A, Bthe amounts the giver and the receiver hold before the transfer
kthe number of items handed over

Worked example

3 pens and 2 notebooks cost ₹110. 2 pens and 5 notebooks cost ₹165. What does one notebook cost?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 23 Sep 2024, Shift 2 · Q74Easy

Example 2 · Linear Equations · Word Problems and Applications

Ten chairs and six tables together cost ₹5,140; three chairs and two tables together cost ₹1,635. The cost of 1 chair and 1 table is:

Taking the transfer as the difference

If giving

k

items makes two people equal, they differed by

2k

, not

k

: the giver loses

k

and the receiver gains

k

Comparing totals over different numbers of days

One group earning ₹4,200 in 6 days and another earning ₹6,300 in 7 days cannot go into the same pair of equations as they stand. Divide each total by its own number of days to get earnings per day first.

Drill 7 more on two unknowns from two statements

Concept 3 of 4: Swapped quantities and the change in the bill

In a delivery mix-up the two counts swap places while the prices stay where they were. Price everything in units of the cheaper item and both bills are short expressions in one unknown count.

Definition

Call the cheaper item's price 1 unit and the dearer one's $r$ units. If $q_1$ of the dearer and $q_2$ of the cheaper were ordered, the ordered bill is $rq_1 + q_2$ units and the swapped bill is $rq_2 + q_1$ units.
"The bill rose by $g\%$ " means swapped bill $= \left(1 + \frac{g}{100}\right) \times$ ordered bill. That is one linear equation in the unknown count.
The change is $(r - 1)(q_2 - q_1)$ units, so the bill rises when more of the cheaper item was ordered and falls when more of the dearer one was.
Notes interchanged: with $x$ notes of ₹ $a$ and $y$ of ₹ $b$ , the amounts are $ax + by$ and $bx + ay$ . Adding gives $(a + b)(x + y)$ ; subtracting gives $(a - b)(x - y)$ .
Finish with what is asked, often the ratio of the two ordered quantities.

Swapped quantities

rq_1 + q_2 \;\to\; rq_2 + q_1 \qquad (a + b)(x + y) = P + Q \qquad (a - b)(x - y) = P - Q

rhow many times the cheaper price the dearer price is
$q_1, q_2$ the ordered quantities of the dearer and the cheaper item
P, Qthe amounts before and after two kinds of notes are interchanged

Worked example

A shop orders 10 large boxes and some small boxes. A large box costs 4 times as much as a small one. The delivery swaps the two quantities, and the bill rises by 50%. How many small boxes were ordered?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 2 · Q58Moderate

Example 3 · Linear Equations · Word Problems and Applications

A company placed an order for 20 high-end laptops and some quantity of standard desktops. The price of a high-end laptop was 5 times that of a standard desktop. Due to a mistake, the number of laptops and desktops was interchanged in the delivery. This caused the total bill to increase by

80\%

. What was the ratio of the number of high-end laptops to standard desktops in the original order?

Swapping the prices instead of the quantities

In a delivery mix-up each item keeps its own price; only the counts change places. The swapped bill is (dearer price × the cheaper item's count) plus (cheaper price × the dearer item's count).

Taking the percentage of the new bill

"The bill rose by 40%" compares with the bill that was ordered: new bill

= 1.4 \times

ordered bill, not ordered bill

= 0.6 \times

new bill.

Drill 4 more on swapped quantities and the change in the bill

Concept 4 of 4: Whole-number answers: counting coins and notes

A count of coins and their total value give two equations, but there are three kinds of coin. The equations alone allow many answers; the fact that every count is a whole number and none is negative narrows them to a short range, and the options do the rest.

Definition

Write the count equation and the value equation, then subtract the count equation (times the smallest coin value) from the value equation. One kind of coin disappears.
Write every other count in terms of the one the question asks about. Each must be 0 or more: that gives a lower and an upper limit.
With one equation in two whole unknowns, such as $5x + 2y = 31$ , step through the values: $31 - 5x$ must be even and not negative, so $x$ is odd and at most 6.
Test the options against the range: usually only one fits.

Coins of ₹5, ₹2 and ₹1

x + y + z = N,\quad 5x + 2y + z = V \;\Rightarrow\; 4x + y = V - N

x, y, zthe numbers of ₹5, ₹2 and ₹1 coins
Nthe total number of coins
Vthe total value in rupees

Worked example

₹50 is paid with ₹5, ₹2 and ₹1 coins, 20 coins in all. Which of 3, 5, 8 and 9 can be the number of ₹5 coins?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 9 Sep 2024, Shift 1 · Q59Moderate

Example 4 · Linear Equations · Word Problems and Applications

R pays ₹

100

to P with ₹

5

, ₹

2

and ₹

1

coins. The total number of coins used for paying are

40

. What is the number of coins of denomination ₹

5

in the payment?

Checking only one count for a negative value

Every count must be 0 or more, and each gives its own limit. In a three-coin question one count sets the upper limit and another the lower limit; an option outside either one is impossible.

Drill 1 more on whole-number answers: counting coins and notes

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)

Turning a word problem into one equation in one unknown
Steady change
$\text{value after } t \text{ steps} = a + dt \qquad d = \frac{\text{later value} - \text{earlier value}}{\text{steps between them}}$
Two unknowns from two statements
Equal after a transfer of k
$A - k = B + k \;\Rightarrow\; A - B = 2k$
Swapped quantities and the change in the bill
Swapped quantities
$rq_1 + q_2 \;\to\; rq_2 + q_1 \qquad (a + b)(x + y) = P + Q \qquad (a - b)(x - y) = P - Q$
Whole-number answers: counting coins and notes
Coins of ₹5, ₹2 and ₹1
$x + y + z = N,\quad 5x + 2y + z = V \;\Rightarrow\; 4x + y = V - N$

Watch out for (8)

Using two letters where one will do
→ Turning a word problem into one equation in one unknown
Miscounting the steps between two values
→ Turning a word problem into one equation in one unknown
Answering x instead of the question
→ Turning a word problem into one equation in one unknown
Taking the transfer as the difference
→ Two unknowns from two statements
Comparing totals over different numbers of days
→ Two unknowns from two statements
Swapping the prices instead of the quantities
→ Swapped quantities and the change in the bill
Taking the percentage of the new bill
→ Swapped quantities and the change in the bill
Checking only one count for a negative value
→ Whole-number answers: counting coins and notes

Test yourself on Linear Equations

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.

Take the Linear Equations chapter test Drill all 23 questions